Question

In: Finance

Suppose that you are the manager of a newly formed retirement fund. You are to set up a series of semiannual payments to accumulate a sum of $1,000,000 in ten years.

I need to know how to solve this with a financial calculator

Suppose that you are the manager of a newly formed retirement fund. You are to set up a series of semiannual payments to accumulate a sum of $1,000,000 in ten years. You assume that the appropriate interest rate for the period is 6 percent annual, compounded semiannually. The first payment into the fund will be made six months from today and the last payment will be at the end of the tenth year.

a. What is the required semiannual payment, to the nearest dollar?

b. Suppose that immediately after the 4th payment is made at the end of the second year, interest rates have risen and you believe that the appropriate rate for the remainder of the period is now 8 percent annual. Assume that you can earn that rate on funds already accumulated and the 16 future payments. Interest is to be compounded semiannually on all funds. Calculate the revised payment that will allow you to reach your investment goal of $1 MM.

Expert Solution

A.

Period is 10 Years i.e 20 Semi Annual Years

r = 6% per anum & 3% per six Months

FV of ANnuity = CF * [ ( 1+r )ⁿ - 1 ] / r

r is Int rate per six months & n is No. of Semi ANnual Periods

$ 1,000,000 = CF * [ ( 1+0.03 )²⁰ - 1 ] / 0.03

= CF * [ ( 1.03 )²⁰ - 1 ] / 0.03

= CF * [ ( 1.8061 ) - 1 ] / 0.03

= CF * [ 0.8061 ] / 0.03

CF = $ 1,000,000 * 0.03 / 0.8061

= $ 37216.23

B .

FV of Annuity after 2 Years = = CF * [ ( 1+r )ⁿ - 1 ] / r

= $ 37216.23 * [ ( 1+0.03 )⁴ - 1 ] / 0.03

= $ 37216.23 * [ ( 1.03 )⁴ - 1 ] / 0.03

= $ 37216.23 * [ ( 1.1255 - 1 ] / 0.03

= $ 37216.23 * [ ( 0.1255 ] / 0.03

= $ 155687.90

The amount accumulated after 2 Years is 155687.90,

The Value of this amount at the end of 8 Years ( 2 + * years amounts to 10 Years)

FV of This Amount after 8 Years = AMount * FVF ( r%, 16 periods)

= $ 155687.90 * FVF ( 4%, 16)

= $ 155687.90 * 1.8730

= $ 291,600.50

The Balance to be Accumulated in last 8 Years = $ 1,000,000 - $ 291,600.50

= $ 708,399.50

FV of ANnuity = CF * [ ( 1+r )ⁿ - 1 ] / r

$ 708,399.50 = CF * [ ( 1+0.04 )¹⁶ - 1 ] / 0.04

= CF * [ ( 1.04 )¹⁶ - 1 ] / 0.04

= CF * [ ( 1.8730 ) - 1 ] / 0.04

= CF * [ 0.8730 ] / 0.04

CF = $ 708399.50 * 0.04 / 0.8730

= $ 32458.17

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