142 Part 2 Fundamental Concepts in Financial Management
- The data to the right of the graph show the value of a $100 annuity when the
interest rate is 10% and the annuity lasts for 25, 50, and 100 years and forever.
The difference between these values shows how much the additional years
contribute to the annuity’s value. The payments for distant years are worth
very little today, so the value of the annuity is determined largely by the pay-
ments to be received in the near term. Note, though, that the discount rate
affects the values of distant cash " ows and thus the graph. The higher the dis-
count rate, the steeper the decline and thus the smaller the values of the dis-
tant " ows.
Figure 5-3 highlights some important implications for! nancial issues. For ex-
ample, if you win a $10 million lottery that pays $500,000 per year for 20 years be-
ginning immediately, the lottery is really worth much less than $10 million. Each
cash " ow must be discounted, and the sum of the cash " ows is much less than $10
million. At a 10% discount rate, the “$10 million” is worth only $4,682,460; and
that’s before taxes. Not bad, but not $10 million.
11 21 31 41 51 61 71 81 91
YearsPV of Each $100 Payment;
Addition to Annuity’s Value$50$0
1$100Bars indicate PV of each payment.
Sum of PVs from 0 to N = Value of the AnnuityValue of 25-Year Annuity: $907.70
Value of 50-Year Annuity: $991.48
Value of 100-Year Annuity: $999.93
Value of Perpetuity: $1,000.00Contribution of Payments to Value of $100 Annuity at 10% Interest Rate
F I G U R E 5! 3SELF^ TEST What’s the present value of a perpetuity that pays $1,000 per year beginning one
year from now if the appropriate interest rate is 5%? What would the value be if
payments on the annuity began immediately? ($20,000, $21,000. Hint: Just
add the $1,000 to be received immediately to the value of the annuity.)
Would distant payments contribute more to the value of an annuity if interest
rates were high or low? (Hint: When answering conceptual questions, it
often helps to make up an example and use it to formulate your answer. PV
of $100 at 5% after 25 years! $29.53; PV at 20%! $1.05. So distant pay-
ments contribute more at low rates.)