Question

You plan to make a deposit today to an account that pays 6% compounded annually that will found the following withdrawals with nothing left in the account after the withdrawals, starting at the end of year 4 you plan to make 5 annual withdrawals (withdraw in year 4,5,6,7,8) that will increase by 2% over the previous year's withdrawals. The amount for withdrawals in year 4 will be $2000, what amount do you need to deposit today?

Answer 1

Given –

Year 1	Year 2	Year 3	Year 4	Year 5	Year 6	Year 7	Year 8
0	0	0	2000	2040	2081	2122	2165

Amounts are rounded to nearest dollars.

Interest rate = 6%

Year 4 cash flow = 2000, continue up to 8 years (5 installments)

It increases by 2% every year (see the table above)

Total number of years = 80 years

What amount do you need to deposit today?

This is a geometric gradient cash flow

Putting the formula for geometric gradient cash flows

Calculating the present value of the gradient cash flow at the end of 3^rd year

P = A1 [1 – (1+g) ^N (1+i) ^–N ÷ i-g], where g = 2% and i=6%, A1 = 2000, n=5 years

P = $2,000 [1 – (1+0.02) ⁵ (1+0.06) ^–5 ÷ 0.06 – 0.02]

P = $8,748

Now calculating the present value of the above amount at 0^th year

Present Value = $8,748 (1+.06) ^-3

Present Value = $7,344.98 or $$7, 345

Alternatively

PW = 2000 (1+.06) ^-4 + 2040 (1+.06) ^-5 + 2081 (1+.06) ^-6+ 2122 (1+.06) ^-7+ 2165 (1+.06) ^-8

PW = $7,345