13.4
4. Ted Turner once offered to donate "1 billion dollars" to the U.N. charities. His actual offer was to give $100 million per year for 10 years, starting next year. For this question, assume that the interest rate is 12%, compounded annually.
a. What is his donation really worth today? (4 pts)
His donation is an annuity of $100,000,000 per year for 10 years with r=12% and m=1 (from "compounded annually")
\( \Large PV = \frac{100000}{.12} \left [ 1 - \frac{1}{\left(1+.12 \right )^{10} } \right ] = 565{,}022{,}302\)
b.What would the donation be worth if he offered to give $75 million per year forever, starting next year? (3 pts)
This is a perpetuity. CF=$75,000,000 and r=12%.
c.What would it be worth if he offered to make the first of the ten $100 million payments immediately? Thus, his offer would be 10 annual payments of $100 million with the first to be paid today. (6 pts)
Well the 9 payments that remain after the immediate payment are still an annuity. The only difference is that it is for 9 years instead of 10. Their value is:
\( \Large PV = \frac{100000}{.12} \left [ 1 - \frac{1}{\left(1+.12 \right )^{9} } \right ] = 532{,}824{,}979\)
of course, we still have the first payment that was made immediately. Its present value is just $100,000,000 since it is made in the present, so the total value of the 10 payments is $632,824,979.
d.What would it be worth if he promised to make the 10 annual payments, but the first one would be made 4 years from now. (7 pts)
This is a deferred annuity question. We already know from part (a) that an annuity of 10 payments of 100,000,000 at 12% is worth $565,022,302. This is what his gift would be worth if it were slated to start next year, so this is what it will be worth 3 yrs from now when there is 1 year left until it is started. If we want to know what it is worth today, we must discount back 3 years.
\( \Large PV = \frac{565{,}022{,}302}{1.12^{3} } = 402{,}171{,}714\)