Question

Colin is 40 years old and wants to retire in 27 years. His family has a history of living well into their 90s. Therefore, he estimates that he will live to age 95. He currently has a salary of $150,000 and expects that he will need about 75% of that amount annually if he were retired. He can earn 8 percent from his portfolio and expects inflation to continue at 3 percent. Some years ago, he worked for the government and expects to receive an annuity that will pay him $20,000 in today’s dollars per year beginning at age 67. The annuity includes a cost of living adjustment, which is equal to inflation. Colin currently has $200,000 invested for his retirement. His Social Security benefit in today’s dollars is $30,000 per year at normal age retirement of age 67. How much does he need to accumulate at age 67 exclusive of his pension and Social Security benefits? A. $2.1 million. B. $2.2 million. C. $2.8 million. D. $2.9 million.

Answer 1

Given,

Current earnings = 150,000

Rate of return = 8%

Retirement component:

No of years till retirement = 27 years

Savings required to cover = 95-67 = 28 years of life

Annual requirement = 75% ($150,000) = $112,500

Social security coverage = $30,000

Annuity expected to receive = $20,000

Amount needed annually (in today’s terms) = 112,500 - 30,000 – 20,000 = 62,500

Inflation adjusted value at 67 years i.e. future value at the end of 67 years

= 62,500 * (1.03)²⁷ = $62,500 * 2.221 = $128830.56

Amount needed at the end of 67 years to provide $128830.56 per annum i.e. Present value of annuity

	P + P *((1-(1/(1+r)(n-1)))/r)

PV of annuity due =

Where,

P = Periodic payment

r = (1+0.08/1+0.03) – 1 = 4.854%

n = 30 years    

Calculation of PV of annuity due

           

= 128830.56 + 128830.56 * (1-(1/1+0.0485)^(28-1)))/0.0485

= 128830.56 + 128830.56 * ((1 – 0.2784)/0.0485)

= 128830.56 + 128830.56 * 14.88

= 128830.56 + 1916998.73

= 2.05 million =2.1 million approximately