Question

John starts his career at 21 years old and expects to retire 44 years later at the age of 65. His first annual salary is $72,000 that will increase at 1.5% per year until he finishes his part-time MBA at 28 years old. With his MBA, John expects salary to increase at 3% per year until retirement. At the end of each year, he deposits 10% of his annual salary into a retirement saving plan that pays 6% interest per year compounded monthly. On the first day of his retirement, John converts his whole retirement saving plan into a registered retirement income fund (RRIF) that earns 8% interest per year compounded quarterly. The RRIF will pay John $Y per quarter, the first payment being paid on the day he buys the RRIF, for 25 years. Find Y.  (Show your work without using MS Excel)

Answer 1

There are two streams of annuity involved here.

First stream from age 21 to 28 when

Annuity = 10% of salary

First annuity, A = 10% x 72,000 = 7,200

Growth rate in annuity, g = 1.5%

EAR = interest rate, i = 6% compounded monthly = (1 + 6% / 12)¹² - 1 = 6.17%

n = 28 - 21 = 7 years

Hence, FV of this annuity at the end when he is 28 years old = FV₂₈ = A / (i - g) x [(1 + i)ⁿ - (1 + g)ⁿ] = 7,200 / (6.17% - 1.5%) x [(1 + 6.17%)⁷ - (1 + 1.5%)⁷] = 63,322.98

FV of this amount on retirement = FV₁ = FV₂₈ x (1 + i)^{(65 - 28)} =  63,322.98 x (1 + 6.17%)³⁷ =  579,819.40

Second stream from age 29 to 65 years:

Salary at the end of 28 years = 72,000 x (1 + 1.5%)⁷ =  79,908.83

Salary at the end of 29th year =  79,908.83 x (1 + 3%) =  82,306.10

Hence, first annuity, A = 10% x  82,306.10 =  8,230.61

Growth rate, g = 3%

n = 65 - 28 = 37

Hence, FV of second stream of payment at the time of retirement = FV₂ =  = A / (i - g) x [(1 + i)ⁿ - (1 + g)ⁿ] = = 8,230.61 / (6.17% - 3%) x [(1 + 6.17%)³⁷ - (1 + 3%)³⁷] = 1,603,446.49

Hence, total kitty size at the time of retirement = FV = FV₁ + FV₂ = 579,819.40 + 1,603,446.49 =  2,183,265.90

On the first day of his retirement, John converts his whole retirement saving plan into a registered retirement income fund (RRIF) that earns 8% interest per year compounded quarterly. The RRIF will pay John $Y per quarter, the first payment being paid on the day he buys the RRIF, for 25 years.

Frequency = Quarterly

Interest rate per period = interest rate per quarter, i = 8% / 4 = 2%

Type = beginning of the period

Annuity = Y

Period = nos. of quarter in 25 years, n = 4 x 25 = 100

Hence, FV = PV of period beginning annuity = Y / i x [1 - (1 + i)^-n] x (1 + i)

Or, 2,183,265.90 = Y / 2% x [1 - (1 + 2%)^-100] x (1 + 2%) =  43.9603Y

Hence, Y = 2,183,265.90 / 43.9603 =  49,664.47