Question

Mr. Smith is contemplating retirement. He has just celebrated his 50 birthday and his net worth is $2.5 million. He hopes that after retirement he can maintain a lifestyle that costs him $110000 per year in today's dollars (i.e., real dollars, inflation adjusted).

For simplicity, assume that all expenses occur at the end of each year; the first expense of $110000 will happen one year from now.

If he retires, he will invest all his net worth in government bonds that yield a safe nominal annual return of 5.0%. Inflation is expected to be 1.5% per year.

(a) If Mr. Smith lives until he is 80, what is the present value of his living expenses?

Is he rich enough to retire today? Yes or No

(b) If Mr. Smith lives until he is 100, what is the present value of his living expenses?

Is he rich enough to retire today? Yes or No

Answer 1

a)

given,

His current worth $2.5 million to fetch 5% returns pa

yearly payment expected at retirement P = $110000

annual inflation rate                            r =1.5% or 0.015

no. of years to live                             n = 30 years (80-50)

we need to find the Present value of these annuity cash stream. it must be discounted at the inflation rates to arrive at the real dollar value in today’s terms.

PV of annuity   =   P X   (1-(1+r)^-n      /r

                       =110000 X (1-(1+0.015)^-30)   /   0.015

                      =110000 X 0.360237570093509   /   0.015

                     = $ 2,641,742.18

the present value of his living expenses   = = $ 2,641,742.18

whether it is sufficient?

he will invest $2.5m for 30 years at 5%

the FV of those investments =   PV X (1+r)ⁿ

                                                         = 2500000 X (1+0.05)³⁰

                                                         ⁼ 10,804,855.94

the PV of above FV will be   =   FV /   (1+r)ⁿ       while discounting, we have to discount at inflation rate

                                             = 10,804,855.94   / ( 1+0.015)³⁰

                                                           = $ 6,912,540.89

since the PV of his investments   $ 6,912,540.89   is greater than the present value of his living expenses, $ 2,641,742.18 the amount is sufficient for his retirement expenses.

b)     if he lives for 100 years

no. of years to live                             n = 50 years (100-50)

we need to find the Present value of these annuity cash stream. it must be discounted at the inflation rates to arrive at the real dollar value in today’s terms.

PV of annuity   =   P X   (1-(1+r)^-n      /r

                       =110000 X (1-(1+0.015)^-50)   /   0.015

                      =110000 X 0.52499532110347   /   0.015

                     = $ 3,849,965.69

the present value of his living expenses   = = $ 3,849,965.69

whether it is sufficient?

he will invest $2.5m for 50 years at 5%

the FV of those investments =   PV X (1+r)ⁿ

                                                         = 2500000 X (1+0.05)⁵⁰

                                                         ⁼ 28,668,499.46

the PV of above FV will be   =   FV /   (1+r)ⁿ       while discounting, we have to discount at inflation rate

                                             = 28,668,499.46   / ( 1+0.015)⁵⁰

                                                           = $ 13,617,671.38

since the PV of his investments  $ 13,617,671.38 is greater than the present value of his living expenses, $ 3,849,965.69 the amount is sufficient for his retirement expenses.