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LECTURE SIX

 a.   Portfolio performance measurement
 b.   Hedge fund risk management
 c.   Credit risk management
 d.   Probability of default



                                          1
Part 1

PORTFOLIO
PERFORMANCE
MEASUREMENT
   a.    Intro. Performance measurement
   b.    Surplus at Risk
   c.    The Market Line (ML) and CAPM
   d.    Several ratios to measure market performance



                                                        2
Part 1. Portfolio Performance Measurement   1. Intro. Performance measurement

                                                              Portfolio management


                                                       Risk                            Return



                                                        Do profits reflect my risk exposure?




                                              HOW?
                                              • Looking at process/strategies in place, and
                                              • Whether outcomes are in line with what was intended
                                                or should have been achieved.
Lecture 6




                                                 • Pure luck?
                                                 • Good strategy
                                                 • Separate effect of the market and active management
1. Intro. Performance measurement
                                             Some definitions
                                                             Sell side                           Buy side
Part 1. Portfolio Performance Measurement




                                             Creation, promotion, analysis and    Final buyers of financial assets
                                               sale of securities.                   (Large portions of securities)


                                             Leverage and speculative             Tend to me more conservative
                                                                                     No leverage


                                             Examples?                            Examples?
Lecture 6
1. Intro. Performance measurement
                                             Some definitions
                                                             Sell side                            Buy side
Part 1. Portfolio Performance Measurement




                                             Creation, promotion, analysis and     Final buyers of financial assets
                                               sale of securities.                    (Large portions of securities)


                                             Leverage and speculative              Tend to me more conservative
                                                                                      No leverage


                                             Institutions such   as                Investing institutions such as
                                               • Investment bankers                   • mutual funds,
                                                 (intermediaries between issuers      • pension funds and
                                                 and public),                         • insurance firms
                                               • Research companies that
                                                 perform stock research and
Lecture 6




                                                 make ratings. Ig. Roubini
                                               • Market makers who provide
                                                 liquidity in the market.
1. Intro. Performance measurement
                                            Absolute and Relative risk
                                                       Absolute risk                        Relative Risk
Part 1. Portfolio Performance Measurement




                                             With respect to the portfolio        With respect to a benchmark
                                             itself
                                                                                  • Tracking error :        e = R P - RB
                                             • Risk factor: σ                     • In dollar terms:        exP
                                             • P: Initial portfolio value
                                             • In dollar terms
                                                                                             𝜎 (𝑅 𝑃 − 𝑅 𝐵 ) 𝑃
                                                            𝜎 ∆𝑃
                                                                                                  ∆𝑃 ∆𝐵
                                                            ∆𝑃                               𝜎(      −   ) 𝑃
                                                                                                   𝑃   𝐵
                                                        𝜎      ∗ 𝑃
                                                             𝑃
                                                                                                    𝜔 ∗ 𝑃
                                                        𝜎 𝑅𝑃 ∗ 𝑃             Tracking Error Volatility
                                               Note that there is only       Where
Lecture 6




                                               the asset or portfolio
                                                                                      2   2 P  2  P B   2 P
1. Intro. Performance measurement
                                            Absolute and Relative risk
Part 1. Portfolio Performance Measurement



                                            Example


                                             SPX:                10% return              -10% return
                                             My trade:           6% return               -4% return




                                                         Absolute risk                    Relative Risk


                                                          What would be the difference between absolute
                                                                       and relative risk?
Lecture 6
What is my risk in the long term?
                                            2. Surplus at Risk                                           Focused on net profits
                                            Performance Measurement
                                            Sort of relative risk measurement
Part 1. Portfolio Performance Measurement


                                            Two assets, long pension assets and short pension liabilities.
                                                          Assets $   120,00              Liabilities $   100,00
                                                         Surplus               $                          20,00

                                            Volatility                  12%               Volatility          3%
                                            Expected R                   8%             Expected R            5%
                                                                Correlation                      0,3

                                            Change of assets                       Change of assets
                                                                 $      9,60                        $     5,00
                                            (per period)         12% of $120           (per period) 5% of $100



                                                  Expected growth of surplus
                                                                               $              4,60 Change assets - liabilities
                                                           Expected surplus $                24,60 Considering the surplus
                                                         Variance of surplus $              190,44 variance(a – b) = variance (a) + variance(b) - (2)cov(a,b)
                                                         Volatility of surplus $             13,80
Lecture 6




                                                           Confidence level                    95%
                                                             Normal deviate                    1,64
                                                         Surplus at Risk $VaR                18,03 –(expected surplus) + (volatility of surplus)*(normal deviate)

                                                          The complete variance formula is: σ12P12+ σ22P12-2 σ1σ2P1P2ρ
3.The Market Line (ML) and CAPM
                                            Small recap
                                            Decompose total return into a component due to market risk premia and
Part 1. Portfolio Performance Measurement



                                            other factors.


                                             E(Ri)
                                                         Overvalued                            ML

                                                                        M
                                             E(RM)


                                                RF
                                                                Undervalued
Lecture 6




                                                                       RiskM                          Riski
                                                     Note: Risk is either b or 
3.The Market Line (ML) and CAPM
                                            Small recap
                                                   E ( Ri )  rf  b [ E ( RM )  rf ]              Ri   i  b i RM  ei
Part 1. Portfolio Performance Measurement



                                                   E ( Ri )  rf  b [ E ( RM )  rf ]              This is the CAPM model
                                                  E ( Ri )  rf   i  b i [ E ( RM )  rf ]
                                             E(Ri)
                                                                                  B
                                                                                                             ML

                                                             A
                                                                              M
                                             E(RM)                                              C

                                                                 E

                                              RFR                        D
Lecture 6




                                                                             RiskM                                   Riski
                                                     Note: Risk is either b or 
3.The Market Line (ML) and CAPM
                                            Small recap
                                            Capital Market Line (CML)
                                            obtained by combining the market portfolio and the riskless asset
Part 1. Portfolio Performance Measurement




                                            • CML specifies the expected return for a given level of risk
                                            • All possible combined portfolios lie on the CML, and all are Mean-Variance
                                              efficient portfolios

                                            • Here we have a clear relation between the risk of my portfolio and the risk
Lecture 6




                                              of the market. This is reflected by beta
                                                                     Cov( Ri , RM )       It measures how much an asset’s return
                                                              bi                         is driven by the market return
                                                                          2M
3.The Market Line (ML) and CAPM
                                            Small recap
                                            Capital Market Line (CML)
Part 1. Portfolio Performance Measurement



                                            If the stock has a high positive β:
                                                  • It will have large price swings driven by the market
                                                  • It will increase the risk of the investor’s portfolio(in fact, will make the
                                                     entire market more risky …)
                                                  • The investor will demand a high Er in compensation.

                                            If the stock has a negative β :
                                                  • It moves “against” the market.
                                                  • It will decrease the risk of the market portfolio
                                                  • The investor will accept a lower Er

                                            Then the SML depicts the relation between β and the Expected Return (Er)
Lecture 6




                                                                                                   For the risk-free security, b = 0
                                                                                                   For the market itself, b=1.
3.The Market Line (ML) and CAPM
                                            Small recap
                                            Capital Market Line (CML)
Part 1. Portfolio Performance Measurement



                                                                   E ( Ri )  rf  b [ E ( RM )  rf ]
                                                                   E ( Ri )  rf  b [ E ( RM )  rf ]


                                             Excess of return of a portfolio is a function of the excess of return of the market
                                                                            W.R. a risk free rate
Lecture 6
4.Treynor Ratio
                                            The Treynor measure calculates the risk premium per unit of risk (bi)
Part 1. Portfolio Performance Measurement



                                                                               [ E ( RP )  RF ]
                                                                      TR 
                                                                                    b ( RP )
                                            Beta measures the investment volatility relative to the market volatility
                                            (systematic risk)

                                            The Treynor Ratio is negative if

                                            • RF > E[RP] AND β > 0 .
                                                    Manager has performed badly: failing to get performance better than
                                                    the risk free rate AND manager made a not good election of
                                                    portfolio

                                            • RF < E[RP] AND β > 0
                                                    Manger has performed well, managing to reduce risk but getting a
Lecture 6




                                                    return better than the risk free rate
                                            Higher Ti generally indicates better performance
4.Treynor Ratio
                                            ADVANTAGE: It indicates the volatility a ASSET brings to an entire portfolio.
                                            The Treynor Ratio should be used only as a ranking mechanism for
Part 1. Portfolio Performance Measurement



                                            investments within the same sector.
                                            .
                                                When presented with investments that have the same return, investments
                                                   with higher Treynor Ratios are less risky and better managed


                                                                                                                        Cov( Ri , RM )
                                                                                                                 bi 
                                                                                                                             2M
Lecture 6
5. Sharpe Ratio
                                            Describes how much excess return you are receiving                 for     the
                                            extra volatility that you endure for holding a riskier asset.
Part 1. Portfolio Performance Measurement



                                                                          [ E ( RP )  RF ]
                                                                   SR 
                                                                                ( RP )

                                               The Sharpe measure is exactly the same as the Treynor measure, except
                                               that the risk measure is the standard deviation:



                                            • Tells us whether a portfolio's returns are due to smart investment
                                              decisions or a result of excess risk.

                                            • The greater a portfolio's Sharpe ratio, the better its risk-adjusted
                                              performance has been.

                                            • A negative Sharpe ratio indicates that a risk-less asset would perform
Lecture 6




                                              better than the security being analysed.
4-5. Sharpe V Treynor Ratio
                                             The Sharpe and Treynor measures are similar, but different:

                                                         • Sharpe uses the standard deviation, Treynor uses beta
Part 1. Portfolio Performance Measurement



                                               X
                                                         • Sharpe is more appropriate for well diversified portfolios, Treynor for
                                            Portfolio Return
                                                       15%
                                                                RFR
                                                                5%
                                                                      Beta Std. Dev. Trenor Sharpe
                                                                      2.50   20%     0.0400 0.5000
                                               Y       8%  individual assets14% 0.0600 0.2143
                                                                5%    0.50
                                               Z
                                            Market
                                                       6%
                                                         • Sharpe and Treynor: The ranking, not the number itself, is what is most
                                                       10%
                                                                5%
                                                                5%
                                                                      0.35
                                                                      1.00
                                                                              9%
                                                                             11%
                                                                                     0.0286 0.1111
                                                                                     0.0500 0.4545
                                                           important
                                                                               Risk vs Return
                                                   15%
                                                   Portfolio                   Return                       RFR             Beta                Std. Dev.             Trenor            Sharpe
                                                                                                                      X
                                                                                M
                                                    X Y                         15%                         5%              2.50                    20%                0.0400           0.5000
                                               Return




                                                   10%
                                                                                                                            Portfolio         Return   RFR            Beta
                                                    Y                            8%                         5%              0.50                    14%                0.0600 Dev.
                                                                                                                                                                           Std.         Trenor Sharpe
                                                                                                                                                                                        0.2143
                                                 5%                                                                            X               15%     5%             2.50   20%        0.0400 0.5000
                                                    Z Z                          6%                         5%              0.35
                                                                                                                               Y               8%    9%5%              0.0286
                                                                                                                                                                      0.50   14%        0.11110.2143
                                                                                                                                                                                        0.0600
                                                Market
                                                 0%                             10%                         5%              1.00
                                                                                                                               Z               6% 11%  5%              0.0500
                                                                                                                                                                      0.35    9%        0.45450.1111
                                                                                                                                                                                        0.0286
                                                        0.00           0.50        1.00          1.50        2.00    2.50   Market            10%           5%        1.00        11%   0.0500   0.4545
                                                                                          Beta


                                                                               Risk vs Return                       Risk vs Return
                                                                                                                    X                                            Risk vs Return
                                                        15%                                                                         15%
                                                                   15%                        M                                                                                                   X
                                                                                                                                                                  M                       X
                                               Return




                                                        10%
                                                                                                                                Return

                                                                                                                      M             10%                 Y
Lecture 6




                                                              Return




                                                                   10%                                  Y   Y
                                                        5%                            Z                                                  5%
                                                                                                                                                    Z
                                                        0%             5%                                                                0%
                                                                                                  Z
                                                              0%              5%            10%             15%      20%                  0.00      0.50           1.00          1.50   2.00     2.50
                                                                                          Std. Dev.                                                                       Beta
                                                                       0%
6. Sortino Ratio
                                            The Sortino ratio generalizes (focus on the downside) from the Sharpe by
                                            using:
Part 1. Portfolio Performance Measurement



                                            • In the numerator, instead of excess return (above riskfree), Sortino uses
                                              excess above hurdle (MAR, minimum acceptable return)

                                            • In the denominator, instead of volatility (annualized standard deviation),
                                              Sortino uses downside deviation.

                                                                           [ E ( RP )  MAR]
                                                                    SR 
                                                                                  L ( RP )
                                            • Appears to resolve several of the issues inherent in the Sharpe ratio:
                                               • It incorporates a relevant return target, in both the numerator and the
                                                  denominator;
                                               • It quantifies downside volatility without penalizing upside volatility; and
                                                  because of its focus on downside risk,
                                               • It is more applicable to distributions that are negatively skewed than
Lecture 6




                                                  measures based on standard deviation.
5-6. Sortino Ratio and Sharpe Ratio
                                             Example
                                                                                  Only consider the
                                                                                  returns below the      Square difference WR
                                                                                  hurdle rate                                      Sq. Difference WR the
Part 1. Portfolio Performance Measurement


                                                                                                         the hurdle rate
                                                                                                                                   average of returns
                                                                                                                                                                                     Sumation of returns
                                                                                   Excess over Hurdle
                                                                     We take a minimun                                                             Averag montly return of P     1,821%
                                                                     acceptable return
                                                                                                                         Squared difference        Average yearly return        21,851%   Times 12
                                            Month        Price        Returns Hurdle rate      R - HUR     Hurdle>R    (R-Hur)^2 (R-Av)^2          Hurdle rate yield return Y   18,000%   1.5% * 12
                                                     1     663,03       -2,539%    1,50%            -4,04%      -4,04%    0,1631%      0,1901%
                                                     2       680,3      -9,834%    1,50%           -11,33%     -11,33%    1,2847%      1,3585%     Rf month                      1,80%
                                                     3       754,5      10,132%    1,50%             8,63%                             0,6907%     RF Y                         21,60%    Times 12
                                                     4     685,09        8,234%    1,50%             6,73%                             0,4113%                                                   Use the formula
                                                     5     632,97        9,120%    1,50%             7,62%                             0,5327%
                                                     6     580,07       -0,136%    1,50%            -1,64%      -1,64%    0,0268%      0,0383%     Volatility                   29,06%
                                                     7     580,86       -3,966%    1,50%            -5,47%      -5,47%    0,2988%      0,3349%     Sharpe ratio                 0,865%
                                                     8     604,85       -5,675%    1,50%            -7,17%      -7,17%    0,5148%      0,5619%
                                                     9     641,24        3,719%    1,50%             2,22%                             0,0360%     Sortino                      19,54%
                                                    10     618,25        6,575%    1,50%             5,07%                             0,2260%     Excess return                3,851%    P-Hurdle
                                                    11     580,11      -10,186%    1,50%           -11,69%     -11,69%    1,3656%      1,4416%     Montly downside VaRianc      19,71%
                                                    12       645,9       7,760%    1,50%             6,26%                             0,3527%
                                                    13     599,39        1,139%    1,50%            -0,36%      -0,36%    0,0013%      0,0047%
                                                    14     592,64       15,067%    1,50%            13,57%                             1,7545%
                                                    15     515,04       -4,791%    1,50%            -6,29%      -6,29%    0,3958%      0,4372%
                                                    16     540,96      -10,391%    1,50%           -11,89%     -11,89%    1,4140%      1,4913%
                                                    17     603,69       19,217%    1,50%            17,72%                             3,0262%
                                                    18     506,38       -4,280%    1,50%            -5,78%      -5,78%    0,3340%      0,3722%
Lecture 6




                                                    19     529,02       -2,772%    1,50%            -4,27%      -4,27%    0,1825%      0,2109%
                                                    20       544,1      -7,270%    1,50%            -8,77%      -8,77%    0,7692%      0,8265%
7. Jensen alpha
                                            Shows by much the returns of an
                                            actively managed portfolio are
Part 1. Portfolio Performance Measurement


                                            above or below market returns.
                                                                                                                        >0
                                            A positive Alpha means that a
                                            portfolio has beaten the market,                                            =0
                                            while a negative value indicates
                                            underperformance                                                            <0




                                                                                  Risk Premium
                                            A fund manager with a negative
                                            alpha and a beta greater than one
                                            has added risk to the portfolio but
                                            has poorer performance than the
                                                                                                                               0
                                            market

                                                                                                     Market Risk Premium

                                                                                             R i  RFR   i  b i  R M  RFR    i
Lecture 6




                                                                         Alpha = Excess of return – (Beta * (Excess of return))
Part 1. Portfolio Performance Measurement   7. Jensen alpha

                                                       R i  RFR   i  b i  R M  RFR    i
                                                     Alpha = Excess of return – (Beta * (Excess of return))
Lecture 6
7. Jensen alpha
                                                                             Portfolio   Portfolio             Market
                                                                                P           Q
Part 1. Portfolio Performance Measurement



                                                    Beta                       0.90          1.6                   1.0
                                                    RM-Rf                      11%           19%                  10%
                                                    Alpha                      2.0%          3.0%                  0%

                                                                R i  RFR   i  b i  R M  RFR    i
                                                                                                         Portfolio P
                                                     Expected Return


                                                                                                                       Portfolio Q

                                                                       19%

                                                                                                                        SML
                                                                       16%
                                                                                 M
                                                                       11%         M2
                                                                                         P
Lecture 6




                                                                       9%



                                                                                 0.9               1.6                  Beta
8. Information Ratio
                                            Measure of the risk-adjusted return of a portfolio.
Part 1. Portfolio Performance Measurement



                                            Defined as expected active return divided by tracking error
                                                • Active return : difference between the return of portfolio and
                                                   the return of a benchmark
                                                • Tracking error is the standard deviation of the active return
                                                                E[ R P  RB ]            Component attributable to the manager’s skill
                                                          IR                 
                                                               VAR( R P  RB )            While Sharpe consider the σ of total
                                                                                           returns, IF consider σ of alpha

                                                • Measures the active return of the manager's (abnormal return)
                                                  portfolio per unit of risk that the manager takes relative to the
                                                  benchmark.

                                                • The higher the information ratio, the higher the active return of
                                                  the portfolio, given the amount of risk taken, and the better the
Lecture 6




                                                  manager.
8. Information Ratio
                                                                        Returns
                                               Date         Portfolio   Market    Excess
Part 1. Portfolio Performance Measurement



                                             01/01/2010        2%          2,06% -0,49%
                                             01/02/2010        1%         -5,62%   6,64%
                                             01/03/2010       0,61%       -3,42%   4,03%
                                             01/04/2010       0,76%        2,84% -2,08%
                                             01/05/2010       9,69%       -5,00% 14,69%
                                             01/06/2010       1,39%        5,30% -3,91%
                                             01/07/2010       3,10%       -2,33%   5,44%
                                             01/08/2010       0,46%        8,57% -8,12%
                                             01/09/2010       6,11%        4,77%   1,34%
                                             01/10/2010       9,37%      14,69% -5,32%
                                             01/11/2010       3,88%       -6,68% 10,56%
                                             01/12/2010       9,54%        1,38%   8,16%
                                                Mean          3,96%      1,38%   2,58% Expected excess of return (Benchmark)
                                            Standard Dev      3,73%      6,40%   6,88%
Lecture 6




                                                           Information Ratio       0,3750                 E[ R P  RB ]   
                                                                                                 IR                    
                                                                                                         VAR( R P  RB ) 
9. Modigliani´s Risk Adjustment Performance
                                            • Also known as M2

                                            • Closely related to the Sharpe Ratio
Part 1. Portfolio Performance Measurement




                                            • Focuses on total volatility as a measure of risk, but its risk adjusted
                                              measure of performance has the interpretation of a differential return
                                              relative to the benchmark index

                                            • The idea is to lever or de-lever a portfolio (i.e., shift it up or down the
                                              capital market line) so that its standard deviation is identical to that
                                              of the market portfolio.

                                            • The formula for M2 is:

                                                                          
                                                                  M 2   M R i  R f   R f
                                                                        
                                                                           i 
                                                                               
Lecture 6




                                            • The M2 of a portfolio is the return that this adjusted portfolio earned.
                                              This return can then be compared directly to the market return for the
                                              period.
9. Modigliani´s Risk Adjustment Performance
                                            • Suppose that
                                                    • Return                    Ri: 35%            RM: 28%
                                                    • Volatility                σi: 42%            σM: 30%
Part 1. Portfolio Performance Measurement




                                            • Find a portfolio combination with the same level of risk than the
                                              benchmark (market)
                                                                                       M   30  0.714
                                                    • Portion of the portfolio           i 
                                                                                               42

                                                      • Portion of risk free asset                   1    M   0.286
                                                                                                         
                                                                                                             i 
                                                                                                                 

                                            • The return of this portfolio will be       [0.286 * 0.06]  [0.714 * 0.35]  0.267
                                            • This portfolio is -1.3% than the market return.
                                                                                                         This is the M2
                                                              Expected Return




                                                                                               Market

                                                                                  M               Portfolio
                                                                                    M2
                                                                                           P
                                                                                      P*
Lecture 6




                                                                                                              •   Reduce the return of the P
                                                                                                              •   Obtain leSs volatility
                                                                                                Volatility
9. Modigliani´s Risk Adjustment Performance
                                                                                 
                                                                         M 2   M R i  R f   R f
                                                                               
                                                                                  i 
                                                                                                                                     SR 
                                                                                                                                             [  ( RP )  RF ]
Part 1. Portfolio Performance Measurement


                                                                                                                                                   ( RP )
                                                           Portfolio         Return           RFR        Beta   Std. Dev.    Trenor      Sharpe
                                                              X               15%             5%         2.50     20%        0.0400      0.5000
                                                              Y               8%              5%         0.50     14%        0.0600      0.2143
                                                              Z               6%              5%         0.35      9%        0.0286      0.1111
                                                           Market             10%             5%         1.00     11%        0.0500      0.4545



                                                                          X     0.200.15  0.05  0.05  0.105
                                                                        M 2  0.11                  Risk vs Return

                                                                              0.11      0.08  0.05  0.05  0.074
                                                                   15%
                                                                        M2                           M
                                                                                                                                              X
                                                                         Y           0.14
                                                                              0.11      0.06  0.05  0.05  0.062
                                                               Return

                                                                   10%                    Y
                                                                         2
                                                                        M
                                                                        5%
                                                                         Z           0.09
                                                                                      Z

                                            • Recall that the market return 0.50 0.10,1.00 only X outperformed. This is the
                                                                            was        so
                                                                        0%
                                                                   0.00                          1.50    2.00    2.50
                                              same result as with the Sharpe Ratio.         Beta


                                                                                                    Risk vs Return                      X
                                            • The M2 of a portfolio is the return that this adjusted portfolio earned. This
                                                                         15%

                                              return can then be compared directly to M market return for the
                                                                                               the
Lecture 6




                                                               Return




                                                                 10%
                                              period.                                                   Y
                                                                            5%                            Z

                                                                            0%
                                                                                 0%            5%           10%             15%          20%
                                                                                                          Std. Dev.
10. Marginal Risk
                                            • Represents the change in risk due to a small increase in one of the
                                              allocations. It is essentially a derivative that measures the rate of change in
                                              some measure of interest given a small change in a variable.
Part 1. Portfolio Performance Measurement




                                                                                   P Cov( Ri , RP )
                                                                   M arg Risk                        b i , P P
                                                                                  wi     P
                                            • Beta represents the marginal contribution to the risk of the total portfolio

                                            • Large values of beta indicates that a small addition will have a relatively large
                                              effect on the portfolio

                                            • Positions with large betas should be cut first to reduce risk

                                            The portfolio risk/standard deviation is the sum of the risk contributions from
                                            each asset.
                                                                    ContributionToRisk  wi b i , P P
Lecture 6
10. Marginal Risk                   M arg Risk 
                                                                                                P Cov( Ri , RP )
                                                                                                                   b i , P P
                                                                                               wi     P
                                            I am running a hedge fund
Part 1. Portfolio Performance Measurement



                                            Bank of America 25% . Beta=3
                                            σp =10%,

                                                             Contribution to Risk =.25 x 3 x 10%= 7.5%

                                            • 7,5% of my portfolio risk is going to be dictated by what happens to
                                              Bank of America

                                            • The risk is too concentrated in one stock. In practice, it is
                                              desirable to spread out total risk contributions across as many stocks
                                              or assets as you can in the most equivalent manner.
Lecture 6
Part 1I

HEDGE FUND RISK
MANAGEMENT

   a. Introduction
   b. Strategies




                     30
1. Introduction
                      What is a H.F.?
Part 2. Hedge Funds
Lecture 6
1. Introduction
                      What is a H.F.?
Part 2. Hedge Funds
Lecture 6
1. Introduction
                      What is a H.F.?
Part 2. Hedge Funds
Lecture 6
Instruments whose prices
                      2. H.F. Strategies                        Achieve a beta as close to zero to
                                                                                                           fluctuate based on the changes
                                                                                                              in economic policies along
                                                                 protect against systematic risk
                      An heterogeneous group                                                                     with the flow of capital



        Long positions in stocks
         expected to increase.

       Short positions in stocks
        expected to decrease


  Exploit pricing inefficiencies
  before or after a corporate
             event:
Bankruptcy, Merger, Acquisition



Find “bargains” and accept risk
Part 2. Hedge Funds




1000 basis points above the risk-
      free rate of return


    Exploits pricing differentials
      between fixed-income
Lecture 6




             securities.             Long position in convertible securities.
                                                      AND                                            Holds a portfolio of other investment
                                      Short position in the same company’s                            funds instead of investing directly in
                                                 common stock.                                                     securities
Part III

CREDIT RISK
MANAGEMENT

   a.   Intro
   b.   Drivers of Credit Risk
   c.   Settlement risk
   d.   Credit losses



                                 35
1. Introduction
                                 Definition
                                 The potential for loss due to failure of a borrower to meet its
                                 contractual obligation to repay a debt in accordance with the agreed terms
                                 • Its effect is measured by the cost of replacing cash flows if the other
                                    party defaults
                                 • Commonly also referred to as default risk
                                 • Credit events include
Part 3. Credit Risk Management




                                      • bankruptcy,
                                      • failure to pay,
                                      • loan restructuring
                                      • loan moratorium
                                 • Example: A homeowner stops making mortgage payments



                                  Market Risk                            Credit Risk
                                                                         Potential loss due to the non
                                  Potential loss due to changes in
                                                                         performance of a financial
Lecture 6




                                  market prices or values
                                                                         contract, or financial aspects of
                                                                         non performance in any contract
2. Drivers of Credit Risk
                                 Default:
                                 Discrete state for the counterparty (Default or not). It has associated the
                                           Probability of Default (PD) defined as the likelihood that the
                                           borrower will fail to make full and timely repayment of its financial
                                           obligations

                                 Exposure At Default (EAD)
Part 3. Credit Risk Management




                                 The expected value of the loan at the time of default

                                 Loss Given Default (LGD)
                                 The amount of the loss if there is a default, expressed as a percentage of the
                                 EAD

                                 Recovery Rate (RR)
                                 The proportion of the EAD the bank recovers
Lecture 6
3. Settlement risk
                                 In initial consideration

                                 Settlement risk: The risk that one party will fail to deliver the terms of a
                                 contract with another party at the time of settlement

                                 Foreign exchange (FX) settlement risk is the risk of loss when a bank in a
                                 foreign exchange transaction pays the currency it sold but does not receive
                                 the currency it bought.
Part 3. Credit Risk Management




                                 Settlement Risk management:
                                     • Real time systems
                                     • Bilateral netting agreements (two institutions)
                                     • Multilateral netting agreements (two industries)
                                                    CLS Bank. In foreign exchange and operates the largest
                                                    multicurrency cash settlement system. It is owned by the
                                                    world's leading financial institutions
Lecture 6
4. Credit losses (overview)
                                 Set up:
                                 The credit losses are defined by

                                                         CL  i 1 bi * CEi * (1  f i )
                                                                    N


                                     Where
                                     • Random Variable bi is a bernoulli trial that takes values of 1 (Def) or 0 (non Def)
                                     • CEi is the Credit Exposure at time of default
                                     • fi is the recovery rate (What means 1-fi )
Part 3. Credit Risk Management




                                 The Expected Credit Loss for a portfolio is:

                                                   E[CL]  i 1 E[bi ] * CEi * (1  f i )
                                                                   N
                                                                                                 Default is affected by
                                                                                                 correlation among
                                                              i 1 pi * CEi * (1  f i )
                                                                    N
                                                                                                 assets:

                                 Example                                     Expected Credit Loss
                                     Asset      Exposure       Prob. Def.
                                                                             (5% * 25)  (10% * 30)  (20% * 45)
Lecture 6




                                       A           $25            5%
                                                                             $13.5M
                                       B           $30            10%
                                       C           $45            20%

                                      TT         $100
4. Credit losses (overview)
                                 Description of the complete distribution
                                                      Issuer     Exposure       P. Def     P. No Def.
                                                         A       £     25.00       5%          95%
                                                         B       £     30.00      10%          90%
                                                         C       £     45.00      20%          80%

                                          Default      Loss     Probability Cumulative Exp. Loss         Variance
                                           None
Part 3. Credit Risk Management




                                             A
                                             B
                                             C
                                            A,B
                                            A,C
                                            B,C
                                           A,B,C
Lecture 6




                                                  1. How can I find the loss
                                                              2. What is the prob. Associated to each scenario?
                                                                             3. Easy
                                                                                          4. Expected loss of each scenario
                                 What could be the VaR of this portfolio?                               5. Variance
Part IV

PROBABILITY OF
DEFAULT
Likelihood that the borrower will fail to make full and timely
repayment of its financial obligations

   a. Actuarial
   b. Market prices methods




                                                                 41
1. Methodologies



                                   Actuarial methods
                                   • Measure default rates using historical data
                                   • Provided by external rating agencies
Part 4. Probability of default




                                   Market price methods
                                   • Infer default risk from market prices of debt, equity
                                     prices, credit derivatives
Lecture 6
1. Actuarial Methods                             Corporate Default Probabilities Increase
                                          Factor 1                                  exponentially across Credit Grades
                                         Credit Ratings            b.


                                    a.
Part 4. Probability of default




                                                                        • Credit rating is a measure of the firm’s
                                                                          credit risk
                                                                        • External credit rating: Standard & Poor’s,
                                                                          Moody’s, Fitch, etc.
                                                                          Factor 2
                                                                        Prob. Of default
                                 • Probability of staying in the
                                   same rating category is
                                   given on the diagonal.
Lecture 6




                                 • Off-diagonal probabilities
                                   present the likelihood that
                                   the rating will change
                                   within a one-year period.
Part 4. Probability of default   1. Actuarial Methods                                           Factor 3
                                                                                             Transition Matrix




                                  • Credit migration or transition matrices use ratings migration
                                    histories.
                                  • One-year horizon.
                                  • Measured using the cohort and the duration method.
Lecture 6




                                  • Generally, the transition matrix is affected by the business cycle:
                                    downgrades including defaults are higher during recessions
Part 4. Probability of default   1. Actuarial Methods                                            Factor 4
                                                                                           Cumulative Default Rates
Lecture 6




                                 • How many companies rated ( ) defaulted in each year
                                 • This measure is cumulative. It necessarily increases with the horizon
1. Actuarial Methods
                                                  d1           Default



                                                                                Default
                                                1 - d1             d2

                                                  No Default
                                                                1 – d2                 d3           Default
Part 4. Probability of default




                                                                   No Default

                                                                                  1 – d3
                                                                                                     No Default
                                  Cumulative        d1          d1+ (1- d1)d2   (1- d1)(1- d2) d3
Lecture 6




                                 Compute the cumulative probability of default:
                                 • The probability of default in the first year is 5%
                                 • The probability of default in the second year is 7%
1. Actuarial Methods
                                  Compute the cumulative probability of default:
                                  • The probability of default in the first year is 5%
                                  • The probability of default in the second year is 7%


                                                d1=5%           Default


                                                                Default in 2=
                                                              95%*7%=6.65%       Default
Part 4. Probability of default




                                    Survival rate = 95%

                                                  No Default
                                                                                        d3           Default
                                                          Survival rate 2=
                                                          95%*97% = 0.883
                                                                    No Default

                                                                                   1 – d3
Lecture 6




                                                                                                      No Default
                                   Cumulative       d1           d1+ (1- d1)d2   (1- d1)(1- d2) d3
Factor 5
                                 1. Actuarial Methods                                           Recovery Rates

                                 Recovery Rates

                                 Amount recovered through foreclosure or bankruptcy procedures in a credit
                                 event (default), expressed as a percentage of face value

                                 Are function of
                                 • The state of the economy. Higher with expansion
                                 • The obligor’s characteristics: Higher when the borrower’s assets are
                                   tangible and when previous rating was high
                                 • The credit event: distressed debt has higher recovery rate than plain
Part 4. Probability of default




                                   default
                                 • The status of the debtor: Higher seniority has higher recovery rates

                                 Credit rating agencies have used two methods to calculate RECOVERY
                                 RATES (Moody’s)

                                 • Average issuer-weighted trading price on a sovereign's bonds 30 days
                                   after its initial missed interest payment
Lecture 6




                                 • Ratio of the value of the old securities to the value of the new
                                   securities received in exchange,
1. Actuarial Methods
                                  Recovery Rates
Part 4. Probability of default
Lecture 6




                                 • Average historical sovereign recovery rate: 53%
                                 • 67% of recovery rate according to the ratio of value
1. Actuarial Methods
                                 Recovery Rates
Part 4. Probability of default




                                 Senior debt has a higher recovery rate.
                                 • According to S&P recovery rates have averaged 51% on a discounted basis and 60%
                                    on a nominal basis, based on a sample from 1987 to 2011.
                                 • If measured on a dollar-weighted basis, which is the sum of all defaulted debt in the
                                    sample divided by the sum of the dollar amount of debt recovered, the averages are
                                    slightly lower: 48%
                                 • Loans and revolving credit facilities, that have seniority in the capital structure and are
Lecture 6




                                    often secured: recovered 74% on a discounted basis and when measured on a dollar-
                                    weighted basis, the average recovery for loans and facilities is 65%
                                 • Bonds have lower average recoveries. The long-run discounted average recovery for
                                    bonds is 38%
Next Lecture
                                 2. Market price methods
                                 Infer default risk from market prices of debt, equity prices, credit
Part 4. Probability of default




                                 derivatives
                                  • Infer default risk from bonds
                                  • Merton's model (Structural Model)
Lecture 6
LECTURE SIX

End of the lecture




                     52

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Lecture 6

  • 1. LECTURE SIX a. Portfolio performance measurement b. Hedge fund risk management c. Credit risk management d. Probability of default 1
  • 2. Part 1 PORTFOLIO PERFORMANCE MEASUREMENT a. Intro. Performance measurement b. Surplus at Risk c. The Market Line (ML) and CAPM d. Several ratios to measure market performance 2
  • 3. Part 1. Portfolio Performance Measurement 1. Intro. Performance measurement Portfolio management Risk Return Do profits reflect my risk exposure? HOW? • Looking at process/strategies in place, and • Whether outcomes are in line with what was intended or should have been achieved. Lecture 6 • Pure luck? • Good strategy • Separate effect of the market and active management
  • 4. 1. Intro. Performance measurement Some definitions Sell side Buy side Part 1. Portfolio Performance Measurement  Creation, promotion, analysis and  Final buyers of financial assets sale of securities. (Large portions of securities)  Leverage and speculative  Tend to me more conservative No leverage  Examples?  Examples? Lecture 6
  • 5. 1. Intro. Performance measurement Some definitions Sell side Buy side Part 1. Portfolio Performance Measurement  Creation, promotion, analysis and  Final buyers of financial assets sale of securities. (Large portions of securities)  Leverage and speculative  Tend to me more conservative No leverage  Institutions such as  Investing institutions such as • Investment bankers • mutual funds, (intermediaries between issuers • pension funds and and public), • insurance firms • Research companies that perform stock research and Lecture 6 make ratings. Ig. Roubini • Market makers who provide liquidity in the market.
  • 6. 1. Intro. Performance measurement Absolute and Relative risk Absolute risk Relative Risk Part 1. Portfolio Performance Measurement With respect to the portfolio With respect to a benchmark itself • Tracking error : e = R P - RB • Risk factor: σ • In dollar terms: exP • P: Initial portfolio value • In dollar terms 𝜎 (𝑅 𝑃 − 𝑅 𝐵 ) 𝑃 𝜎 ∆𝑃 ∆𝑃 ∆𝐵 ∆𝑃 𝜎( − ) 𝑃 𝑃 𝐵 𝜎 ∗ 𝑃 𝑃 𝜔 ∗ 𝑃 𝜎 𝑅𝑃 ∗ 𝑃 Tracking Error Volatility Note that there is only Where Lecture 6 the asset or portfolio  2   2 P  2  P B   2 P
  • 7. 1. Intro. Performance measurement Absolute and Relative risk Part 1. Portfolio Performance Measurement Example SPX: 10% return -10% return My trade: 6% return -4% return Absolute risk Relative Risk What would be the difference between absolute and relative risk? Lecture 6
  • 8. What is my risk in the long term? 2. Surplus at Risk Focused on net profits Performance Measurement Sort of relative risk measurement Part 1. Portfolio Performance Measurement Two assets, long pension assets and short pension liabilities. Assets $ 120,00 Liabilities $ 100,00 Surplus $ 20,00 Volatility 12% Volatility 3% Expected R 8% Expected R 5% Correlation 0,3 Change of assets Change of assets $ 9,60 $ 5,00 (per period) 12% of $120 (per period) 5% of $100 Expected growth of surplus $ 4,60 Change assets - liabilities Expected surplus $ 24,60 Considering the surplus Variance of surplus $ 190,44 variance(a – b) = variance (a) + variance(b) - (2)cov(a,b) Volatility of surplus $ 13,80 Lecture 6 Confidence level 95% Normal deviate 1,64 Surplus at Risk $VaR 18,03 –(expected surplus) + (volatility of surplus)*(normal deviate) The complete variance formula is: σ12P12+ σ22P12-2 σ1σ2P1P2ρ
  • 9. 3.The Market Line (ML) and CAPM Small recap Decompose total return into a component due to market risk premia and Part 1. Portfolio Performance Measurement other factors. E(Ri) Overvalued ML M E(RM) RF Undervalued Lecture 6 RiskM Riski Note: Risk is either b or 
  • 10. 3.The Market Line (ML) and CAPM Small recap E ( Ri )  rf  b [ E ( RM )  rf ] Ri   i  b i RM  ei Part 1. Portfolio Performance Measurement E ( Ri )  rf  b [ E ( RM )  rf ] This is the CAPM model E ( Ri )  rf   i  b i [ E ( RM )  rf ] E(Ri) B ML A M E(RM) C E RFR D Lecture 6 RiskM Riski Note: Risk is either b or 
  • 11. 3.The Market Line (ML) and CAPM Small recap Capital Market Line (CML) obtained by combining the market portfolio and the riskless asset Part 1. Portfolio Performance Measurement • CML specifies the expected return for a given level of risk • All possible combined portfolios lie on the CML, and all are Mean-Variance efficient portfolios • Here we have a clear relation between the risk of my portfolio and the risk Lecture 6 of the market. This is reflected by beta Cov( Ri , RM ) It measures how much an asset’s return bi  is driven by the market return  2M
  • 12. 3.The Market Line (ML) and CAPM Small recap Capital Market Line (CML) Part 1. Portfolio Performance Measurement If the stock has a high positive β: • It will have large price swings driven by the market • It will increase the risk of the investor’s portfolio(in fact, will make the entire market more risky …) • The investor will demand a high Er in compensation. If the stock has a negative β : • It moves “against” the market. • It will decrease the risk of the market portfolio • The investor will accept a lower Er Then the SML depicts the relation between β and the Expected Return (Er) Lecture 6 For the risk-free security, b = 0 For the market itself, b=1.
  • 13. 3.The Market Line (ML) and CAPM Small recap Capital Market Line (CML) Part 1. Portfolio Performance Measurement E ( Ri )  rf  b [ E ( RM )  rf ] E ( Ri )  rf  b [ E ( RM )  rf ] Excess of return of a portfolio is a function of the excess of return of the market W.R. a risk free rate Lecture 6
  • 14. 4.Treynor Ratio The Treynor measure calculates the risk premium per unit of risk (bi) Part 1. Portfolio Performance Measurement [ E ( RP )  RF ] TR  b ( RP ) Beta measures the investment volatility relative to the market volatility (systematic risk) The Treynor Ratio is negative if • RF > E[RP] AND β > 0 . Manager has performed badly: failing to get performance better than the risk free rate AND manager made a not good election of portfolio • RF < E[RP] AND β > 0 Manger has performed well, managing to reduce risk but getting a Lecture 6 return better than the risk free rate Higher Ti generally indicates better performance
  • 15. 4.Treynor Ratio ADVANTAGE: It indicates the volatility a ASSET brings to an entire portfolio. The Treynor Ratio should be used only as a ranking mechanism for Part 1. Portfolio Performance Measurement investments within the same sector. . When presented with investments that have the same return, investments with higher Treynor Ratios are less risky and better managed Cov( Ri , RM ) bi   2M Lecture 6
  • 16. 5. Sharpe Ratio Describes how much excess return you are receiving for the extra volatility that you endure for holding a riskier asset. Part 1. Portfolio Performance Measurement [ E ( RP )  RF ] SR   ( RP ) The Sharpe measure is exactly the same as the Treynor measure, except that the risk measure is the standard deviation: • Tells us whether a portfolio's returns are due to smart investment decisions or a result of excess risk. • The greater a portfolio's Sharpe ratio, the better its risk-adjusted performance has been. • A negative Sharpe ratio indicates that a risk-less asset would perform Lecture 6 better than the security being analysed.
  • 17. 4-5. Sharpe V Treynor Ratio The Sharpe and Treynor measures are similar, but different: • Sharpe uses the standard deviation, Treynor uses beta Part 1. Portfolio Performance Measurement X • Sharpe is more appropriate for well diversified portfolios, Treynor for Portfolio Return 15% RFR 5% Beta Std. Dev. Trenor Sharpe 2.50 20% 0.0400 0.5000 Y 8% individual assets14% 0.0600 0.2143 5% 0.50 Z Market 6% • Sharpe and Treynor: The ranking, not the number itself, is what is most 10% 5% 5% 0.35 1.00 9% 11% 0.0286 0.1111 0.0500 0.4545 important Risk vs Return 15% Portfolio Return RFR Beta Std. Dev. Trenor Sharpe X M X Y 15% 5% 2.50 20% 0.0400 0.5000 Return 10% Portfolio Return RFR Beta Y 8% 5% 0.50 14% 0.0600 Dev. Std. Trenor Sharpe 0.2143 5% X 15% 5% 2.50 20% 0.0400 0.5000 Z Z 6% 5% 0.35 Y 8% 9%5% 0.0286 0.50 14% 0.11110.2143 0.0600 Market 0% 10% 5% 1.00 Z 6% 11% 5% 0.0500 0.35 9% 0.45450.1111 0.0286 0.00 0.50 1.00 1.50 2.00 2.50 Market 10% 5% 1.00 11% 0.0500 0.4545 Beta Risk vs Return Risk vs Return X Risk vs Return 15% 15% 15% M X M X Return 10% Return M 10% Y Lecture 6 Return 10% Y Y 5% Z 5% Z 0% 5% 0% Z 0% 5% 10% 15% 20% 0.00 0.50 1.00 1.50 2.00 2.50 Std. Dev. Beta 0%
  • 18. 6. Sortino Ratio The Sortino ratio generalizes (focus on the downside) from the Sharpe by using: Part 1. Portfolio Performance Measurement • In the numerator, instead of excess return (above riskfree), Sortino uses excess above hurdle (MAR, minimum acceptable return) • In the denominator, instead of volatility (annualized standard deviation), Sortino uses downside deviation. [ E ( RP )  MAR] SR   L ( RP ) • Appears to resolve several of the issues inherent in the Sharpe ratio: • It incorporates a relevant return target, in both the numerator and the denominator; • It quantifies downside volatility without penalizing upside volatility; and because of its focus on downside risk, • It is more applicable to distributions that are negatively skewed than Lecture 6 measures based on standard deviation.
  • 19. 5-6. Sortino Ratio and Sharpe Ratio Example Only consider the returns below the Square difference WR hurdle rate Sq. Difference WR the Part 1. Portfolio Performance Measurement the hurdle rate average of returns Sumation of returns Excess over Hurdle We take a minimun Averag montly return of P 1,821% acceptable return Squared difference Average yearly return 21,851% Times 12 Month Price Returns Hurdle rate R - HUR Hurdle>R (R-Hur)^2 (R-Av)^2 Hurdle rate yield return Y 18,000% 1.5% * 12 1 663,03 -2,539% 1,50% -4,04% -4,04% 0,1631% 0,1901% 2 680,3 -9,834% 1,50% -11,33% -11,33% 1,2847% 1,3585% Rf month 1,80% 3 754,5 10,132% 1,50% 8,63% 0,6907% RF Y 21,60% Times 12 4 685,09 8,234% 1,50% 6,73% 0,4113% Use the formula 5 632,97 9,120% 1,50% 7,62% 0,5327% 6 580,07 -0,136% 1,50% -1,64% -1,64% 0,0268% 0,0383% Volatility 29,06% 7 580,86 -3,966% 1,50% -5,47% -5,47% 0,2988% 0,3349% Sharpe ratio 0,865% 8 604,85 -5,675% 1,50% -7,17% -7,17% 0,5148% 0,5619% 9 641,24 3,719% 1,50% 2,22% 0,0360% Sortino 19,54% 10 618,25 6,575% 1,50% 5,07% 0,2260% Excess return 3,851% P-Hurdle 11 580,11 -10,186% 1,50% -11,69% -11,69% 1,3656% 1,4416% Montly downside VaRianc 19,71% 12 645,9 7,760% 1,50% 6,26% 0,3527% 13 599,39 1,139% 1,50% -0,36% -0,36% 0,0013% 0,0047% 14 592,64 15,067% 1,50% 13,57% 1,7545% 15 515,04 -4,791% 1,50% -6,29% -6,29% 0,3958% 0,4372% 16 540,96 -10,391% 1,50% -11,89% -11,89% 1,4140% 1,4913% 17 603,69 19,217% 1,50% 17,72% 3,0262% 18 506,38 -4,280% 1,50% -5,78% -5,78% 0,3340% 0,3722% Lecture 6 19 529,02 -2,772% 1,50% -4,27% -4,27% 0,1825% 0,2109% 20 544,1 -7,270% 1,50% -8,77% -8,77% 0,7692% 0,8265%
  • 20. 7. Jensen alpha Shows by much the returns of an actively managed portfolio are Part 1. Portfolio Performance Measurement above or below market returns. >0 A positive Alpha means that a portfolio has beaten the market, =0 while a negative value indicates underperformance <0 Risk Premium A fund manager with a negative alpha and a beta greater than one has added risk to the portfolio but has poorer performance than the 0 market Market Risk Premium R i  RFR   i  b i  R M  RFR    i Lecture 6 Alpha = Excess of return – (Beta * (Excess of return))
  • 21. Part 1. Portfolio Performance Measurement 7. Jensen alpha R i  RFR   i  b i  R M  RFR    i Alpha = Excess of return – (Beta * (Excess of return)) Lecture 6
  • 22. 7. Jensen alpha Portfolio Portfolio Market P Q Part 1. Portfolio Performance Measurement Beta 0.90 1.6 1.0 RM-Rf 11% 19% 10% Alpha 2.0% 3.0% 0% R i  RFR   i  b i  R M  RFR    i Portfolio P Expected Return Portfolio Q 19% SML 16% M 11% M2 P Lecture 6 9% 0.9 1.6 Beta
  • 23. 8. Information Ratio Measure of the risk-adjusted return of a portfolio. Part 1. Portfolio Performance Measurement Defined as expected active return divided by tracking error • Active return : difference between the return of portfolio and the return of a benchmark • Tracking error is the standard deviation of the active return E[ R P  RB ]  Component attributable to the manager’s skill IR   VAR( R P  RB )  While Sharpe consider the σ of total returns, IF consider σ of alpha • Measures the active return of the manager's (abnormal return) portfolio per unit of risk that the manager takes relative to the benchmark. • The higher the information ratio, the higher the active return of the portfolio, given the amount of risk taken, and the better the Lecture 6 manager.
  • 24. 8. Information Ratio Returns Date Portfolio Market Excess Part 1. Portfolio Performance Measurement 01/01/2010 2% 2,06% -0,49% 01/02/2010 1% -5,62% 6,64% 01/03/2010 0,61% -3,42% 4,03% 01/04/2010 0,76% 2,84% -2,08% 01/05/2010 9,69% -5,00% 14,69% 01/06/2010 1,39% 5,30% -3,91% 01/07/2010 3,10% -2,33% 5,44% 01/08/2010 0,46% 8,57% -8,12% 01/09/2010 6,11% 4,77% 1,34% 01/10/2010 9,37% 14,69% -5,32% 01/11/2010 3,88% -6,68% 10,56% 01/12/2010 9,54% 1,38% 8,16% Mean 3,96% 1,38% 2,58% Expected excess of return (Benchmark) Standard Dev 3,73% 6,40% 6,88% Lecture 6 Information Ratio 0,3750 E[ R P  RB ]  IR   VAR( R P  RB ) 
  • 25. 9. Modigliani´s Risk Adjustment Performance • Also known as M2 • Closely related to the Sharpe Ratio Part 1. Portfolio Performance Measurement • Focuses on total volatility as a measure of risk, but its risk adjusted measure of performance has the interpretation of a differential return relative to the benchmark index • The idea is to lever or de-lever a portfolio (i.e., shift it up or down the capital market line) so that its standard deviation is identical to that of the market portfolio. • The formula for M2 is:  M 2   M R i  R f   R f   i   Lecture 6 • The M2 of a portfolio is the return that this adjusted portfolio earned. This return can then be compared directly to the market return for the period.
  • 26. 9. Modigliani´s Risk Adjustment Performance • Suppose that • Return Ri: 35% RM: 28% • Volatility σi: 42% σM: 30% Part 1. Portfolio Performance Measurement • Find a portfolio combination with the same level of risk than the benchmark (market)   M   30  0.714 • Portion of the portfolio  i    42 • Portion of risk free asset 1    M   0.286   i   • The return of this portfolio will be [0.286 * 0.06]  [0.714 * 0.35]  0.267 • This portfolio is -1.3% than the market return. This is the M2 Expected Return Market M Portfolio M2 P P* Lecture 6 • Reduce the return of the P • Obtain leSs volatility Volatility
  • 27. 9. Modigliani´s Risk Adjustment Performance  M 2   M R i  R f   R f   i   SR  [  ( RP )  RF ] Part 1. Portfolio Performance Measurement  ( RP ) Portfolio Return RFR Beta Std. Dev. Trenor Sharpe X 15% 5% 2.50 20% 0.0400 0.5000 Y 8% 5% 0.50 14% 0.0600 0.2143 Z 6% 5% 0.35 9% 0.0286 0.1111 Market 10% 5% 1.00 11% 0.0500 0.4545 X  0.200.15  0.05  0.05  0.105 M 2  0.11 Risk vs Return  0.11 0.08  0.05  0.05  0.074 15% M2 M X Y 0.14  0.11 0.06  0.05  0.05  0.062 Return 10% Y 2 M 5% Z 0.09 Z • Recall that the market return 0.50 0.10,1.00 only X outperformed. This is the was so 0% 0.00 1.50 2.00 2.50 same result as with the Sharpe Ratio. Beta Risk vs Return X • The M2 of a portfolio is the return that this adjusted portfolio earned. This 15% return can then be compared directly to M market return for the the Lecture 6 Return 10% period. Y 5% Z 0% 0% 5% 10% 15% 20% Std. Dev.
  • 28. 10. Marginal Risk • Represents the change in risk due to a small increase in one of the allocations. It is essentially a derivative that measures the rate of change in some measure of interest given a small change in a variable. Part 1. Portfolio Performance Measurement  P Cov( Ri , RP ) M arg Risk    b i , P P wi P • Beta represents the marginal contribution to the risk of the total portfolio • Large values of beta indicates that a small addition will have a relatively large effect on the portfolio • Positions with large betas should be cut first to reduce risk The portfolio risk/standard deviation is the sum of the risk contributions from each asset. ContributionToRisk  wi b i , P P Lecture 6
  • 29. 10. Marginal Risk M arg Risk   P Cov( Ri , RP )   b i , P P wi P I am running a hedge fund Part 1. Portfolio Performance Measurement Bank of America 25% . Beta=3 σp =10%, Contribution to Risk =.25 x 3 x 10%= 7.5% • 7,5% of my portfolio risk is going to be dictated by what happens to Bank of America • The risk is too concentrated in one stock. In practice, it is desirable to spread out total risk contributions across as many stocks or assets as you can in the most equivalent manner. Lecture 6
  • 30. Part 1I HEDGE FUND RISK MANAGEMENT a. Introduction b. Strategies 30
  • 31. 1. Introduction What is a H.F.? Part 2. Hedge Funds Lecture 6
  • 32. 1. Introduction What is a H.F.? Part 2. Hedge Funds Lecture 6
  • 33. 1. Introduction What is a H.F.? Part 2. Hedge Funds Lecture 6
  • 34. Instruments whose prices 2. H.F. Strategies Achieve a beta as close to zero to fluctuate based on the changes in economic policies along protect against systematic risk An heterogeneous group with the flow of capital Long positions in stocks expected to increase. Short positions in stocks expected to decrease Exploit pricing inefficiencies before or after a corporate event: Bankruptcy, Merger, Acquisition Find “bargains” and accept risk Part 2. Hedge Funds 1000 basis points above the risk- free rate of return Exploits pricing differentials between fixed-income Lecture 6 securities. Long position in convertible securities. AND Holds a portfolio of other investment Short position in the same company’s funds instead of investing directly in common stock. securities
  • 35. Part III CREDIT RISK MANAGEMENT a. Intro b. Drivers of Credit Risk c. Settlement risk d. Credit losses 35
  • 36. 1. Introduction Definition The potential for loss due to failure of a borrower to meet its contractual obligation to repay a debt in accordance with the agreed terms • Its effect is measured by the cost of replacing cash flows if the other party defaults • Commonly also referred to as default risk • Credit events include Part 3. Credit Risk Management • bankruptcy, • failure to pay, • loan restructuring • loan moratorium • Example: A homeowner stops making mortgage payments Market Risk Credit Risk Potential loss due to the non Potential loss due to changes in performance of a financial Lecture 6 market prices or values contract, or financial aspects of non performance in any contract
  • 37. 2. Drivers of Credit Risk Default: Discrete state for the counterparty (Default or not). It has associated the Probability of Default (PD) defined as the likelihood that the borrower will fail to make full and timely repayment of its financial obligations Exposure At Default (EAD) Part 3. Credit Risk Management The expected value of the loan at the time of default Loss Given Default (LGD) The amount of the loss if there is a default, expressed as a percentage of the EAD Recovery Rate (RR) The proportion of the EAD the bank recovers Lecture 6
  • 38. 3. Settlement risk In initial consideration Settlement risk: The risk that one party will fail to deliver the terms of a contract with another party at the time of settlement Foreign exchange (FX) settlement risk is the risk of loss when a bank in a foreign exchange transaction pays the currency it sold but does not receive the currency it bought. Part 3. Credit Risk Management Settlement Risk management: • Real time systems • Bilateral netting agreements (two institutions) • Multilateral netting agreements (two industries) CLS Bank. In foreign exchange and operates the largest multicurrency cash settlement system. It is owned by the world's leading financial institutions Lecture 6
  • 39. 4. Credit losses (overview) Set up: The credit losses are defined by CL  i 1 bi * CEi * (1  f i ) N Where • Random Variable bi is a bernoulli trial that takes values of 1 (Def) or 0 (non Def) • CEi is the Credit Exposure at time of default • fi is the recovery rate (What means 1-fi ) Part 3. Credit Risk Management The Expected Credit Loss for a portfolio is: E[CL]  i 1 E[bi ] * CEi * (1  f i ) N Default is affected by correlation among  i 1 pi * CEi * (1  f i ) N assets: Example Expected Credit Loss Asset Exposure Prob. Def.  (5% * 25)  (10% * 30)  (20% * 45) Lecture 6 A $25 5%  $13.5M B $30 10% C $45 20% TT $100
  • 40. 4. Credit losses (overview) Description of the complete distribution Issuer Exposure P. Def P. No Def. A £ 25.00 5% 95% B £ 30.00 10% 90% C £ 45.00 20% 80% Default Loss Probability Cumulative Exp. Loss Variance None Part 3. Credit Risk Management A B C A,B A,C B,C A,B,C Lecture 6 1. How can I find the loss 2. What is the prob. Associated to each scenario? 3. Easy 4. Expected loss of each scenario What could be the VaR of this portfolio? 5. Variance
  • 41. Part IV PROBABILITY OF DEFAULT Likelihood that the borrower will fail to make full and timely repayment of its financial obligations a. Actuarial b. Market prices methods 41
  • 42. 1. Methodologies Actuarial methods • Measure default rates using historical data • Provided by external rating agencies Part 4. Probability of default Market price methods • Infer default risk from market prices of debt, equity prices, credit derivatives Lecture 6
  • 43. 1. Actuarial Methods Corporate Default Probabilities Increase Factor 1 exponentially across Credit Grades Credit Ratings b. a. Part 4. Probability of default • Credit rating is a measure of the firm’s credit risk • External credit rating: Standard & Poor’s, Moody’s, Fitch, etc. Factor 2 Prob. Of default • Probability of staying in the same rating category is given on the diagonal. Lecture 6 • Off-diagonal probabilities present the likelihood that the rating will change within a one-year period.
  • 44. Part 4. Probability of default 1. Actuarial Methods Factor 3 Transition Matrix • Credit migration or transition matrices use ratings migration histories. • One-year horizon. • Measured using the cohort and the duration method. Lecture 6 • Generally, the transition matrix is affected by the business cycle: downgrades including defaults are higher during recessions
  • 45. Part 4. Probability of default 1. Actuarial Methods Factor 4 Cumulative Default Rates Lecture 6 • How many companies rated ( ) defaulted in each year • This measure is cumulative. It necessarily increases with the horizon
  • 46. 1. Actuarial Methods d1 Default Default 1 - d1 d2 No Default 1 – d2 d3 Default Part 4. Probability of default No Default 1 – d3 No Default Cumulative d1 d1+ (1- d1)d2 (1- d1)(1- d2) d3 Lecture 6 Compute the cumulative probability of default: • The probability of default in the first year is 5% • The probability of default in the second year is 7%
  • 47. 1. Actuarial Methods Compute the cumulative probability of default: • The probability of default in the first year is 5% • The probability of default in the second year is 7% d1=5% Default Default in 2= 95%*7%=6.65% Default Part 4. Probability of default Survival rate = 95% No Default d3 Default Survival rate 2= 95%*97% = 0.883 No Default 1 – d3 Lecture 6 No Default Cumulative d1 d1+ (1- d1)d2 (1- d1)(1- d2) d3
  • 48. Factor 5 1. Actuarial Methods Recovery Rates Recovery Rates Amount recovered through foreclosure or bankruptcy procedures in a credit event (default), expressed as a percentage of face value Are function of • The state of the economy. Higher with expansion • The obligor’s characteristics: Higher when the borrower’s assets are tangible and when previous rating was high • The credit event: distressed debt has higher recovery rate than plain Part 4. Probability of default default • The status of the debtor: Higher seniority has higher recovery rates Credit rating agencies have used two methods to calculate RECOVERY RATES (Moody’s) • Average issuer-weighted trading price on a sovereign's bonds 30 days after its initial missed interest payment Lecture 6 • Ratio of the value of the old securities to the value of the new securities received in exchange,
  • 49. 1. Actuarial Methods Recovery Rates Part 4. Probability of default Lecture 6 • Average historical sovereign recovery rate: 53% • 67% of recovery rate according to the ratio of value
  • 50. 1. Actuarial Methods Recovery Rates Part 4. Probability of default Senior debt has a higher recovery rate. • According to S&P recovery rates have averaged 51% on a discounted basis and 60% on a nominal basis, based on a sample from 1987 to 2011. • If measured on a dollar-weighted basis, which is the sum of all defaulted debt in the sample divided by the sum of the dollar amount of debt recovered, the averages are slightly lower: 48% • Loans and revolving credit facilities, that have seniority in the capital structure and are Lecture 6 often secured: recovered 74% on a discounted basis and when measured on a dollar- weighted basis, the average recovery for loans and facilities is 65% • Bonds have lower average recoveries. The long-run discounted average recovery for bonds is 38%
  • 51. Next Lecture 2. Market price methods Infer default risk from market prices of debt, equity prices, credit Part 4. Probability of default derivatives • Infer default risk from bonds • Merton's model (Structural Model) Lecture 6
  • 52. LECTURE SIX End of the lecture 52