# 1.4 Interest-Rate calculations and returns

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1 .4 Interest-Rate calculations and returns Effective Annual Rate (EAR) The Effective Annual Rate (EAR) is the actual rate paid (or received) after accounting for compounding that occurs during the year Compounding (calculating interest income) may occur more than once during the year For example, if you invest \$00 for year at 0% annual rate compounded semi-annually, your investment will grow to: \$ % -----> \$ % -----> \$0.5 0 / 0.0 FV \$00 + \$00 (.05) \$00 (.05) \$0.5 This is equivalent to investing \$00 for year at 0.5% annual rate compounded annually (interest is calculate once a year): \$ % > \$0.5 0

2 Effective Annual Rate (EAR) The actual interest you earn during the year (0.5%) includes: simple interest (0%) and interest on interest (5% 5%0.5%) If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison. For example: choose the best of the following two investments: A savings account that pays 4% annual rate compounded monthly A certificate that pays 4.% annual rate compounded quarterly Saving account : Certificate : FV FV 0.4 \$ Annual Percentage Rate (APR) APR is the annual rate that is quoted by law By definition: APR period rate number of compounding periods per year Consequently, to get the period rate we rearrange the APR equation: Period rate APR / number of comp. periods per year For example: find the APR if the monthly rate is %: find the APR if the semiannual rate is 4%: What is the monthly rate if APR 8% with monthly compounding? Can you divide the above APR by to get the semiannual rate? NO!!! You need an APR based on semiannual compounding to find the semiannual rate.

3 EAR - Formula EAR where: APR + m m m is the number of compounding periods per year APR is the quoted rate per year, compounded m times a year Effective Annual Rate: Example Find the Effective Annual Rate of an 8% APR loan that is compounded monthly What we have is a loan with a monthly interest rate rate of ½ percent This is equivalent to a loan with an annual interest rate of 9.56 percent: r EAR + m m

4 EAR on a financial Calculator Hewlett Packard 0B keys: display: description: [gold] [P/YR].00 Sets P/YR. 8 [gold] [NOM%] 8.00 Sets 8 APR. [gold] [EFF%] 9.56 Texas Instruments BAII Plus keys: description: [nd] [ICONV] Opens interest rate conversion menu [ ] [C/Y] Sets payments per year [ ][NOM] 8 [ENTER] Sets 8 APR. [ ] [EFF] [CPT] 9.56 IMPORTANT: When finished, don t forget to set your calculator back to payment per year Effective Annual Rate: Example Recall the choice between the following two investments: A savings account that pays 4% annual rate compounded monthly A certificate that pays 4.% annual rate compounded quarterly The best alternative is the one that pays the highest EAR: Savings account (APR4% and m): EAR The best alternative is the one that pays the highest EAR: Certificate (APR4.% and m4): EAR

5 Things to Remember You always need to make sure that the interest rate and the time period match. If you are looking at annual periods, you need an effective annual rate. If you are looking at monthly periods, you need a monthly rate. If you have an APR based on monthly compounding: you have to use monthly periods for monthly payments, or adjust the interest rate appropriately if you have payments other than monthly Computing APRs from EARs If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: APR [ ] ( + EAR) m m

6 APR - Example Suppose you want to earn an effective rate of % You are looking at an account that compounds on a monthly basis What APR must it pay? APR Computing Loan Payments with APRs You have \$30,000 in student loans that call for monthly payments over 0 years. \$5,000 is financed at 9 percent APR \$8,000 is financed at 8 percent APR and \$7,000 at 5 percent APR What is the monthly payment on each loan? For the \$5,000 loan: Monthly rate is: 9%/ 0.75% Monthly payment is given by solving: This gives: [ ] 5,000 PMT (.0075)

7 Computing Payments with a Calculator We find the monthly payment on each loan: N I/Y PV 5,000 8,000 7,000 PMT FV Future Values and Payments with Monthly Compounding You have just landed a job and are going to start saving for a down-payment on a house. You want to save 0 percent of the purchase price and then borrow the rest from a bank. You have an investment that pays 0 percent APR. Houses that you like and can afford currently cost \$00,000. Real estate has been appreciating in price at 5 percent per year and you expect this trend to continue. How much should you save every month in order to have a down payment saved five years from today?

8 Future Values and Payments with Monthly Compounding First we estimate that in 5 years, a house that costs \$00,000 today will cost \$7,68.6 Next we estimate the monthly payment required to save up that much in 60 months. N 5 N 60 I/Y 5 I/Y 0 PV 00,000 PV 0 PMT 0 PMT FV 7,68.6 FV \$5, \$7,68.6 Mortgages You would like to buy a house. You have negotiated a 30- year, \$300,000 mortgage at an APR of 3.6% with monthly payments. What is your monthly payment? We have: PV 0 \$300,000, monthly rate 3.6%/ 0.3%, and T months. We solve for PMT in the following equation: This gives: PMT [ ] 300,000 PMT (.003)

9 Mortgages How much of the first three mortgage payment, goes toward principal and interest? In general, to calculate the interest portion of each monthly payment, use: (monthly rate) (balance of Principal at the Beginning of Month) The principal portion of each monthly payment is given by: PMT - Interest Payment For the first three mortgage payments: Month 3 A Principal at the beginning of the month 300, , , B Interest charged during the month C Monthly payment,363.94,363.94, D Principal reduction C-B E Principal at the end of the month A-D 99, , , Mortgages After that you have paid two-thirds of your monthly payments, what is the amount still remaining to be paid on the mortgage? Two-thirds of your monthly payments will be paid right after the 40 th payment. The remaining value of the mortgage at that time is given by the present value of the remaining 0 monthly payments. We have: monthly rate 3.6%/ 0.3%, PMT, (rounded), and T 0 months. We use the present value of annuity formula to get: PV 40

10 Continuous Compounding Sometimes investments or loans are figured based on continuous compounding Under continuous compounding interest is calculated (and accumulated) on a constant basis. This means that the compounding period is infinitely small (interest is compounded infinite times): r m r Therefore: lim EAR lim[ ] e m m e is a transcendental number approximately equal to.78. Example: What is the effective annual rate of 7% compounded continuously? EAR The TVM relationship with continuous compounding: m FV t+t PV t e (r T) Factors Influencing Rates Supply of funds by savers Households Demand for funds to invest in real assets Businesses Government s Net Supply and/or Demand Federal Reserve Actions

11 Equilibrium Level of Interest Rates Interest Rates Supply Gov t budgetary deficit will shift the demand curve to the right r r 0 Demand Q 0 Q Q Funds Equilibrium Level of Interest Rates Interest Rates Supply Fed s expansionary monetary policy will shift the supply curve to the right r 0 r Demand Q 0 Q Q Funds

12 Inflation, Real and Nominal Interest Rates You found your dream apartment on Fifth Avenue. The owner gives you the option to buy the apt. now for its market value (\$,000,000), or buy it one year from now at the prevailing market value at that time You have exactly \$ million and you can invest it at an annual rate of R 5% p.a. (nominal rate) You expect Fifth Avenue real-estate prices to grow at i 0% p.a. (inflation rate) Based on this information, should you buy the apt. now or next year? Inflation, Real and Nominal Interest Rates If you wait one year, and invest your money at R 5%, you will have: The cost of the apt. in one year is expected to be: Conclusion: the real value of your money has depreciated (you should buy now) Considering inflation, what is the real interest rate (r) that you earn during the year?

13 The Fisher Relation The Fisher Relation: (+ R) (+ r)(+i) Your real rate is: r [(+ R)/(+i)] - Conclusion: in real terms the value of your money will depreciate over the year by 4.55% Real vs. Nominal Cash Flows Real CF (C t,real ) vs. Nominal CF (C t,nom ): C t,nom C t,real (+i) t In our example: t, i 0.0, C,n \$,00,000 (actual cost), and: C t,real C t,nom / (+i) t [\$,00,000 /.0 ] \$,000,000 What rate should we use? C t,nom should be discounted with R: PV C t,real should be discounted with r: PV

14 Rates of Return: Single Period HPR P P P D HPR Holding Period Return P 0 Beginning price P Ending price D Dividend during the period The HPR has two components: HPR P P P0 0 + D P 0 capital gain yield dividend yield Single Period Returns - Example Suppose a stock had an initial price of \$4 per share, paid a dividend of \$0.84 per share during the year, and had an ending price of \$46.. Calculate: a. HPR HPR b. Dividend yield DY c. Capital gains yield CGY

15 Single Period Returns - Example Dividends \$0.84 Ending Market Value \$46.0 Total inflows \$47.04 Time: 0 Outflows \$4.00 Historic Rates of Return: Multi Period - Example The following are ABC stock single-period returns for the 0X- 0X4 period: Year (t) Return (R t ) 0X -0.8% 0X X X Calculate: a. holding period return (HPR) b. geometric average return (GAR) c. arithmetic average return (R)

16 Historic Rates of Return: Multi Period - Example a. holding period return HPR ( + R 0X )( + R 0X )( + R 0X3 )(+ R 0X4 ) - b. geometric average return GAR [( + R 0X )( + R 0X )( + R 0X3 )(+ R 0X4 )] /4 - [ + HPR] /4 - c. arithmetic average return R ( R0 X + R0 X + R0 X 3 + R0 X 4 ) 4 Historical Return Statistics The history of capital market returns can be summarized by describing the average return R R R ( ) + L+ T T the standard deviation of those returns SD VAR ( R R) + ( R R) T + L( R T R) the frequency distribution (histogram)of the returns.

17 Other Return Statistics - Example Historic Return Variance: ( R0X R) + ( R0X R) + ( R0X3 R) + ( R0X4 R) ˆ σ 4- ( ) + ( ) + ( ) + ( ) Historic Standard Deviation: σˆ Historical Returns, Average Standard Series Annual Return Deviation Distribution Large Company Stocks.% 0.5% Small Company Stocks Long-Term Corporate Bonds Long-Term Government Bonds U.S. Treasury Bills Inflation % 0% + 90% Source: Stocks, Bonds, Bills, and Inflation 003 Yearbook, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.

18 The Risk-Return Tradeoff 8% 6% Small-Company Stocks Annual Return Average 4% % Large-Company Stocks 0% 8% 6% T-Bonds 4% T-Bills % 0% 5% 0% 5% 0% 5% 30% 35% Annual Return Standard Deviation Rates of Return Common Stocks Long T-Bonds T-Bills Source: Stocks, Bonds, Bills, and Inflation 000 Yearbook, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.

19 Risk Premiums Rate of return on T-bills is essentially risk-free. Investing in stocks is risky, but there are compensations. The difference between the return on T-bills and stocks is the risk premium for investing in stocks. An old saying on Wall Street is You can either sleep well or eat well. Risk Statistics The measures of risk that we discuss are variance and standard deviation. The standard deviation is the standard statistical measure of the dispersion of a sample around its mean, and it will be the measure we use most. Its interpretation is facilitated by a discussion of the normal distribution.

20 Normal Distribution A large enough sample drawn from a normal distribution looks like a bell-shaped curve. Probability 3s 49.3% s 8.8% s 8.3% 0.% 68.6% + s 3.7% + s 53.% + 3s 73.7% Return on large company common stocks 95.44% 99.74% Normal Distribution s.d. s.d. mean Symmetric distribution

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