Chapter 2: The basic theory of interest Investment Science D.G. Luenberger

Interest Interest is the time value of money. Almost surely, a dollar tomorrow is less than a dollar today. Interest rate is the exchange rate between the money in the future and the money today. Interest can be simple interest or compound interest. Most financial institutions employ some form of compounding.

Basic formulas FV = PV * (1 + r)n PV = FV / (1 + r)n PV = PV1 + …+ PVn FV = FV1 + … + FVn

(Constant) ideal bank An ideal bank applies the rate of interest to both deposits and loans, and it has no service charges or transactions fees. Its interest rate applies equally to any size of principal. If an ideal bank has an interest rate that is independent of the length of time for which it applies (r1 month = r12 month ), it is said to be a constant ideal bank. In the rest of the chapter, we assume that interest rates are constant.

Constant ideal banks and equivalent streams Suppose that a 10% constant ideal banks exists. We can deposit $100 today and get $110 in a year; or we can payback $110 in a year for a $100 loan today. Thus, a 10% bank can change the stream (1, 0) into the stream (0, 1.1), and change the stream (0, 1.1) into (1, 0). Two cash flows streams of the same length, X and Y, are equivalent if and only if the present values of the two streams are equal. See proof on page 22. When the horizon is fixed, the present value is the only number needed to make investment decisions.

An investment decision Suppose that a 10% ideal bank exists (or the cost of capital is 10%). Suppose the initial cost of an investment project is $1. The expected payoff of the investment in a year is $1.21. Would you accept the project? An equivalent stream of the expect payoff, (0, 1.21), is (1.1, 0). This equivalent stream is higher than the initial cost, (1, 0). Thus, benefits out-weight costs. We accept the project.

An alternative cash flow presentation Sometimes, it is more convenient to include the entire cash flow streams, i.e., costs and payoffs. If this approach is chosen, we need to give the cost components a negative sign. For the previous example, the cash flow stream is (-1, 1.21).

(-1, 1.21) The present value of the entire cash flows, both negative and positive elements, is frequently called net present value (NPV). The NPV of the (-1, 1.21) @10% is 0.1, which is larger than zero. Thus, we accept the project. The internal rate of return (IRR) is 21%, that is, the value of interest rate that renders the NPV equal to zero. Based on the IRR criterion, we also accept the project because the IRR is greater than the required 10% return.