Having recently graduated, Jack and Jill just started working for the same company. The company has offered each of them a retirement savings plan (e.g., an Individual Retirement Annuity, sometimes called an IRA) in which for each year that they save $5000 in the retirement plan, the company will also contribute $5000 to their plan. Contributions to the plan, if any, are made at the end of each year, and the money in the plan is invested in a broad index of stocks. Jill has decided to start the plan immediately-contributing $5000 at the end of each year for the next 30 years. Jack has decided postpone his starting the plan for 6 years because of his loan payments for a Tesla. Thus, Jack will make his first contribution to the plan in year 7. Use Excel to answer the following:
(a) If the return each year is exactly 10%, how much will Jill have in the account at the beginning of year 31?
(b) If the return each year is exactly 10%, how much will Jack have in the account at the beginning of year 31? (Recall Jack only contributed for 25 years)
(c) Suppose that instead the returns alternate between 30% and-10% (i.e., 30% for year 1, -10% for year 2, 30% for year 3, -10% for year 4, 30% for year 5, etc. Observe that the average return is 10%. How much will Jill have in the account at the beginning of year 31?
(d) If, similar to (c), the returns alternate between 30% and -10%, how much will Jack have in the account at the beginning of year 31?
(e) What is the difference between your answers to (a) and (b)? Also what is the difference between your answers (c) and (d)? What insight does this give?
(f) What is the difference between your answers to (a) and (c)? Also what is the difference between your answers (b) and (d)? What insight does this give?

Jack takes quite a hit by not investing early. He could have increased his balance by about 86% (constant 10%) or 71% (alternating returns) had he started when Jill did. Ultimately, his Tesla costs him in excess of $260,000.

f) a-c $177,203.66

b-d $65,011.49

The alternating returns year to year mean the average rate for the annuity is lower than the highest return and higher than the lowest return. Over the long term, it tends toward the geometric mean of the annual rates. Here, that limit value is about √(1.30·0.90) -1 ≈ 8.17%. The returns realized by Jack and Jill in this scenario are 8.88% and 8.71%, respectively. (Jill's is lower because of the longer term.)

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Comment on average return

The alternating 30%, -10% returns cannot be averaged by adding them together and dividing by 2. Rather, the interest rate that corresponds to the resulting amount must be computed based on actual results.