A level perpetuity-immediate is to be shared by three charities providing medica

Question

A level perpetuity-immediate is to be shared by three charities providing medical research and a fourth charity providing assistance to children of veterans. For n years, the three research charities will receive the payments equally. Thereafter, all the payments will go to the charity aiding children of veterans. It is reported that the present value of each of the charities' bequest is equal to a common amount when calculated with an annual effective interest rate of 12.25%

1) Find n?
2) If the interest rate is a more modest 6%, what proportion of the total bequest is directed to the charity aiding children of veterans?

*****The answers are n=12, and 49.69694% of the total for part 1 and 2 respectively, BUT I NEED TO KNOW HOW THEY GOT THESE ANSWERS***

Thanks in advance

Explanation / Answer

(a) let C= annuity paid to each of the three charities from year 1 to year n
PV = (C/i)[1-1/(1+i)n]

(b) after year n all payment = 3C will go to the forth charity in terms of perpetuity

PV = (3C/i)[1/(1+i)n]

Since (a) is equal to (b)

(C/i)[1-1/(1+i)n] = (3C/i)[1/(1+i)n]

1-1/(1+i)n = 3C/(1+i)n

Solve for n=12

if the interest rate is now 6%

1-1/(1+i)n = 3C/(1+i)n

Solve for n = 24

PV of the three charities = (C/i)[1-1/(1+i)n] = 12.550C

PV of the forth charities = (3C/i)[1/(1+i)n] = 12.349C

Portion of the forth charities = 12.349C / (12.349C+12.550C) = 49.69694% (some error from rounding)

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