Intuitive understanging of re-investment.

Question

There was an interesting problem that I would like to have some input from people who knows a bit of finance.

The following is the situation.

Smith loans $\$10,000$ for $i=5\%$ for $10$ years. There are two schemes in reinvestment. A: Smith receives a level annual payment of

$$\frac{10,000}{a_\bar{10\rceil}}=1295.05$$ each year, which is reinvested in a bank annually with $i_2=3\%$

or B: Smith receives $5\%$ of the principal each year, and receives the principal at the end of the $10$th year, reinvesting $\$500$ yearly in the same bank.

What's really bugging me is the fact that if one wants to make investments, the larger the principal is the more money you get at the end, assuming that the person is investing the money for the same amount of time and interest rate.

Algebraically, I see that scheme A will end up having a less rate of return. However, according to my argument this is counter intuitive because Smith is receiving more money under scheme A, so why would he lose money?

Answer 1

Let's do the calculation: $$a_{\overline{10}\rceil.05} = \frac{1-(1.05)^{-10}}{.05} = 7.72173$$ so the level payments returned to Smith are $1295.05$ as claimed. Now, if Smith reinvests these at $j = 0.03$ as they are received, then the total accumulated value at the end of the 10-year period is $$1295.05 s_{\overline{10}\rceil j} = 1295.05 \frac{(1.03)^{10}-1}{.03} = 14846.25.$$

In the second scenario, Smith receives interest-only payments of $500$ at the end of each year, plus a lump sum of $10000$ at the end, so this has accumulated value $$500 s_{\overline{10}\rceil j} + 10000 = 15731.94.$$

To understand why the second scenario has a larger rate of return, it helps to see what happens if the reinvestment interest rate varies below or above the loan interest rate. First consider the case where the money is not reinvested at all: Smith merely gets the payments on the loan. Then the total return in the first case is $12950.46$. But in the second case, he gets $15000$, substantially higher, because the borrower, paying interest only at 5% per year, is not paying off any of the principal. Now consider the case where the money is reinvested at a rate that is higher than the loan rate: in this case, we can intuitively understand that the larger the payments Smith gets back from the lender, the better the reinvestment return, thus it makes more sense that the level payments, with a higher reinvestment rate, will accrue more money for Smith.

So, in particular, it is an instructive exercise to show that algebraically, $$\frac{s_{\overline{n}\rceil j}}{a_{\overline{n}\rceil i}} = i s_{\overline{n}\rceil j} + 1$$ if and only if $i = j$.