Problem 15.1
Consider an investment of $5,000 at 6% convertible semiannually. How much can be withdrawn each half−year to use up the fund exactly at the end of 20 years?
Solution.
5,000=Xa¯40|0.03⟹X=216.31
Problem 15.2
The annual payment on a house is $18,000. If payments are made for 40
years, how much is the house worth assuming annual interest rate of 6%?
Solution.
18,000a¯40|=270,833.34
Problem 15.3
If d=0.05, calculate a¯12| .
Solution.
Note: i = d1−dSubstituting the value of d, we havei=5.26%Solving for a¯12|, we havea¯12|=8.733
Problem 15.4
Calculate the present value of 300 paid at the end of each year for 20 years
using an annual effective interest rate of 8%.
Solution.
300a¯20|=2,945.44
Problem 15.5
If a¯n|=x and a¯2n|=y, express d as a function of x and y.
Solution.
a¯2n|=a¯n|+vna¯n|y=x+vnxy−xx=vn1−y−xx=1−vnx−(y−x)x=1−vn2x−yx=1−vn2x−yix=1−vni2x−yix=a¯n|2x−yix=x2x−yd1−dx=x2x−y=x2d1−d(1−d)(2x−y)=x2d(2x−y)−d(2x−y)=x2d2x−y=x2d+d(2x−y)2x−y=d(x2+2x−y)d=2x−yx2+2x−y
Problem 15.6
(a) Given: a¯7|=5.153, a¯11|=7.036, a¯18|=9.180. Find i.
(b) You are given that a¯n|=10.00 and a¯3n|=24.40. Determine a¯4n| .
Solution.
a.) Recall: a¯18|=a¯7|+v7a¯11|Substituting the given values, we have9.180=5.153+(1+i)−7(7.036)⟹i=8.29%
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