# Jack retires and takes a lump sum payment of \$260,000 from the company pension plan. He purchases an annuity from which he will receive equal semi-annual payments for 10 years. If money is worth 8% compounded semi-annually, how much is each payment?

Each of the payments received by Jack is equal to \$17820. Jack receives a lump sum payment of \$260,000 from the company pension plan which is used to purchase an annuity that gives equal semi-annual payments for 10 years. The rate of interest is 8% compounded semi-annually. The amount of each of the payments has to be determined.

The present value of P after n terms at the rate of interest r is equal to `PV = P/(1+r)^n` .

Let the semi-annual payments received by Jack be P.

As rate of interest is 8% compounded semi-annually, r = 8/2 = 4%. The total number of terms is 10*2 = 20. The present value of a payment received after n terms is given by `P/(1+0.04)^n`.

The sum of the present value of all the payments received by Jack is equal to the sum

`P/(1.04)^20 + P/(1.04)^19 + ... P/(1.04)^1`

= `P*(1/(1.04)^20 + 1/(1.04)^19 + ... 1/(1.04)^1)`.

The sum of a geometric series `a, ar, ar^2 ... ar^(n-1)` is given by the formula Sum = `a*(r^n-1)/(r - 1)`.

`P*(1/(1.04)^20 + 1/(1.04)^19 + ... 1/(1.04)^1)`

= `P*((1.04)^-21 - 1)/(1.04^-1 - 1)`

= P*14.59

This sum is equal to the initial amount that had been invested by Jack equal to \$260,000

This gives: \$260,000 = P*14.59

=> P = 17820

Each of payments received by Jack is equal to \$17820.

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