Question

Q3 Consider the FV and PV of an annuity made up of a series of annual cash payments (starting at the end of year 1). Prove mathematically that the formulae are such that PV=FV/(1+r)^n.

Answer 1

Let it be a constant annuity of n payments and rate be r

PV=C/(1+r)+C/(1+r)^2+C/(1+r)^3...C/(1+r)^n

As this is a geometric progression with first term(a) as C/(1+r) and common ratio(d) as 1/(1+r) and number of terms(x) as n

Sum becomes=a*(1-d^x)/(1-d)=C/(1+r)*(1-1/(1+r)^n)/(1-1/(1+r))=C/r*(1-1/(1+r)^n)

FV=C*(1+r)^(n-1)+C*(1+r)^(n-2)+C*(1+r)^(n-3)...C*(1+r)^0

As this is a geometric progression with first term(a) as C*(1+r)^(n-1) and common ratio(d) as 1/(1+r) and number of terms(x) as n

Sum becomes=a*(1-d^x)/(1-d)=C*(1+r)^(n-1)*(1-1/(1+r)^n)/(1-1/(1+r))=C/r*(1+r)^n*(1-1/(1+r)^n)=PV*(1+r)^n

PV=FV/(1+r)^n