Question

2. Carl, age 34, currently makes $70,000. His wage replacement ratio is determined to be 90 percent. He expects inflation will average 3 percent for his entire life expectancy. He expects to earn 5.5 percent on his investments and retire at age 67. Based on family history, he expects to live to age 90. He has received his Social Security benefit statement, which indicated that his Social Security retirement benefit in today's dollars is $12,000 per year. A. Calculate Carl's capital needed at retirement at age 67.

Answer 1

Current income= $70,000

Wage replacement ratio= 90%

Hence the retirement income needed in today's dollar= $70,000*90% = $63,000

Social security retirement benefit in today's Dollar=$12,000

Therefore, net amount needed in today's dollar= $63,000-$12,000= $51,000

Inflation rate=3%

Hence the net income needed in the first year after retirement (ie., after 67-34=33 years) = $51,000*(1+0.03)^33

=$51,000*2.652335= $135,269.10

Capital needed at the retirement age of 67 is the PV of growing annuity with the following variables:

First payment payment=$135,269.10

Growth rate= inflation rate= 3%

Discount rate= Expected rate of return on investment=5.5%

Period= Life expectancy-Retirement age= 90-67= 23 years

Capital needed is calculated at $2,293,969.36 as follows:

1 A B C 1 Present Value of Annuity D Payments at the end of each period 7 8 23 3 Present value of growing annuity is calculated using the formula PVA=[P/(r-g)]*{1-[(1+g)/(1+r)]^n]} 4 Where P= First payment, 5 n= Number of payments. 6 r= Rate of interest per period in decimals and g= Growth rateper period in decimals All amounts in $ Given that 9 First payment (P)= 135269.1 No. of years (N)= 10 Rate of interest (% p.a)=R= 5.5 No. of payments a year (t)= 11 Growth rate (% p.a.) =G= 12 13 Calculations: 14 Formula Cell reference Value 15 Interest rate per period in decimals (r )=R/(100*t) D10/(100*G10) 0.055 16 No. of periods (n)= N*t G9*G10 23 17 Growth rate (g) G/(100*t) D11/(100*G10) 0.03 18 ((1+g)/(1+r)^n ((1+G17)/(1+G15))^G16 0.576035961 19 1-[(1+g)/(1+r)]^n 1-G18 0.423964039 r-g G15-G17 0.025 P/(r-g) D9/20 5410764 22 Present value of growing annuity PV= [P/(r-g)]*(1-((1+g)/(1+r)^n]} G21*G19 2293969.359 23 Rounded to $2,293,969.36 24 Dounded to