Question

Assume that you are 30 years old and expect to retire when you reach 65. If you were to retire today, you would like a fixed (pretax) income of 60,000 per year (in addition to the social security) for a period of 15 years ( your approximate life expectancy at age 65). However you realized that price inflation will erode the purchasing power of the dollar over the next 35 years and you want to adjust your desired retirement income at age 65 to reflect the decline in the purchasing power of the dollar. In addition to the fixed annual income, payable at the beginning of each year starting at age 65, you want to have assets (that is, security investments) of 1,000,000 either for your own needs or to donate to heirs, when you reach 80 years old.

Empirical studies have estimated the average compound rate price inflation and returns on stock and bonds over the past 70 years to be approximately:

	Compound rate
Inflation	3%
Common stock	11
Corporate bonds	6
Equity weighted portfolio	8.5
50% common stocks, 50% bonds

Assume that these rates will remain the same over the next 50 years and that you can earn these rates of return, after transactions costs, by investing in stock and/or bond index mutual funds. Also, assume that contributions to your retirement fund are made at the end of each year. Finally, assume that income taxes on the returns from any retirement (example IRA, or 401 k plans) can be deferred until you withdraw the funds beginning at 65.

1.Determine the amount you must accumulate by age 65 to meet your retirement goal, assuming that you invest in: Equally weighted portfolio (50 percent common stocks, 50% bonds)

2. Determin the annual investment in an equally weighted portfolio (50% common stocks, 50 percent bonds) required to accumulate the funds determined in question 1, assuming that the first payment is made at age

a.30

b.40

c.50

3. (apply to question #2) What conclusion can be drawn from the answers to question 2?

PLEASE USED CALCULATIONS IF NEED IT

Answer 1

Sol. 1. The total amount to be accumulated at age 65 includes 2 parts: a) security investments and b) amount for annual fixed income.

a) Investing in equally weighted portfolio will get 8.5% p.a. (for 15 years till 80)

Hence, amount required at 65 = 1000000 / (1.085^15) = 294140

b) Current annual income required = 60000

Inflation rate = 3%

Time till retirement = 65 - 30 = 35

Amount at the beginning of 65 = 60000 * (1.03^35) = 168831.7

Now we will calculate the Present Value of all the annual amounts required till 80 (beginning of age 79):

Total PV = 1804313.6

Hence, amount to be accumulated = a+b = 168831.7 + 1804313.6 = 1973145

Sol. 2: Now we need to calculate the annual investments required to accumulate 1973145 at age 65 through investments in equally weighted portfolio.

As the contributions are made in the end of each year till age 64, this is the case of ordinary annuity payments.

We already have the amount to accumulate i.e. Future Value.

So, we can calculate the yearly contribution by the below formula:

Here, 'C' is annual contribution.

'i' is the interest rate

'n' is the time period in years.

We have FV as 1973145, i as 8.5% and n as 35, 25 & 15.

We get, C =

a) 10239

b) 25082

c) 69890

Sol. 3. By the above values, we can conclude that for every reduction in the investment horizon there is a exponential increase in the amount to be saved. This is because of the time value of money.

If we divide b by a, we get 2.45

If we divide c by b, we get 2.79

Although, the investment horizon reduces by 10 years in both the cases, the additional annual savings required increases at a greater rate.

Hence, we can conclude that - earlier the investment start, lesser the savings burden.