Investment Analysis HW Week 2 Solutions
September 2, 2020
1. While talking long-distance to your parents on the phone, you hear how their local bank is offering loans at 2% APR with monthly compounding. The Astor Place Bank located down the street from you, has a CD paying an interest rate of 3% APR with semiannual compounding. Maybe you can make a riskless profit by borrowing and lending. Suppose you borrowed $10,000 from your parents' local bank, and invested it in the Astor Place Bank. How much money will you have after one year?
First ask, what are the effective rates of each bank? Recall that to get the effective rate from the APR you divide the APR by the number of compounding periods in a year. For your parents' local bank, the compounding is monthly, and so there are 12 periods in a year. The effective rate is 2/12 = 0.1666% percent per month. Recall also that when you divide the APR by the number of compounding periods in a year, you get the effective rate per compounding period, so here the effective monthly rate. Proceeding similarly with the Astor Place Bank, the effective semiannual rate is 3/2 = 1.5% per six months effective. Suppose you borrowed the 10,000 and invested it in the Astor Place Bank. At the end of one year, that 10,000 would grow to 10000(1 + r)7 = 10000(1 + .015)2 = 10000(1.030225) = 10302.25. Here I used the effective semiannual rate, and I am moving the cashflow one year, or two (semiannual) periods, so T = 2.
But what do you owe on the loan after one year? You owe 10000(1+r)T = 10000(1.020184) = 10201.84. Here I use the effective monthly rate, and I am moving the cashflow twelve monthly periods (so T = 12).
Therefore, after I get the money and pay off the loan, I will have $100.41.
2. You are trying to decide how much to invest in your 401(K). You know that the IRS allows you to invest a maximum of 18,000 every year until retirement. There are 25 years until you retire. Suppose you invest the maximum amount each year. You are considering two alternatives.
(a) You can invest the 18,000 at one go at the end of the year, starting one year from now, making 25 payments in all.
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(b) You can invest the 18,000 broken up by month (so 18, 000/12 = 1500 each month - the IRS doesn't care about time value for this calculation), with the first payment made one month from now, making 25 * 12 = 300 payments in all.
If the interest rate you earn on your investments in the 401(K) is 8% EAR, how much will you have at the end of 25 years under each alternative? Why is the value higher under one alternative than the other?
Start with the first. Here is the stream of investments End of year Amount paid in 0 0 1 18,000 2 18,000