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Learning Unit #9: Interest Rate

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Annualized rate of return = [($160 + $990)/$960]^(1/2) – 1 = 9.4%

- 1. Learning Unit #9 Interest Rates
- 2. Objectives of Learning Unit #9 • Yield to Maturity • Rate of Return • Real and Nominal Interest Rates
- 3. Yield to Maturity • Yield to maturity: The interest rate that equates the present value of payments received from a debt instrument with its value today. – Given cash flows in future from a debt instrument such as coupon bond and fixed payment loan, a present value of the cash flows can be computed with a choice of an interest rate. Depending on your choice of interest rate, the present value could be high or low. – If the present value is exactly equal to the value (price) of the debt instrument today, then that interest is the yield to maturity.
- 4. Yield to Maturity as Interest Rate • There are many interest rate formula available. Only “yield to maturity” is considered as the interest rate in economics and finance. • In finance, the yield to maturity is called “internal rate of return.” • All other formula to compute interest rates are approximations of actual interest rate and they are NOT the interest rate! • Remember when we say “interest rate,” it means the yield to maturity!
- 5. Example of Yield to Maturity • You borrow $100 today and promise to pay back $110 one year later. – A future cash flow is $110 one year later. – A value of instrument (simple loan) is $100 today. • Cash flows on timeline should look like 0 1 $100 $110
- 6. Example of Yield to Maturity • A present value of future cash flow is i PV + = 1 110$ • Equate the present value to the simple loan today: i PVtodayValue + === 1 110$ 100$ • Solve for i: i = 10%
- 7. Another Example of Yield to Maturity • You borrow $100 today and promise to pay back $121 two years later. Cash flows on timeline looks as 0 1 2 $121 $100 • Equate a present value of cash flow with the value today. 2 )i1( 121$ PV100$TodayValue + === • Solve for i: i = 10%
- 8. Yield to Maturity Formula • Formula #4: i = (FV/PV)1/n - 1 This formula can be applied only on simple loan or discount bond where there is only one cash flow in future. • Ex. A 2-year simple loan with $100 principal and $121 payback. Here, PV=$100, FV=$121, and n=2, so the yield to maturity is %101.01 100$ 121$ i 2 1 ==− =
- 9. Yield to Maturity of Bonds • When there are more than one cash flows in future, you cannot use Formula #4. • Instead, you may use a business calculator, Microsoft Excel, or simply try-and-error. • Ex. A coupon bond with 5% coupon rate and 2 year maturity is priced at $950 today. • Cash flows on timeline look like 0 1 2 $50 $1,050 $950
- 10. Yield to Maturity of Bonds • Equate a present value of future cash flows with a value of bond today: 2 )i1( 1050$ i1 50$ PV950$todayValue + + + === • Solve for i: i = 7.8% • If you have a business calculator, simply provide cash flows including today’s value and year, then compute for IRR (internal rate of return).
- 11. Trial-and-Error Method You can find an approximate yield to maturity by trial-and-error method. – First, use the formula of current yield to make an initial guess: current yield = $50/$950 = 5.2% – Second, use the rate of return formula to make another guess: rate of return = ($50 + $1050 - $950)/$950 = 15.8% – An actual yield to maturity is between these two numbers, so let’s start with i = 10% (middle point). Then, PV = $913.22 < $950 = value today, it is too small. To make PV higher, i must be lower (an inverse relationship). – Choose i = 7%, then PV = $963.84 > $950, so i is little too small, and i should be little greater than 7%. – Choose i = 8%, then PV = $946.50 < $950, it is very close, and i should be little less than 8%. – Choose i = 7.8%, then PV = $949.93 ≈ $950, it is a good approximation! – So, an approximate yield to maturity is 7.8%.
- 12. Using Microsoft Excel You can use Microsoft Excel to compute a yield to maturity. – First, input cash flows: -$950, $50, $1050 Notice that the value today is negative! – Then, use IRR function under “Financial formula” where you select three numbers (cells) within the formula. – You should get 7.796% as a result.
- 13. Other Measures of Interest Rates The yield to maturity is the most accurate measure of interest rates. Two approximation measures of interest rates are • Current Yield • Yield on A Discount Basis
- 14. Current Yield • You cannot use the yield to maturity formula #4 on any debt instrument with more than one future cash flows. • Instead, you can use the current yield to approximate a yield to maturity. Formula #5 (Current Yield): i = C/P C: Annual coupon payment P: Price of console today • Current yield is an approximation, it is NOT a yield to maturity or the interest rate.
- 15. Console • Console: A perpetual coupon bond with no maturity. A console has a face value and a coupon rate like a regular coupon bond. Its issuer pays a fixed annual coupon payment each year and forever. • Console were issued and traded in U.K. many many years ago. No one issues console anymore. • The longest maturity of Treasury security is 30 years, while the longest maturity of corporate bonds is 100 years. Who wants to have console? Who can promise to pay forever?
- 16. Console and Current Yield • For a perpetual fixed payment like console, the current yield is exactly equal to the yield to maturity. So, you can use the current yield formula to compute a yield to maturity on console. • A console is priced at $1000 today and provides $100 annual coupon payments forever. The yield to maturity of the console is its current yield, that is, %10 1000$ 100$ i == • This is a special case. For any other fixed maturity bonds (even with 100 years maturity) or variable cash flows, you cannot use the current yield formula to compute the yield to maturity!
- 17. Console alike in the U.S. • Console is not traded in the U.S., so why is it useful to know? • There are some financial instrument which acts like console. Do you know any financial instrument which does not have maturity and promises to pay a fixed amount each year? • Preferred stocks issued by regulated monopoly such as Duke Power. – Preferred stocks like common stocks do not have maturity. – Preferred stocks promise to pay a fixed dividend as long as issuing corporations make profits. – Regulated monopoly like Duke Power is almost guaranteed to make profits each year by the government. – Utility companies like Duke Power is expected to continue its business almost forever.
- 18. Current Yield on Preferred Stock • Because some preferred stocks act like console, you can use the current yield formula to compute a yield to maturity on the preferred stocks. • Ex. A preferred stock costs $90 today and pays $6 dividend every year. The yield to maturity on the preferred stock is i = $6/$90 = 6.7%
- 19. Yield on A Discount Basis • Yield on a discount basis is used to approximate an interest rate on U.S. Treasury bills. • U.S. Treasury bills were first issued in 1929. How could traders know their interest rates? Did they have business calculators or Microsoft Excel? Of course not. They used the yield on a discount basis to find quickly approximate interest rates with paper and pen!
- 20. Formula of Yield on A Discount Basis maturitytodays 360 x F PF i − = F: Face value P: Purchase price Formula #6: • U.S. Treasury bills have maturities of 1 month (4 weeks), 3 months (13 weeks), 6 months (26 weeks), and one year (52 weeks). How many days in one month, 3 months, 6 months, or one year approximately? • U.S. Treasury bills has face values of $100, $1,000, or $10,000 (sold in increments of $100).
- 21. Example of Yield on A Discount Basis %95.4 364 360 x 000,10$ 500,9$000,10$ i = − = • Ex. A T-bill has one year maturity, $10,000 face value, and is sold at $9,500. • Note that a number of days to maturity is 364, which is 52 weeks (one year T-bill).
- 22. Yield to Maturity and Saving • As the yield to maturity formula takes into account all cash flows from a debt instrument, it implicitly assumes that a buyer of the instrument to hold it until its maturity and to receive all cash flows. • The yield to maturity formula is useful to evaluate a return an investor may receive from a security if he holds it until maturity. • However, if an investor sells the debt instrument before its maturity, the yield to maturity formula will not tell how much return he gets from the debt instrument.
- 23. Rate of Return • If an investor sells a security before its maturity, he can evaluate how much return he earned by holding the security for a given period by using “Rate of Return” formula. • The rate of return takes into account both purchase price and sales price of a security and any payments between.
- 24. Formula of Rate of Return C + Pt+1 - Pt R = —————— x 100 Pt R: Rate of Return Pt: Price of bond in year t Pt+1: Price of bond in year t+1 C: Total coupon payment between year t and year t+1 Formula #6:
- 25. Use of Rate of Return Formula • The rate of return formula is often used to evaluate a return from investment on a particular security by a saver. • In general, in year t, a saver purchased a security at Pt and in t+1 (any time after year t) he sells it at Pt+1. • For example, a saver purchased a security at $950 (Pt) last year (t), has received $80 annual coupon payment (C), and sells it at $980 (Pt+1) today (t+1).
- 26. Example of Rate of Return You bought a bond at $990 last year, received $50 annual coupon payment, and just sold at $1,000 today. R = (50+1,000-990)/990 x 100 = 6.06%
- 27. Components of Rate of Return The rate of return formula can be decomposed into two terms (sources of returns): C + Pt+1 - Pt C Pt+1 - Pt R = —————— = ——— + ———— Pt Pt Pt • The first term C/Pt is a current yield. It tells a part of return coming from coupon payments. • The second term Pt+1-Pt/Pt is a rate of capital gain. It tells the other part of return coming from a change in a price of security over time. If there is no change in price, this term
- 28. Example of Decomposition of Rate of Return You bought a bond at $990 last year, received $50 annual coupon payment, and just sold at $1,000 today. R = 50/990 + (1,000-990)/990 = 5.05% + 1.01% = 6.06% • Of 6.06% rate of return, 5.05% comes from coupon payment, while 1.01% comes from an increase in price of bond over one year period.
- 29. Capital Gain • Capital gain: You sell a security at higher price than the price at which you bought. • Rate of capital gain: (Pt+1 - Pt)/Pt • Ex. You bought a bond at $990 last year, received $50 coupon payment, and just sold at $1,000 today. RCG = (1,000-990)/990 = 1.01%
- 30. Capital Loss • You may not always have a capital gain. Often a price of financial instrument or asset falls, then you loose its value and end up to sell at well lower price than what you paid for (Capital loss). • Will you get a capital gain or capital loss from your textbook when you sell it? How about your car? • Often, prices of financial instruments such as stocks and bonds fall over a certain period of time and investors are suffered from capital losses.
- 31. Rate of Return on Investment • You can use the rate of return formula on any financial instrument and assets. – Example: Stock – Michelle purchased Apple stock at $396.75 on August 1, 2011 and sold it at $614.32 on July 19, 2012, and received no dividend over one year period. – Example: House – Mr. Jones purchased a house at $100,000 in 1991 and sold it at $158,000 this year. – Example: Textbook – you purchased a textbook at $120 at the beginning of the semester and plan to sell it at $40 at the end of the semester.
- 32. Internal Rate of Return • One problem of the rate of return formula is that it does not take into account of time span of the investment. It does not matter how long a saver holds it, the resulting rate of return is the same. • The internal rate of return takes into account cash flows on timeline, and is the most accurate measurement of rate of return on investment. • First, draw cash flows on timeline, including purchase price (as outflow) and sales price (as inflow). Then, apply the business calculator or Microsoft Excel to compute the internal rate of return.
- 33. Example of Internal Rate of Return Steven purchased a U.S. Treasury bond at $960 in 2013, received an annual coupon payment of $80 in 2014 and 2015, and sold it at $990 in 2015. The cash flows on timeline should look 2013 2014 2015 $80 $80+$990 $960 2 )i1( 990$80$ i1 80$ 960$ + + + + = Like the yield to maturity, you solve for i: i = 9.8%
- 34. Real and Nominal Interest Rates • Real interest rate: The interest rate that is adjusted for the inflation rate. • Formula #7 (Fisher Equation): i = r + π i: Nominal interest rate r: Real interest rate π: Inflation rate
- 35. Inflation and Value of Money • As the price level increases (inflation), a value of future cash flows (purchasing power of money in future) decreases. – A dollar in future can buy less than a dollar today. • An interest rate that a borrower promises to pay tells a lender how much a principle increases over time or how much dollar he will pay back. – 5% interest rate means your $100 will increase its value to $105 next year. • Due to inflation, future cash flows will not buy as many goods and services as they could without inflation, so a lender is actually getting less than 5% in terms of value. – An interest rate promised by a borrower does not promise whether you can buy more or less in future from that payment.
- 36. Inflation and Interest Rate • If you can get 20% interest rate on your saving, will it be a good deal? What will happen if an inflation rate is 50%? – Your $100 can buy 50 Big Macs (at $2 each) today. – If you loan your $100 at 20% annual interest rate, you will get $120 next year. – With 50% inflation rate, a price of Big Mac will increase to $3 (= $2 x 1.5). – Then, your $120 future cash flow can buy 40 Big Macs (= $120/$3) next year. – Are you really getting 20% return from your loan or loosing it? • What it really matters is not how much dollar ($120) you get in future, but how much (40 Big Macs) you can buy from that cash flow in future.
- 37. Real Interest Rates • Real interest rate: The interest rate that is adjusted for the inflation rate. • Formula #7 (Fisher Equation): i = r + π i: Nominal interest rate r: Real interest rate π: Inflation rate • A real interest rate on a security tells you how much more goods and services you can purchase in future out of payments from that security.
- 38. Example of Real Interest Rate You bought a 1-year CD at 5% (nominal) interest rate last year. During the last one year, the inflation rate was 3%. How much is a real interest rate on the CD? i = 5% and π = 3% ⇒ 5% = r + 3% ⇒ r = 2% Although you get 5% more cash from this CD than what you put in last year, its value decreased by 3%, so you can actually purchase only 2% more goods and services this year than last year.
- 39. Nominal vs. Real Interest Rate • Both lenders and borrowers must be concerned with the real interest rate rather than the nominal interest rate. – Even if a nominal interest rate is high, if an inflation rate is also high, the real interest rate may be low. • Lenders want high real interest rate, while borrowers want low real interest rate. – Higher the real interest rate, more the lenders are willing to lend their funds. – Lower the real interest rate, more the borrowers are willing to borrow funds.
- 40. Examples of Real and Nominal Interest Rates • You loaned $100 to your brother at 10% interest rate one year ago, and he returns $110 today. Last one year, the inflation rate was 10%. Did you gain from this investment (loan) to your brother? Did your brother loose from this loan from you? How much was a real interest rate? • If you expect an inflation rate will be 3%, are you willing to loan your funds at 1%, 3%, or 5% of nominal interest rate? How much will be a real interest rate?
- 41. Real Interest Rate, Nominal Interest Rate, and Inflation • Because we live in an economy with continuous inflation, a nominal interest rate is always greater than a real interest rate. • Inflation rate changes from year to year, so a difference between a nominal interest rate and a real interest rate also changes.
- 42. Figure 1: Real and Nominal Interest Rates, 1953-2011 A vertical distance between a blue line (nominal interest rate) and a brown line (real interest rate) is an inflation rate (as indicated in a red arrow) on this chart.
- 43. Inflation Rate and Actual Real Interest Rate • When a saver purchases a bond or loans a fund, he knows future cash flows and a nominal interest rate, but uncertain how much those future cash flows actually worth (how much goods and services you can buy with cash flows). • A saver must make a guess on inflation rate (i.e. 3%). If he wants 5% real interest rate, then he must ask 8% nominal interest rate on a bond. • However, no one can predict future inflation rate precisely, so he will get more or less actual real interest rate in reality. – If an actual inflation rate happens to be 3%, then he will get 5% real interest rate as he expected. – If an actual inflation rate happens to be 1%, then he will get 7% real interest rate – If an actual inflation rate happens to be 6%, then he will get only 2% real interest rate
- 44. Inflation-Indexed Bonds • Inflation-Indexed bonds guarantee a real interest rate by adjusting coupon and principal payments for changes in price level (inflation). – Indexed bonds have a fixed real interest rate, but nominal interest rate varies as price level in economy changes. – TIPS (Treasury Inflation Protection Securities) are inflation-indexed bonds issued by the U.S. government. – Ex. A borrower guarantees 5% real interest rate on $100 loan. A lender will receive $108 (8% nominal interest rate) if an actual inflation rate is 3%, or she will receive $112 (12% nominal interest rate) if an actual inflation rate is 7%.
- 45. Example of Bond Quotations on Barron’s • Dow Jones’ publishes quotes of U.S. Treasury securities (Treasury bills, notes, and bonds) and corporate bonds every day on the Wall Street Journal (online only) and weekly on Barron’s. – See “How to Interpret Bond Quotations of Barron’s” on Blackboard
- 46. Disclaimer Please do not copy, modify, or distribute this presentation without author’s consent. This presentation was created and owned by Dr. Ryoichi Sakano North Carolina A&T State University

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