Question

James inherits a $20,000 life insurance benefit from his father. The benefit will be paid to him in an annuity. James has three options how he will receive the payment.

Option 1: James will receive a level payment of $2,000 at the end of each period for n periods

Option 2: He will receive a level payment of $819 at the end of each period for 3n periods

Option 3: He will receive a level payment of $X at the end of each period for 4n periods.

If James earns the same yield rate for all 3 options, calculate X.

Answer 1

The problems is an present value annuity problems with two unknown variables i.e. r (yield rate) and n (periods).

Present value annuity formula with the above variables and present value as 20,000 and Annuity payment PMT is as below:

P = PMT * [ 1 - { 1 / (1+r) }ⁿ ] / r  

Annuity factor =  [ 1 - { 1 / (1+r) }ⁿ ] / r = P / PMT

As per Option 1, 2 and 3, we get three equations:

Option 1,

20,000 / 2000 = [ 1 - { 1 / (1+r) }ⁿ ] / r

Option 2,

20,000 / 819 = [ 1 - { 1 / (1+r) }³ⁿ ] / r

Option 3,

20,000 / X = [ 1 - { 1 / (1+r) }⁴ⁿ ] / r

Here, lets assume { 1 / (1+r) }ⁿ = Y

so, the equations above equations becomes,

Equation 1: 20,000 * r / 2000 = 1 - Y ;

Equation 2: 20,000 * r / 819 = 1 - Y³ ; and

Equation 3: 20,000 * r / X = 1 - Y⁴

Divide Equation 2 by Equation 1, we get:

1 + Y + Y² = 2.442 or, Y² + Y - 1.442 = 0

Solving the quadratic equation we get

Y = 0.8

Placing the Y value in equation 1, we get the value of r

r = 0.02 ~ 2%

So, the yield rate is 2%

Now let us put the value of Y and r in equation 3,

we get the value of X as

20,000 * 0.02 / (1 - Y⁴) = X

So, X = 677.50

Hence, the value of X is $ 677.50.

Hope this resolves the query.