Question

Thomas is considering the purchase of two different annuities.  The first begins in five years and pays $5,000 per year for three years.  The second begins in 10 years and pays $3,000 per year for five years.  Thomas is currently very risk-averse as he has a young family and little income, and thus his current opportunity cost of capital is 5.7%.  In nine years, Thomas expects to be more financially secure and his cost of capital will increase to 6.2% from then on.  If each annuity costs $5,000 today, which annuity (or annuities), if any, should Thomas purchase?  How much value will Thomas realize from his purchase(s), if he makes any purchases? Purchase both; $8,439.59 from first, $7,568.63 from second Purchase both; $5,766.79 from first, $7,568.63 from second Purchase both; $8,439.59 from first, $2,631.56 from second Purchase both; $5,766.79 from first, $2,631.56 from second

Answer 1

The underlying inference in this problem is to realize that Thomas has two distinct discount rates. The first discount rate of 5.7 % is for the first 9 years and the second discount rate of 6.2 % is for the time beginning from the end of Year 10.

First Annuity: Begins at the end of Year 5 and continues for three years, which implies that annuities come in at the end of Year 5, Year 6 and Year 7. Each annuity is worth $ 5000 and since cash flows happen before the end of year 9, the appropriate discount rate is 5.7 %

Therefore, PV of Annuities = 5000 x (1/0.057) x [1-{1/(1.057)^(3)] x 1/(1.057)^(4) = $ 10766.79

Cost of Annuity = $ 5000

Annuity of NPV = 10766.79 - 5000 = $ 5766.79

Second Annuity: The second annuity worth $ 3000 each begins at the end of year 10, continues for five years. This implies that annuity payments occur at the end of Year 10, Year 11, Year 12, Year 13 and Year 14. This annuity series will be first discounted to the end of Year 9 at a discount rate of 6.2 % as the discount rate switches to the higher value (changes from 5.7 % to 6.2 %) at that point in time (end of Year 9). Post this the sum of the annuities is discounted back to the current time at the lower discount rate of 5.7 %.

Therefore, PV of Annuity = 3000 x (1/0.062) x [1-{1/(1.062)^(5)] x 1/(1.057)^(9) = $ 7631.56

Cost of Annuity = $ 5000

Annuity NPV = 7631.56 - 5000 = $ 2631.56

Hence, the correct option is (d).