Question

In detail show and explain all steps

Martha is 61 years old (born January 2, 1958) and will receive disability payments of $3500. per month till she reaches 65.  

After age 65 she will receive $1750. per month for life. What is the present worth of this policy if she lives until:

a) 80 years

b) 85 years

c) 90 years

Annual interest is 2.5% compounded semi-annually

and what should the insurance company offer her as a payout to make it worth her while

Answer 1

We have the following information

Interest rate (i) = 2.5% semi-annually

No. of interest periods per year, (C) = 2

Effective interest rate (R) = ((1 + (i/C))^C – 1

i = the nominal Interest rate

C = the number of interest periods in a year

R = ((1 + (2.5%/2))² – 1

R = 2.52% compounded annually

Martha is 61 years old and for the next 4 years she will receive $3,500 per month. This means that she wil receive (3,500 × 12 × 4) $168,000 per year for the next four years.

After ataining the age of 65 year she will start to receive $1,750 per month or $21,000 per year

Case 1: Marths lives till 80 years

Present Worth = 168,000(P/A, 2.52%, 4) + 21,000(A/F, 2.52%, 5)(P/A, 2.52%, 16)

Present Worth = 168,000[((1+0.0252)⁴ – 1)/0.0252(1+0.0252)⁴] + {21,000 × [0.0252/((1 + 0.0252)⁵ – 1)] × [((1+0.0252)¹⁶ – 1)/0.0252(1+0.0252)¹⁶]}

For the payments received after turning 65 years, we have first converted the 5^th year payment into annualized payment for the 5 years and then calculated the present worth for the continuing annual payment.

Present Worth = $683,744.40

Case 2: Marths lives till 85 years

Present Worth = 168,000(P/A, 2.52%, 4) + 21,000(A/F, 2.52%, 5)(P/A, 2.52%, 21)

Present Worth = 168,000[((1+0.0252)⁴ – 1)/0.0252(1+0.0252)⁴] + {21,000 × [0.0252/((1 + 0.0252)⁵ – 1)] × [((1+0.0252)²¹ – 1)/0.0252(1+0.0252)²¹]}

Present Worth = $696,198.31

Case 3: Marths lives till 90 years

Present Worth = 168,000(P/A, 2.52%, 4) + 21,000(A/F, 2.52%, 5)(P/A, 2.52%, 26)

Present Worth = 168,000[((1+0.0252)⁴ – 1)/0.0252(1+0.0252)⁴] + {21,000 × [0.0252/((1 + 0.0252)⁵ – 1)] × [((1+0.0252)²⁶ – 1)/0.0252(1+0.0252)²⁶]}

Present Worth = $707,194.35

Case 4: Infinite Period

Present Worth = 168,000(P/A, 2.52%, 4) + 21,000(A/F, 2.52%, 5)(P/A, 2.52%, n = infinite)

Present Worth = 168,000[((1+0.0252)⁴ – 1)/0.0252(1+0.0252)⁴] + {21,000 × [0.0252/((1 + 0.0252)⁵ – 1)] × (1/0.0252)}

Present Worth = $790,180.00