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**Unformatted text preview: **Computing Effective Annual Rates 1 1. Suppose we are told that our investment in a bank will grow at the rate of 8% per annum, compounded quarterly. Say we invest some amount P . What this means is that in 1 year, we will have P 1 + . 08 4 4 = F. F is the future value of our investment. Here, 0.08 is called the quoted rate and . 08 / 4 = 0 . 02 is called the periodic rate . In this example, our investment earns 2% for 4 quarters. More generally, say r is the stated annual interest rate (SAIR), and r is compounded n times a year. Then the following describes how our initial investment of P grows up to be F : P 1 + r n n = F. The periodic rate here is r/n . That’s the rate we earn per period. In general, if we invest for t years, we have P 1 + r n tn = F. We could also use this formula for discounting: P = F 1 + r n- tn . This last formula describes what it is worth for us to have F dollars in t years, compounded n times a year, with an SAIR of r . 2. What could n be? Besides semiannual compounding ( n = 2) and quarterly com- pounding ( n = 4), we could have monthly compounding ( n = 12), daily compound- ing, hourly compounding, and on and on. As n gets larger, we approach something 1 Notes for Finance 100 (sections 301 and 302) prepared by Jessica A. Wachter. known as continuous compounding. In fact 1 + r n n → e r as n gets large. Under continuous compounding, Pe rt = F and, P = Fe- rt Thus e- rt is a discount factor , just like 1 / (1 + r ) in the annual compounding case. 3. To see how different rates of compounding affect the present discounted value, consider the following question. What is $100 one year from now worth today? Annual Compounding: $100 1 . 06 = $94 . 3 Monthly Compounding: $100 (1+ . 06 / 12) 12 = $94 . 19 Continuous Compounding: $100 e- . 06 = $94 . 18 This exercise illustrates an interesting point. As the compounding frequency in- creases, the present value falls. Why is the present value of $100 less when com- pounding is continuous than when compounding is annual or monthly? To answer this question, think about the meaning of present value. The present value is the amount you must put away today to have $100 in one year. Under continuous compounding, you can put away fewer dollars today because the money you invest will grow at a faster rate. 4. We would like to have some way of comparing these different rates of compounding. This is where the effective annual rate, the EAR, comes in. The EAR is defined to be the rate that makes the following statement true: 1 + EAR = 1 + r n n . In words, this equation says that the EAR is the interest rate that, when compounded annually, produces the same value as r , compounded n times a year....

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