Question

You deposit $10,000 annually into a life insurance fund for the next 10 years, after which time you plan to retire.

a.	If the deposits are made at the beginning of the year and earn an interest rate of 6 percent, what will be the amount in the retirement fund at the end of year 10? (Do not round intermediate calculations. Round your answer to 2 decimal places. (e.g., 32.16))

b.

Instead of a lump sum, you wish to receive annuities for the next 20 years (years 11 through 30). What is the constant annual payment you expect to receive at the beginning of each year if you assume an interest rate of 6 percent during the distribution period? (Do not round intermediate calculations. Round your answer to 2 decimal places. (e.g., 32.16))

c.	Repeat parts (a) and (b) above assuming earning rates of 5 percent and 7 percent during the deposit period and earning rates of 5 percent and 7 percent during the distribution period. (Do not round intermediate calculations. Round your answers to 2 decimal places. (e.g., 32.16))

Deposit Period	Value at 10 Years	Distribution    Period	Annual payment
5 percent	$	5 percent	$
		7 percent	$
7 percent	$	5 percent	$
		7 percent	$

Answer 1

Given information:

Payment per month = P = 10,000

duration = n = 10 years

interest rate = 6% per annum

a.

Payments are made at the beginning of the year, then the accumulated amount at the end of 10 years will be:

P*(1+r)¹⁰ + P*(1+r)⁹ + P*(1+r)⁸+......+P(1+r)

This is a geometric progression and can be simplified to {The total sum of geometric progression ar, ar², ar³...arⁿ is ar (rⁿ-1)/r}

P* (((1+r)ⁿ - 1) / r) * (1+r) = 10000*(((1+0.06)^10- -1)/0.06)*(1+0.06) = $139,716.43

Substituting the given values we get the future value = $139,716.43

b. Calculate the annual payment for next 20 years.

Here we need to calculate future cash flows for which the PV is the value calculated in part a.

PV = P + P /(1+r) + P/(1+r)² + P/(1+r)³ +.....+ P/(1+r)¹⁹

Based on the geometric progression sum formula, we get (geometric term is 1/1+r and firsst term is P)

PV = P( (1/1+r)ⁿ -1) / ( (1/1+r) -1)

Substitute the value of r and PV to get

$139,716.43 = P ((1/1+0.06)²⁰ -1) / ( (1/1+0.06) -1)

We get P = $139,716.43 / ((1/1+0.06)²⁰ -1) / ( (1/1+0.06) -1) = $3798.13

Answer: Constant annual payment expected to receive = $3798.13

c.

In this case we use different interest rate at the time of deposit and at the time of withdrawal,

For deposit period with 5% interest rate: Value at 10 years = 10000*(((1+0.05)^10- -1)/0.05)*(1+0.05) = 132,067.87

Distribution period: with 5% interest rate = 132,067.87 / ((1/1+0.05)²⁰ -1) / ( (1/1+0.05) -1) = $3994.074

Distribution period: with 7% interest rate = 132,067.87 / ((1/1+0.07)²⁰ -1) / ( (1/1+0.07) -1)= $3803.88

For deposit period with 7% interest rate: Value at 10 years = 10000*(((1+0.05)^10- -1)/0.05)*(1+0.05) = 147,835.99

Distribution period: with 5% interest rate = 147,835.99/ ((1/1+0.05)²⁰ -1) / ( (1/1+0.05) -1) = $4470.94

Distribution period: with 7% interest rate = 147,835.99 / ((1/1+0.07)²⁰ -1) / ( (1/1+0.07) -1)= $3606.15