Saturday, October 8, 2016

Present and Accumulated Values of an Annuity-Immediate

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 How much can be withdrawn each half−year to use up the fund exactly at the end of 20 years?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|0.03X=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 = d1dSubstituting 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+vnxyxx=vn1yxx=1vnx(yx)x=1vn2xyx=1vn2xyix=1vni2xyix=a¯n|2xyix=x2xyd1dx=x2xy=x2d1d(1d)(2xy)=x2d(2xy)d(2xy)=x2d2xy=x2d+d(2xy)2xy=d(x2+2xy)d=2xyx2+2xy

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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