4.10 Average Rates
Often, interest rates change over the life of an investment. In this case, we often try to find the average rate. To illustrate, here is an example:
Example 4.10.1
7 years ago, Warren’s grandmother gave each of her grandchildren different amounts of money. Warren invested $4,000 of his money with CJC investments. For the first year, CJC put his money in a GIC which paid an effective rate of 5.9%. For the next 2 years, CJC moved the money into Canada Savings Bonds, which paid out a semi-annual rate of 7.8%. To capitalize on the markets, CJC moved the money into a mutual fund, where it earned an average monthly interest rate of 4.8% for three years. For the final year, the money was put it into stocks which paid out 5.6% compounded quarterly. How much money does Warren have today?
To solve this, we will take the Present value, and find its value at each point:
| YEAR | P/Y |
C/Y | N | I/Y | PV | PMT | FV |
| 1 | 1 | 1 | 5.9 | 4,000 | 0 | CPT: -4236 | |
| 2-3 | 2 | 2 ×2 = 4 | 7.8 | 4236 | 0 | CPT: -4936.488637… | |
| 4-6 | 12 | 12× 3=36 | 4.8 | 4936.488637… | 0 | CPT: -5699.43497… | |
| 7 | 4 | 4 | 5.6 | -5699.43497… | 0 | CPT: 6025.36864… |
We can also look at this algebraically:
[latex]FV = $4,000(1.059)^1(1.039)^4(1.004)^{36}(1.014)^4= $6025.36864...[/latex]
So Warren has $6,025.37.
Now we can work out what Warren’s average effective rate is. We take the PV and FV, and use the calculator to find out what effective rate will turn $4,000 to $6025.37 in 7 years. Make sure to use your memory buttons to store all of the decimal points.
| P/Y | C/Y | N | I/Y | PV | PMT | FV | |
|---|---|---|---|---|---|---|---|
| 1 | 7 | CPT | 4,000 | 0 | -6025.36864.. |
We get an effective rate of j1 = 6.027290102%. This is Warren’s average rate.
Once we have the effective average rate, we can use it to solve other problems:
Example 4.10.2
| P/Y | C/Y | N | I/Y | PV | PMT | FV | |
|---|---|---|---|---|---|---|---|
| 1 | 7 | 6.027290102 | CPT | 0 | -8,150 |
Which tells us that Amy invested $5,410.46, 7 years ago.
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