Part A: Future values of cash flows at the end of the seventh year at different rates
Section a: 6% interest rate
The first rate that has been used is 6%. In the first year, the amount is $ 15,000. Since these amounts are received at the end of each year, the interest is gained during the following year. The formula for future value is; FV = A (1+r) ^ n. In this formula, FV is the future of the annuity; A is the annuity itself while r is the interest rate. On the other hand, n represents the time. In the first instance the future value for $ 15,000 in the entire period will be $ 15,000 (1 + 0.06) ^ 6 = $ 15,000 * 1.4185 = $ 21,277.79. The future value for the $ 20,000 received in year two will amount to $ 20,000 (1 + 0.06) ^ 5 = $ 20,000 * 1.3382 = $ 26,764.51 in the entire period. The future value for the $ 30,000 in year three will be $ 30,000 (1 + 0.06) ^ 4 = $ 30,000 * 1.2625 = $ 37,874.31. Between the fourth and sixth year there are no cash flows hence no future cash flows. In the seventh year, the amount is received at the end of the year hence its value will remain as $ 150,000. This is because the calculation of $ 150,000 (1 + 0.06) ^ 0 = $ 150,000 * 1 = $ 150,000. The total future value will be = $ 21,277.79 + $ 26,764.51 + $ 37,874.31 + $ 150,000 = $ 235,916.61.
Section b: 9% interest rate
At the interest rate of 9%, the future value for the amount received in the first year will be $ 15,000 (1 + 0.09) ^ 6 = $ 15,000 * 1.6771 = $ 25,156.50. The future value for the $ 20,000 received in the second year will be $ 20,000 (1 + 0.09) ^ 5 = $ 20,000 * 1.5386 = $ 30,772.48.The future value for $ 30,000 received in year three will be $ 30,000 (1 + 0.09) ^ 4 = $ 30,000 * 1.4116 = $ 42,347.45. There are no cash flows between the fourth and sixth year. However, in the seventh year, there is a cash flow of $ 150,000 whose value remains intact since its value will be $ 150,000 * (1 + 0.09) ^ 0 = $ 150,000 * 1 = $ 150,000. Therefore, the total future value will be = $ 25,156.50 + $ 30,772.48 + $ 42,347.45+ $ 150,000 = $ 248,276.43.
Section c: 15% rate of interest
At the interest rate of 15%, the future value for $ 15,000 in the first year will be $ 15,000[(1 + 0.15) ^ 6 = $ 15,000 * 2.3131 = $ 34,695.91. The future value for the $ 20,000 in the second year will be $ 20,000 (1 + 0.15) ^ 5 = $ 20,000 * 2.0114 = $ 40,227.14. The future value for the $ 30,000 received in the third year will be $ 30,000 (1 + 0.15) ^ 4 = $ 30,000 * 1.7490 = $ 52,470.19.There are no cash flows between year four and six. In year seven, there is a cash flow of $ 150,000 at the end of the year hence it retains its value. Therefore, the total future value will be = $ 34,695.91 + $ 40,227.14 + $ 52,470.19 + $ 150,000 = $ 277,393.24.
Part B: County Ranch Annuity
The present value of this annuity will be given by PV = A [1 - (1 + r) ^ -n]/ r. In this case, A is the annuity of $ 500 while r is the discount rate of 6%. In addition, n is the time of 25 years. Therefore, the present value is = $ 500[1 – (1 + 0.06) ^ -25]/ 0.06 = $ 6,391.68.
Part C: Local government lottery
The solution in this case is the usage of the calculator’s TVM keys. In this case, P/y is equal to one whereas C/y is also equal to one. It will be paid in end mode.
Therefore, the investment rate for this trust will be 5.5619%.
Part D: State of Tranquility’s Lottery
Section a: investment rate of 8%
Basing the calculations on the first rate of 8%, the aim is to determine the present value of the annuity payment of $ 250,000 over a 20 year period. Therefore, the formula which will be applied is PV = A [1 - (1 + r) ^ -n]/ r. In this case, A is the annuity of $ 250000 while r is the investment rate of 8%. On the other hand, n is the time of 20 years. Therefore, the PV = $ 250,000 [1 – (1 + 0.08) ^ - 20]/ 0.08 = $ 250,000 * 9.8181 = $ 2,454,536.85. Since there is an option of taking a lump sum of $ 2,867,480; the present value amounting to $ 2,454,536.85 will be less rational. Therefore, the best option is taking the lump sum of $ 2,867,480.
Section b: an investment rate of 5%
If the rate is 5%, the present value of the $ 250000 annuity will be $ 250,000 [1 – (1 + 0.05) ^ -20]/ 0.05 = $ 250,000 * 12.4622 = $ 3,115,552.59. In this case, the best payoff is taking the $ 250,000 annuity.
Section c: investment rate equating annuity stream and lump sum payment
In order to find the investment rate that equates these values, the formula to be applied is PV = A [1 - (1 + r) ^ -n]/ r. The value that is unknown in the formula is r. $ 2,867,480 = $ 250,000 * [1 - (1 + r) ^ - 20]/ r. In this case, the present value interest factor (PVIFA) = [1 – (1 + r) ^ -20]/ r = $ 2,867,480/ $ 250,000 = 11.4699. By using a calculator, there are values to be fed which will yield the following output.
These results imply that the indifference rate between the lump sum and annuity is 6%.
