Question

Norma receives $500, 000 from a life insurance policy with which she purchases an annuity-certain. The annuity will pay 10 equal annual installments, with the first payment made immediately. On the day she receives her fourth payments she is offered, in lieu of the future annual payments, a new payments scheme: (a) 4 annual payments of $40, 000, beginning in one year, followed by a monthly perpetuity of X. (b) The first monthly perpetuity payment would occur one month after the fourth annual payment of $40, 000. The effective annual rate of interest is 9% for the entire time period. Determine the value of X.
Answer: X = 1, 943.71
Not sure how to get this answer

Answer 1

Investment amount = $500,000

The initial annuity can be treated as an annuity due. The present value of annuity due is given by the formula.

Here PV = Investment amount = $500,000

C = annual payment

i = effective rate of interest = 9%

n = number of annual installments = 10

C = $500,000 / 6.995246894 = $71,477.1055

She receives 4 payments of $71,477.1055 and has to receive 6 more payments of $71,477.1055 with the first of those 6 payments starting a year from now. This can be treated as an ordinary annuity and the present value of the annuity can be calculated using the formula.

C = annual payment = $71,477.1055

i = effective rate of interest = 9%

n = number of annual installments = 6

PV = $320,640.4762

Present value of remaining payments = $320,640.4762 --------------(1)

a)

In the new scheme annual payments of $40,000 are made for 4 years which begin after a year

The present value of these payments can be calculated using the PV for ordinary annuity formula

C = annual payment = $40,000

i = effective rate of interest = 9%

n = number of annual installments = 4

PV = $129,588.7951

Present value of first 4 payments in the new scheme = $129,588.7951 -----------(a)

b)

Effective annual interest rate = 9%

Effective annual interest rate = (1+monthly interest rate)¹² -1

1+monthly interest rate = (1+Effective annual interest rate)^1/12

1+monthly interest rate = (1+9%)^1/12

1+monthly interest rate = 1.007207323

monthly interest rate = 1.007207323 - 1 = 0.007207323 = 0.720732%

After 4 years, X payments are made every month for perpetuity starting after a month

The formula for calculating present value of perpetuity is given as

PV = C/r

Here C = monthly payment = X

r = monthly interest rate = 0.720732%

PV = X / 0.720732%

But this PV is 4 years from now. Discounting this to the present will give,

Present value of perpetuity payments in the new scheme = Earlier PV / (1+Effective annual interest rate)⁴

= X / [ 0.720732% *(1+9%)⁴] = X / 1.017373% ------------------(b)

Total PV of the new scheme = (a) + (b) = $129,588.7951 + X / 1.017373% -----------(2)

This will be equal to PV in (1). Equating (1) and (2) gives you

$129,588.7951 + X / 1.017373% = $320,640.4762

X / 1.017373% = $320,640.4762 - $129,588.7951 = $191,051.6811

X = $191,051.6811 * 1.017373% = $1,943.7073 = $1,943.71 (rounded to 2 decimal places)