Answer to Question #228419 in Macroeconomics for Chilu

Question

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 formul a gives the price of a consola 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. a) b) Suppose that i = 10%. Let $z = 100. What is the present value of the consol? If i = 10%, what is the expected present discounted value of a bond that pays $z over the years? 20 years? 30 years? 60 years? c) Repeat the calculations in (a) and (b) for i = 2% and i = 5%

Answer 1

(a)

Present value $= \frac{\$100 }{ 0.1} = \$1000$

(b)

i=10%

N = 20 years

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

N = 30 years

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

N = 60 years

Present discounted value = $z $\times$ P/A(10%, 60) = $z $\times$ 9.9671** = $9.9671z

(c)

i = 2%

For perpetuity, present discounted value $= \frac{ \$100 }{ 0.02} = \$5000$

N = 20 years

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

N = 30 years

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

N = 60 years

Present discounted value = $z $\times$ P/A(2%, 60) = $z $\times$ 34.7612** = $34.7612z

i = 5%

For perpetuity, present discounted value $= \frac{\$100 }{ 0.05} = \$2000$

N = 20 years

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

N = 30 years

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

N = 60 years

Present discounted value = $z $\times$ P/A(5%, 60) = $z $\times$ 18.9304** = $18.9304z