Question

Graham receives $640,000 at his retirement. he invests x in a twenty-year annuityimmediate with annual payments and the remaining $640, 000 − x is used to purchase a perpetuity-immediate with annual payments. his total annual payments received during the first twenty years are twice as large as those received thereafter. the annual effective interest rate is 5%. find x.

Answer 1

For the amount invested in the 20 year annuity immediate,

the return will be;
 r/(1 - (1+r)^-n) = 0.05/(1- 1.05^-20)
= 0.0802425872
= 8.02425872% 

Now, return on perpetuity-immediate = 5% 

So, 5% + 8.02425872% = 13.02425872

for equal returns from both investments,
X = 5/(13.02425872) x 640,000

= $245,695.365 

= $ 245,695.36