Question

# A perpetuity-immediate pays X per year. Kevin receives the first n payments, Jeffrey receives the next...

A perpetuity-immediate pays X per year. Kevin receives the first n payments, Jeffrey receives the next n payments and Hal receives the remaining payments. The present value of Kevin's payments is 20% of the present value of the original perpetuity. The present value of Hal's payments is K of the present value of the original perpetuity. Calculate the present value of Jeffrey's payments as a percentage of the original perpetuity.

If r is the interest rate per year

PV of the perpetuity = X/r

PV of amount to Kevin = X/r* (1-1/(1+r)^n) = 20% of PV of original perpetuity

=> X/r*(1-1/(1+r)^n) = 0.2* X/r

=> (1-1/(1+r)^n) = 0.2 ....................... (1)

=> (1+r)^n =1.25 ............................. (2)

PV of amount to jeffrey = X/(1+r)^(n+1) + X/(1+r)^(n+2)+ .... + X/(1+r)^(n+n)

=1/(1+r)^n * (X/(1+r) + X/(1+r)^2+ .... +X/(1+r)^n )

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

Putting the values from (1) and (2)

= 1/1.25 * X/r * 0.2

= 0.16* X/r

So, the % of PV of Jeffrey's payments to PV of original perpetuity =  0.16* X/r / (X/r) = 0.16 or 16%

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