Question

5. (3pts) Given the following three annuities: i. the present value of a 4n-year annuity-immediate of 1 at the end of every year is X; ii. the present value of a 4n-year annuity-immediate of 1 at the end of every second year is 10.2; iii. the present value of a 4n-year annuity-immediate of 1 at the end of every fourth year is 5. Find X, assuming a positive annual interest rate. (Hint: Use Fission/Fusion formula. Answer: 20.6)

Math Interest Theory/ Financial Math

Please Use Formulas

Answer 1

If the Interest rate is r, then

PV of annuity immediate every year for 4n years =X

=> 1/r* (1-(1/(1+r))^(-4n)) = X ............................(1)

PV of annuity immediate every 2 years ( 2n payments) = 10.2

=> 1/((1+r)^2-1)* (1-(1/(1+r)^2)^(-2n)) = 10.2

=>1/((1+r)^2-1)*(1-(1/(1+r))^(-4n)) = 10.2 ...................... (2)

PV of annuity immediate every 4 years (n payments) =5

=> 1/((1+r)^4-1)* (1-(1/(1+r)^4)^(-n)) = 5

=>1/((1+r)^4-1)*(1-(1/(1+r))^(-4n)) = 5 ...................... (3)

Dividing equation (2) by (3) gives

((1+r)^4-1) / ((1+r)^2-1) = 10.2/5 = 2.04

=> ((1+r)^2 +1)* ((1+r)^2-1)/((1+r)^2-1) = 2.04

=> ((1+r)^2 +1) = 2.04

=> r^2+2r +2=2.04

=> r^2+2r-0.04 = 0

r= (-2+(4-2*1*(-0.04))^0.5)/2

Taking positive r

r= (-2+2.039608)/2 = 0.019804

Putting the value of r in equation (2)

  1/((1+0.019804)^2-1)*(1-(1/(1+r))^(-4n)) = 10.2

=> 25* (1-(1/(1+r))^(-4n)) = 10.2

=> (1-(1/(1+r))^(-4n)) = 10.2/25 = 0.408

Putting the above value and value of r in equation (1)

=1/0.019804*0.408 = X

=> X = 20.60