Unit 9.5 Financial applications
When writing equations for balance sheets of loans, annuities, endowments and other investment schemes, it helps to be able to convert quickly between continuous growth and discrete-time growth such as monthly, quarterly or yearly. Recall from Example 6.6 that an interest rate (written as a real number, so you need to divide by 100 if it is quoted as a percent) of \(r\) corresponds to a growth factor of \(e^r\) over the course of a year. The increase in an initial amount \(M\) over a year is \(e^r M - M = (e^r - 1) M\text{.}\) This increase is proportional to \(M\) of course, and the constant of proportionality \(e^r - 1\) is called the annual yield. Denoting this by \(y\text{,}\)we can state either in terms of the other:
\begin{equation}
y = e^r - 1 \qquad ; \qquad r = \ln (1+y) \, .\tag{9.1}
\end{equation}
For those committed to stating things in terms of percentages, let \(Y = 100 y\) denote the annual percentage yield (APY) and \(R = 100 r\) denote the percentage interest rate, this becomes
\begin{equation*}
Y = 100 (e^{R/100} - 1) \qquad ; \qquad R = 100 \ln (1 + Y/100) \, .
\end{equation*}
Checkpoint 144.
Consider a mortgage loan (loan for a house) or car loan. Typically payments on these are made monthly, which we will take to be every \(1/12\) of a year. In this case the factor by which your debt grows each month is \(e^{r/12}\text{,}\)where \(r\) is the (annual) interest rate. That’s only if you don’t pay off the loan. Actually, these loans are typically configured so you pay a fixed amount every month until the loan is paid off in an integer number of months (usually, in fact, an integer number of years). To agree on some notation, let \(r\) be the annual interest rate, \(P\) be the principal, that is the initial debt, and let \(M\) be the monthly payment.
In order to deal successfully with used car sales people, it’s helpful to understand how these determine your balance over the successive months. The key relation is to understand what happens from one month to the next. We will discuss this, then leave the rest of the balance sheet computation for in-class discussion and homework. To determine your debt after a month, just take your initial debt \(P\text{,}\) multiply by the factor \(e^{r/12}\) for the growth of the debt over the first month, and subtract the amount you just paid off, namely \(M\text{.}\) We can write this as \(P_1 = e^{r/12} P_0 - M\text{.}\) It holds equally from any month to the next: \(P_{n+1} = e^{r/12} P_n - M\text{,}\)where \(P_n\) is your debt after \(n\) months.
How about your retirement account? Say you put \(M\) dollars every month into an interest bearing account. How much do you have after \(n\) months? It’s the same formula, with an opposite sign because you’re adding to your balance, not subtracting.
Checkpoint 145.
Write a formula for your retirement balance after \(n+1\) months, \(P_{n+1}\text{,}\) in terms of your balance \(P_n\) after \(n\) months.
A guaranteed rate annuity works similarly. By the time you retire you have put \(P\) dollars into an account. (How did this happen? See Checkpoint 145) You hand this over to a company who guarantees you a certain APY every year, call it \(Y\text{.}\) Each year you also withdraw a fixed amount to live on, call it \(M\text{.}\)
Checkpoint 146.
The University of Pennsylvania’s endowment works something like this. The balance increases by roughly 5% each year due to the growth of the investments and new donations. Meanwhile, during the year, the university spends roughly 3.4% of the present endowment. Unlike the formula for growth of a retirement fund or reduction of debt, this one is only approximate because the actual return varies. Nevertheless, it is useful for forecasting. Let \(E_n\) denote the size of the endowment after \(n\) years.
