Present Value
Prof. P.V. Viswanath, 1997

Key Concept: A dollar today is not worth the same as a dollar later. Why?

Discount Rate: Suppose A values one dollar payable one year from now at 90 cents today. Then we say that his discount rate is given by (1/0.9)-1 or 11.11%

Interest Rate: An equilibrium discount rate set in the marketplace is called an interest rate. Let's see what this means. Clearly, a discount rate can vary from one person to another. Suppose A values a future dollar more than B, then it will be worthwhile for A to buy it from B in return some number of current dollars. We have seen that A values a dollar to be paid one year in the future at $0.90 today. Suppose B values it at $0.80. Then, assuming that B has access to future dollars (say, for example, B has been promised some money by his aunt next year), he will sell those future dollars to A and A will buy them. The price at which the transaction will take place will be greater than $0.80 (otherwise, B won't sell them) and less than $0.90 (otherwise, A won't buy them). Other individuals will also participate in this market, on one side or the other depending on their valuations of next year's dollar.

Suppose all the people who want to trade finish trading. Now, if a new person enters the market, the best price at which s/he can buy will be the maximum of the valuations of all the sellers in the market. The best price at which s/he can sell will be the minimum of the valuations of all the buyers in the market. If there are enough people trading, both these prices will be the same. This is called the market price. In a market where both A and B are trading with their valuations of 90 and 80 cents respectively, the market price will be between 80 and 90 cents. Suppose the market price for next year's dollar is 86 cents. Then, the market interest rate is computed at 1/0.86 = 16.28%. Just as we use market prices to value other goods and assets (e.g. your car), rather than our own personal valuations, similarly, in order to value future dollars, we use this market interest rate because by trading at this market interest rate, we can obtain better terms than our own valuation, and certainly no worse. (And so can anybody else.)

Furthermore, just as we can convert future dollars into present dollars (which we call discounting), we can also convert present dollars into future dollars (which we call compounding). Following the definition of a discount rate, we can derive the following:

Present Value of a simple cash flow = CFt/(1+r)t

where CFt is the cash flow at the end of time period t, and r is the discount rate.

Of course, from our previous discussion, it is clear that the appropriate discount rate to use is the relevant market interest rate.

Similarly, Future Value at tof Simple Cash flow = CF0(1+r)t.

The Effective Interest Rate is the true interest rate, taking into account, the compounding effects of frequent interest payments. It is the rate at which money grows per year. Thus, if the interest rate is 1% per month, then at the end of 1 month, $1 would grow to $1(1.01). At the end of two months, it would grow to $1(1.01)2. At the end of 12 months, it would grow to $1(1.01)12 or $1.1268. Hence, the money is growing at the rate of 12.68% per year. The effective interest rate is 12.68% per year, or per annum.

If the interest rate is 1% per month, then this rate corresponds to a 12.68% effective annual interest rate, as we have seen. However, this is frequently expressed, by convention as 1 x 12 or 12% APR. In other words, a 12% APR corresponds to a 12.68% effective annual rate if interest is compounded monthly. If interest were compounded more frequently with the same APR, the effective annual interest rate would be higher. Other formulae:

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