1 The Present value of an infinite stream of dollar payments

1. The Present value of an infinite stream of dollar payments of $z (that starts next year) is $z/I when the nominal interest rate, I, is constant. This formula gives the price of a consol -a bond paying a fixed nominal payment each year, forever. It is also a good approximation for the present discounted value of a stream of constant payments over long but not infinite periods, as long as i is constant 1.1. Suppose i-1096. Let $z-100, what is the present value of the consol? 12. ifi-10%, what is the expected present discounted value of a bond that pays $z over the next 10 years? 20 years? 30 years? (Hint: Use the formula from Chapter 14, but remember to adjust for the first payment. 1.3. Repeat the calculations in (1.1) and (1.2) for i-2% and 1-5%.

Solution

(1.1)

Present value = $100 / 0.1 = $1,000

(1.2)

(a) N = 10 years

Present discounted value = $z x P/A(10%, 10) = $z x 6.1446** = $6.1446z

(b) N = 20 years

Present discounted value = $z x P/A(10%, 20) = $z x 8.5136** = $8.5136z

(c) N = 30 years

Present discounted value = $z x P/A(10%, 30) = $z x 9.4269** = $9.4269z

(1.3)

(A) i = 2%

(i) For perpetuity, present discounted value = $100 / 0.02 = $5,000

(ii)

(a) N = 10 years

Present discounted value = $z x P/A(2%, 10) = $z x 8.9826** = $8.9826z

(b) N = 20 years

Present discounted value = $z x P/A(2%, 20) = $z x 16.3514** = $16.3514z

(c) N = 30 years

Present discounted value = $z x P/A(2%, 30) = $z x 22.3965** = $22.3965z

(B) i = 5%

(i) For perpetuity, present discounted value = $100 / 0.05 = $2,000

(ii)

(a) N = 10 years

Present discounted value = $z x P/A(5%, 10) = $z x 7.7217** = $7.7217z

(b) N = 20 years

Present discounted value = $z x P/A(5%, 20) = $z x 12.4622** = $12.4622z

(c) N = 30 years

Present discounted value = $z x P/A(5%, 30) = $z x 15.3725** = $15.3725z

**From P/A factor table