Question

A 40 year annuity-due will pay 10 in each of the first 4 years, 9 in each of the next 4 years, etc., until payments of 1 are made in each of the last 4 years. The present value of the annuity payments at an annual effective rate of 5% is X. Determine X.

Answer 1

Simple basic Formula:

Present value= Future value/(1+r)^n
r =interest rate 5% or 0.05
n = number of periods. (n=0 to 39)

In Annuity due, we get the payment at the beginning of the year.

So n=0, we get 10 and it ends with getting 1units at n= 39.

So the present value of annuity due=
{[10/(1.05)^0]+[10/(1.05)^1]+[10/(1.05)^2]+[10/(1.05)^3]+[9/(1.05)^4]+[9/(1.05)^5]+[9/(1.05)^6]+[9/(1.05)^7]+[8/(1.05)^8]+[8/(1.05)^9]+[8/(1.05)^10]+[8/(1.05)^11]+.......................[1/(1.05)^36]+[1/(1.05)^37]+[1/(1.05)^38]+[1/(1.05)^39]}

=126.3967

So X=126.3967