3. The Core Question of
Finance
Congratulations!!!
You have won a cash prize!
There are two optional payment
schedules:
A - receive $100,000 now
B - receive $100,000 in five years.
Which option would you choose?
3
4. Time Value of Money
Concept
In simple terms
the concept
implies that money today
is always better than
money tomorrow.
4
5. Why Time Value of Money
Exists?
Risk and Uncertainty-future always involves some risk, especially in
respect to cash inflows of company as they are highly uncontrollable;
Inflation-in an inflationary economy a dollar today has always more
purchasing power in compared to a dollar some point in future;
Consumption Preference- individuals generally prefer current
consumption to a future one;
Investment Opportunities-an investor can profitably use money received
today by investing it immediately;
5
6. Allows investors to adjust cash
flows for the passage of time;
It’s an integral part of Capital
Budgeting Processes;
Applied in present and future
value calculations;
6
10. SI = P0(i)(n)
= $1,000(.07)(2)
= $140$140
Simple Interest Example
Assume that you deposit $1,000 in an
account earning 7% simple interest for 2
years. What is the accumulated interest at
the end of the 2nd year?
10
11. Compound Interest
Yields higher return for
investors or deposit
holders;
Cumbersome for
borrowers;
Makes borrowers to be
more adhere to their
payment schedule,for
example.credit cards;
11
12. Assume that you deposit $1,000$1,000 at a
compound interest rate of 7% for 22
yearsyears.
Compound Interest
Example
0 1 22
$1,000$1,000
FVFV22
7%
12
13. At the end of first year
PP00 (1+i)1
= $1,000x$1,000x (1.07)
= $1,070$1,070
Compound Interest
You earned $70 interest on your $1,000
deposit over the first year.
This is the same amount of interest you
would earn under simple interest.
Compound Interest
Example
13
14. At the end of first yearAt the end of first year = $1,000$1,000 (1.07)
= $1,070$1,070
At the end of second yearAt the end of second year = 1070 (1+i)1
1070x(1.07)
$1,144.90$1,144.90
You earned an EXTRA $4.90$4.90 in Year 2 with
compound over simple interest.
Compound Interest
Example(cont.)
14
17. Future Value
The value at some future time of a present
amount of money, or a series of payments
evaluated at a given interest rate;
The interest earned on the initial principal
amount becomes a part of the principal at
the end of the compounding period;
17
18. Future Value Example
Problem
Suppose you invest $1000 for three years in a saving account that pays 10 %
interest per year. If you let your interest income be reinvested, your investment
will grow as follows:
First year : Principal at the beginning $1000
Interest for the year ($1,000 × 0.10) $100
Principal at the end $1,100
Second year : Principal at the beginning $1,100
Interest for the year ($1,100 × 0.10) $110
Principal at the end $1,210
Third year : Principal at the beginning $1,210
Interest for the year ($1210 × 0.10) $121
Principal at the end $1,331
18
19. FormulaFormula
FV = P0(1+i)n
FV: Future Value
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods
In the previous example
FV=1000(1+0.1)3
=1,331
19
20. Double Your Money!
We will use
the““Rule-of-72Rule-of-72””
Quick! How long
does it take to
double $5,000 at a
compound rate of
12% per year
(approx.)?
20
21. Approx. Years to Double = 7272 / i%
7272 / 12% = 6 Years6 Years
[Actual Time is 6.12 Years]
The “Rule-of-72”
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?
21
22. Present Value
Which one would you prefer assuming that
the rate is 8%?
a)$1000 today or,
b)$2000 10 years later?
22
To answer this question we have to
express $2000 in today’s money.
PV=FV/(1+i)n
$926=2000/(1+0.8)10
23. Types of Annuities
• Ordinary AnnuityOrdinary Annuity: Payments or receipts
occur at the end of each period(coupon);
• Annuity DueAnnuity Due: Payments or receipts occur at
the beginning of each period(rent);
An AnnuityAn Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
23
24. Parts of an Annuity
0 1 2 3
$100 $100 $100
(Ordinary Annuity)
EndEnd of
Period 1
EndEnd of
Period 2
Today
EqualEqual Cash Flows
Each 1 Period Apart
EndEnd of
Period 3
24
25. Parts of an Annuity
0 1 2 3
$100 $100 $100
(Annuity Due)
BeginningBeginning of
Period 1
BeginningBeginning of
Period 2
Today
EqualEqual Cash Flows
Each 1 Period Apart
BeginningBeginning of
Period 3
25
26. FVAFVA33 = $1,000(1.07)2
+
$1,000(1.07)1
+ $1,000(1.07)0
= $1,145 + $1,070 + $1,000
= $3,215 or R(FVIFA$3,215 or R(FVIFAi,ni,n))
Example of an
Ordinary Annuity -- FVA
$1,000 $1,000 $1,000
0 1 2 33 4
$3,215 = FVA$3,215 = FVA33
7%
$1,070
$1,145
Cash flows occur at the end of the period
26
27. PVAPVA33 = $1,000/(1.07)1
+
$1,000/(1.07)2
+
$1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32 or R(PVIFA$2,624.32 or R(PVIFAi,ni,n))
Example of an
Ordinary Annuity -- PVA
$1,000 $1,000 $1,000
0 1 2 33 4
$2,624.32 = PVA$2,624.32 = PVA33
7%
$934.58
$873.44
$816.30
Cash flows occur at the end of the period
27
28. FVADFVAD33 = $1,000(1.07)3
+
$1,000(1.07)2
+ $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440 or R(FVIFA$3,440 or R(FVIFAi,ni,n)(1+i))(1+i)
Example of an
Annuity Due -- FVAD
$1,000 $1,000 $1,000 $1,070
0 1 2 33 4
$3,440 = FVAD$3,440 = FVAD33
7%
$1,225
$1,145
Cash flows occur at the beginning of the period
28
29. PVADPVADnn = $1,000/(1.07)0
+ $1,000/(1.07)1
+
$1,000/(1.07)2
= $2,808.02$2,808.02
oror R(PVIFAR(PVIFA’,n-1’,n-1+1)+1)
Example of an
Annuity Due -- PVAD
$1,000.00 $1,000 $1,000
0 1 2 33 4
$2,808.02$2,808.02 = PVADPVADnn
7%
$ 934.58
$ 873.44
Cash flows occur at the beginning of the period
29
30. Julie Miller will receive the set of cash
flows below. What is the Present ValuePresent Value at
a discount rate of 10%10%.
Mixed Flows Example
0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
PVPV00
10%10%
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31. How To Solve
1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
10%
$545.45$545.45
$495.87$495.87
$300.53$300.53
$273.21$273.21
$ 62.09$ 62.09
$1677.15$1677.15 == PVPV00 of the Mixed Flowof the Mixed Flow 31
32. The actual rate of interest earned (paid)
after adjusting the nominal rate for
factors such as the number of
compounding periods per year.
(1 + [ i / m ] )m
- 1
Effective Annual
Interest Rate
32
33. Basket Wonders (BW) has a $1,000 CD at
the bank. The interest rate is 6%
compounded quarterly for 1 year. What
is the Effective Annual Interest Rate
(EAREAR)?
EAREAR = ( 1 + 6% / 4 )4
- 1
= 1.0614 - 1 = .0614 or 6.14%!6.14%!
BW’s Effective
Annual Interest Rate
33
34. Julie Miller is borrowing $10,000$10,000 at a
compound annual interest rate of 12%.
Amortize the loan if annual payments are
made for 5 years.
Step 1: Payment
PVPV00 = R (PVIFA i%,n)
$10,000$10,000 = R (PVIFA 12%,5)
$10,000$10,000 = R (3.605)
RR = $10,000$10,000 / 3.605 = $2,774$2,774
Amortizing a Loan Example
34
35. Amortizing a Loan Example
End of
Year
Payment Interest Principal Ending
Balance
0 --- --- --- $10,000
1 $2,774 $1,200 $1,574 8,426
2 2,774 1,011 1,763 6,663
3 2,774 800 1,974 4,689
4 2,774 563 2,211 2,478
5 2,775 297 2,478 0
$13,871 $3,871 $10,000
