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I know how to find PV of perpetuity; PV = C/r

But, I am not sure about this question since this states "Starting at year 1 you expect to receive $100 at the end of every year except in years 3,6,9,12.." Some exception there, so I don't know how to show it on formula or answer.

Please and help me :(

***** I attached the question as picture above *****

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    I attached the picture of the question -> "enter image description here" – 2017-01-29

2 Answers 2

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Take the value of a perpetuity that pays $100$ every year and subtract from it the value of a perpetuity that pays $100$ every three years.

The value of the perpetutity that pays $100$ at the end of a year and every year thereafter is $$\sum_{i=1}^\infty \frac{100}{(1+r)^i}.$$ To compute this, use the geometric sum formula $$ \sum_{i=1}^\infty \frac{1}{x^i} = \frac{1}{x-1}$$ valid for $x>1.$ Plugging in $x = (1+r)$ gives a value of $$ \frac{100}{(1+r)-1} = \frac{100}{r}.$$

The value of a perpetuity that pays $100$ every three years is $$\sum_{i=1}^\infty \frac{100}{(1+r)^{3i}}$$ since $100/(1+r)^3$ is the discounted value of the first payment, $100/(1+r)^6$ for the second payment after $6$ years, etc.

Since $1/(1+r)^{3i} = 1/((1+r)^3)^i$ we can evaluate the sum with the geometric sum formula plugging in $x=(1+r)^3.$ So the value is $$ \frac{100}{(1+r)^3-1}.$$

The perpetuity in question pays only in years not divisible by three, so it is the same as a perpetuity that pays every year minus one that pays every three years. So the value is $$ \frac{C}{r} - \frac{C}{(r+1)^3-1}$$

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    So find PV = C/r and then subtract PV = C/(1+r)^3 ? kinda confused :( – 2017-01-30
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    PV = C/r = 1000 and then 1000 - 75.13148009 = 924.87. Is it correct? – 2017-01-30
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    @Eric No. For the payout every year, the discounted payoff is C/(1+r)^i in year $i$ so the total value is $\sum_{i=1}^\infty C/(1+r)^i = C/r$ by the geometric sum formula. If it pays out every three years you would need to take $\sum_{i=1}^\infty C/(1+r)^{3i}$ using the geometric sum formula. As you might expect intuitively the value is not too far from $1/3$ of value of the perpetuity that pays every year. – 2017-01-30
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    @Eric I get 697.88 for the final answer – 2017-01-30
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    So, I understood the formula, but I still confused about 'every three years' one. – 2017-01-30
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    ∑ ∞ i=1 C/(1+r) i =C/r = 1000. ∑ ∞ i=1 C/(1+r) 3i -> idk how to solve this one well :( sry for many questions – 2017-01-30
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    how did u get the final answer, can you show me the steps? sry – 2017-01-30
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    @Eric ok, edited – 2017-01-30
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    Thank you so much!!!!!!!!!!! – 2017-01-30
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You can consider the perpetuity as the difference of a perpetuity that pays $C=100$ every year discounted at rate $r=10\%$ and a perpetuity that pays $C=100$ every 3 years discounted at rate $i=(1+r)^3-1=33.1\%$ that is the present value is $$ PV=\frac{C}{r}-\frac{C}{i}=697.88 $$