No Free Lunch: Economics for a Fallen World: Third Edition, Revised

Chapter Sixteen: Valuing the Future - Concepts in Capital & Finance 403 COMPOUND INTEREST In our examples so far, we have used a simple interest rate, which applies the interest to the principle owed once per year. So if you borrow $100 at 5% interest for one year, then you owe $105 at the end of the year: the return of the $100 principle and also a $5 interest payment. However, some loans have compound interest , where interest is applied more frequently. In these loans the interest payment is paid earlier such that later payments during that year pay interest on the interest earned. Equation 2 from earlier in this chapter can be modified to capture additional payments, such that if interest is paid N times per year, the formula is… [Equation 6] FV = PV(1 + i/n) nt If we deposit $100 in our local bank paying interest semi-annually at 5%, then at the end of the first six months we would have $100 X 1.025, or $102.50. After the second six-month period, we would apply the interest rate again, but now to the $102.50 rather than the initial $100, or $102.50 X 1.025, totaling $105.06. In this case the “interest on the interest” only results in six additional cents, but for larger amounts of money it may be substantial. What if you compounded the interest more than twice per year? You can apply equation 6 to more frequent compounding; if interest is compounded monthly then the saver would have $105.12 at the end of the year. We define the effective interest rate as the interest rate that would have to be applied as simple interest to achieve the same interest earned by more frequent compounding. In the case of the 5% interest compounded monthly, the effective interest rate would be 5.12%, since a simple interest rate of 5.12% applied to $100 would result in the same interest earned after one year. Can interest be compounded more frequently? Indeed it can; the limit is continuously compounding, which uses the exponential function as in equation 7, which would lead to only fractionally higher ($105.1271, or $105.13) interest. [Equation 7] FV = PVe it RISK VS. RETURN Related to the risk premium is a fundamental fact, easily illustrated by our present value calculation: higher investment returns can only be obtained (prospectively) by taking higher levels of risk. Let’s assume we have two investments, each paying identical cash flows; yet, investment B is risker than investment A. Given that B is riskier than A, it will have a higher risk premium and associated Effective interest rate: The rate that would have to be applied as simple interest to achieve the same interest earned by more frequent compounding. Higher investment returns can only be obtained (prospectively) by taking higher levels of risk. Simple interest: Occurs when the interest rate is applied only to the principle of a loan. Compound interest: Occurs when the interest rate is applied more frequently than once per year such that interest payments earlier in the year also accrue interest.