2 3 4 400 400 400 400 11 11 Perpetuity Perpetuity is a

2 3 4 400 400 400 400 11 11 perpetuity perpetuity is a

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2 3 4 400 400 400 400 400 1.1 1.1 1.1 1.1 1,667.95 PV
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58 4.4 Perpetuity Perpetuity is a special case of an annuity in which the number of equal cash flows is infinite . The formula for the present value of a perpetuity is: Present Value of a Perpetuity C r Example 4.10 In the early 1900's the Canadian Government issued $100 par value 2% Consol bonds. The holder of these bonds is entitled to receive a coupon (or interest) payment of $2 per year forever. If the current appropriate discount rate is 5% p.a. and the next coupon is due one year from now, how much is one of the Consols worth? 2 0.05 40 C PV r 4.5 Comparing Rates Suppose a bank offers you two deals: (1) pays you 10% interest per year or (2) pays you 5% interest compounded every six months. Which deal would you prefer? If you invest $1, then after a year, Option (1) will give you: $1 1.1 $1.1 Option (2) will give you: 2 $1 1.05 $1.1025 Obviously, option 2 is better as you can enjoy the interest on interest . As the example illustrates, 10% compounded semiannually is actually equivalent to 10.25% per year.
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59 4.5.1 Effective Annual Rate (EAR) In the example, the 10% is called the quoted interest rate . The 10.25%, which is actually the rate that you can earn, is called the effective annual rate (EAR). If you want to compare two alternative investments with different compounding periods, you need to compute the EAR and use that for comparison. To get the effective annual rate, Quoted Rate 1 1 m EAR m Where m is the number of times the interest is compounded during the year. Example 4.11 Suppose a bank offers a nominal interest rate of 5% on your time deposit. Compare the different EARs with various times the interest is compounded each year. Compounding Formula Effective Annual Rate Annually 1 0.05 1 1 1 r 5.0000% Semiannually 2 0.05 1 1 2 r 5.0625% Quarterly 4 0.05 1 1 4 r 5.0945% Monthly 12 0.05 1 1 12 r 5.1162% Weekly 52 0.05 1 1 52 r 5.1246% Daily 365 0.05 1 1 365 r 5.1267% Hourly 8760 0.05 1 1 8760 r 5.1271% Continuously 0.05 1 r e 5.1271% You will always prefer more compounding periods to less.
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60 Example 4.12 You are looking at two savings accounts. HSBC pays you 5.25%, with daily compounding. BOC pays 5.3% with semiannual compounding. Which account should you use? HSBC: 365 0.0525 1 1 365 5.3899% EAR BOC: 2 0.053 1 1 2 5.3702% EAR 4.5.2 Annual Percentage Rate (APR) Another rate we often calculate is the annual percentage rate (APR). APR is the interest rate charged per period multiplied by the number of periods per year. Since the law requires that lenders disclose an APR on all loans, this rate must be displayed on a loan document in an unambiguous way. Example 4.13 What is the APR if (1) the monthly rate is 0.5%; (2) the semiannual rate is 0.5%? For (1): 0.5% 12 6% APR For (2): 0.5% 2 1% APR Remember, APR is only an annual rate that is quoted by law. In order to figure out the actual rate, you need to compute the EAR.
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61 The relationship between EAR and APR: 1 1 m APR EAR m If you have an effective rate, you can compute the APR.
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