The formula for a perpetuity is P = C + C + C … + … C + ... (1 + i) (1 + i) 2 (1

Question

The formula for a perpetuity is

P   =          C         +         C        +      C         … + …       C        +   ...                           

              (1 + i)              (1 + i)2            (1 + i)3                   (1 + i)N

which is an infinite series. This can be reduced to the following simple equation for the yield to maturity: i = C/P or P = C/i. Show that the equation for a perpetutiy can be reduced to P = C/i. HINTS: You can take a C out of each of the terms in the equation above. You can solve for an infinite series by plugging the series back into itself.

Explanation / Answer

P = C/(1+i) + C/(1+i)^2 + C/(1+i)^3 + C/(1+i)^4 + ..............+ C/(1+i)^N    -------> (1)

multiply both sides by (1 + i)

=> P*(1 + i) = [C/(1+i) + C/(1+i)^2 + C/(1+i)^3 + C/(1+i)^4 + ..............+ C/(1+i)^N]*(1 + i)

P*(1+i) = [C + C/(1+i) + C/(1+i)^2 + C/(1+i)^3 + ..............+ C/(1+i)^(N-1)]       -------------> (2)

now subtract equation (1) from (2)

=> P*(1+i) - P = [C + C/(1+i) + C/(1+i)^2 + C/(1+i)^3 + ..............+ C/(1+i)^(N-1)] - [C/(1+i) + C/(1+i)^2 + C/(1+i)^3 + C/(1+i)^4 + ..............+ C/(1+i)^N]

now we'll simplify the above equation

=> P[1 + i - 1] = C

or P*i = C

or P = C/i

Hence proved

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