Question

In lecture, Professor Gruber explained discrete compounding interest. Interest can also be compounded continuously. Here we explain the difference. Professor Gruber calculated future value as FV = P(1+r)", where P is the principal, r is the interest rate, and t is the term of the contract (often in years). This formula can be generalized to FV = P(1+r/m)mt, where m is the number of compounding periods per year (in lecture, this was 1). That is, after every compounding period, more interest accrues on both the principle and the previous accrued interest. This is discrete compounding. Suppose that m becomes large. For example, the interest rate could be 10% per year, but compounded each minute. Future value rises as m increases, but it rises at a diminishing rate. It turns out that as m goes to infinity, future value is described by an exponential function, FV = Pert, where e is the base of the natural logarithm. This is continuous compounding. Problem PS9.1.1a 1/4 points (graded) You are planning to invest in fine wine. Each case costs $100 (at time 0), and you know from experience that the (future) value of a case of wine held for t years is 100t fort > 1. (Suppose that the value is 100 for () <t<1. Why do you think we make this assumption?) Wine is thus increasing in value over time. One case of wine is available to purchase, and the annual interest rate you can earn by keeping money in the bank is 10 percent. Should you buy the case of wine?

Answer 1

Firstly, we say wine is valued at 100 if t is between 0 and 1, because if we apply the formula blindly we would end up saying that the wine is valued at less than 100 if it is kept for less than one year (since square root of less than 1 will be less than 1).

Now, let's compare the values of a bank deposit and wine.

For instance, if a bank deposit is kept for 2 years @10%, $100 will be come $100*(1.1^2) = $121

Whereas wine will be worth 100*(2^0.5) = $141

So win is worth more if kept for 2 years, in comparison to a bank deposit.

Now let's see what happens if we keep them for, say, 100 years.

Bank deposit will be worth $100*(1.1^100) = $1,378,061

Wine, however, will be worth much less, i.e., 100*(100^0.5) = $1,000

So, the answer is not straightforward. It depends on the duration of each investment.