Question

4. When you retire, you would like to have a monthly pension of $8,000 per month for 30 years. Assume you have just had your 25th birthday and you intend to contribute monthly to your retirement fund until you are 60. The month after that you will start taking your pension. Your investment advisor has found a guaranteed investment for your fund that will yield 8% per year compounded quarterly (hint: you need to find equivalent effective monthly rate to be used for calculation purposes) for the duration of your pension needs. How much should you contribute each month to your retirement fund, assuming your contributions start one month after your 25th birthday?

7. Five months from today your company will begin receiving cash flows of $10,000 paid every eight months. The last of these $10,000 flows will occur 69 months from today. Calculate the present value (value today, time 0) of these cash flows. The discount rate provided to you by your accountant is 5% per year compounded quarterly

Answer 1

4]

First, we calculate the effective annual rate (EAR) of 8% compounded quarterly.

EAR = (1 + (r/n))ⁿ - 1

where r = annual nominal rate

n = number of compounding periods per year

EAR of 8% compounded quarterly = (1 + (8%/4))⁴ - 1 = 8.2432%

Next, we calculate the annual nominal rate (r), such that, with monthly compounding, the EAR equals 8.2432%

EAR = (1 + (r/n))ⁿ - 1

8.2432% = (1 + (r/12))¹² - 1

r = ((1 + 8.2432%)^1/12 - 1) * 12

r = 7.9473%

equivalent effective monthly rate =  7.9473% / 12 = 0.6623%

We calculate the amount required at retirement to enable the yearly withdrawals during retirement.  

PV of annuity = P * [1 - (1 + r)^-n] / r,

where P = periodic payment. This is $8,000.

r = interest rate per period. This is 0.6623%

n = number of periods. This is 30 * 12 = 360.

PV of annuity = $8,000 * [1 - (1 + 0.6623%)^-360] / 0.6623%

PV of annuity = $1,095,754.25

Next, we calculate the yearly saving required to accumulate the required amount at retirement.

Future value of annuity = P * [(1 + r)ⁿ - 1] / r,

where P = periodic payment. We need to calculate this.

r = periodic rate of interest. This is 0.6623%

n = number of periods. This is 35 * 12 = 420

$1,095,754.25 = P * [(1 + 0.6623%)⁴²⁰ - 1] / 0.6623%

P =  $1,095,754.25 * 0.6623% / [(1 + 0.6623%)⁴²⁰ - 1]

P =  $483.90

You should contribute $483.90 each month