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The present value of a perpetuity (cash flows paid at the end of each year) is $PV = CF / r$ where $r$ is the interest rate. This formula is proved in the book that I'm studying, Principles of Corporate Finance.

However, then it is stated that if instead the cash flows are spread evenly throughout each year like a continuous stream of payments, we can use the same formula but replace $r$ with the continuously compounded rate $r_c$. This is not proved in the book and I cannot see how this would automatically hold. Can someone please show how this holds?

Mike Pierce
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puk300
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4 Answers4

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If we have continuous compounding the future value after one year is $C\cdot e^r$

For $n$ yearly payments we have the series

$FV=C+Ce^r+Ce^{2r}+Ce^{3r}+\ldots+Ce^{(n-1)r}$

Using the closed form of a geometric series we get

$FV=C\cdot \frac{e^{rn}-1}{e^r-1}$

To get the present value the FV has to be discounted $n$ times

$PV=C\cdot\frac{1}{e^{rn}}\cdot \frac{e^{rn}-1}{e^r-1}=C\cdot \left( 1-\frac{1}{e^{rn}}\right)\cdot \frac{1}{e^r-1}$

Now let $n$ go to infinity

$$PV=\lim_{n \to \infty} C\cdot\left( 1-\frac{1}{e^{rn}}\right)\cdot \frac{1}{e^r-1}$$

$=C\cdot(1-0)\cdot \frac{1}{e^r-1}=\boxed{C\cdot\frac{1}{e^r-1}}$

callculus42
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  • Thank you for your answer. Can you please explain the linkage between your answer and the answer provided by the author "Satish Ramanathan". In contrast to your calculation, he integrate cash flows, which yield $C/r$, which quite different from your answer: $C/(e^r-1)$. I am confused, which method is correct. Thank you for your support. – ssane Aug 31 '19 at 21:02
  • @sane I don´t see a direct linkage between our answers. I think my method is correct. – callculus42 Aug 31 '19 at 21:21
  • Thank you for your comment. The only difference that I see between two methods is that in you answer cash flows are summed, i.e. $C+Ce^r+...$, but in the later answer, cash flows are integrated. Actually, I don't see the reason why one should use integration, instead of summation. If you can give me any guess on this, it would be highy appreciated. Thanks! – ssane Sep 01 '19 at 07:23
  • @sane "Actually, I don't see the reason why one should use integration, instead of summation" Neither I do. – callculus42 Sep 01 '19 at 13:44
  • Thanks! By the way, in the textbook, the right answer for this question is $C/r$. This is why I am in trouble. – ssane Sep 01 '19 at 16:09
  • @sane Can you give me a link where I can see that the right answer is $C/r$? – callculus42 Sep 01 '19 at 16:21
  • Brealey, Myers, Allen, "Principles of Corporate Finance, 12th edition", chapter 2, page 37, example 2. Please see the screenshot from the book (link).

    https://i.stack.imgur.com/ubxYC.jpg

     – ssane Sep 02 '19 at 10:30
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Let the cashflow be evenly spread across time to perpetuity. Then the present value of such a stream would be ( here the discount rate is continuously compounded rate $r_c$)

$$ PV = \int_{0}^{\infty} CF.e^{-r_ct}dt$$

$$ PV = -\frac{CF}{r_c}e^{-r_ct}\|_0^\infty$$

$$ PV = \frac{CF}{r_c}e^{-r_ct}\|_\infty^0$$

$$PV = \frac{CF}{r_c}\left(e^0 - e^{-\infty}\right)$$

$$PV = \frac{CF}{r_c}(1-0) = \frac{CF}{r_c}$$

Edit:

The discount factor for discrete compouding is $\frac{1}{(1+r)}$. The discount factor for continuous compounding is $e^{-r_c}$. Equating these you have $\frac{1}{(1+r)}. = e^{-r_c}$

=> $e^{r_c} = (1+r)$

=>$r_c = ln(1+r)$

Goodluck

Satish Ramanathan
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  • Thanks, but this is obvious. What I am not convinced about is that when this holds, $r_c = ln(1+r)$. Can you show this? – puk300 Jun 09 '16 at 15:05
  • Please see the edit. I don't kwow if your please, please is genuine. But I take it that way. If not you will be advised by someone else at this site. – Satish Ramanathan Jun 09 '16 at 15:21
  • Thanks, my "please, please" is genuine. What I do not understand is why the discrete discount factor and the continuous discount factor should necessarily be equal? Note that the present value of the continuous cash flow stream is higher than the present value of the discrete cash flow. – puk300 Jun 09 '16 at 15:47
  • @SatishRamanathan Thank you for your comprehensive answer. I have a question regarding your answer and the first answer provided by the author "callculus". I do understand both answers, but I don't see the linkage between them. In the first answer PV of perpetuity is equal to $CF/(e^r-1)$, which is quite different from your answer, i.e $CF/r$. Please, can you explain it? Thank you in advance. – ssane Aug 30 '19 at 08:45
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The integration answer is correct. An integral is a sum. The Integrand is discounted correctly. The answer that sums the cash flow adds dollars in different units, Ce^r + Ce^2r for example adds dollars of two different years which have different values. If r is a continuously compounded yield like from zero yield Treasury curve just plug that rate into C/r. If r is an annual rate compounded once, use r’= ln(1+r) with r’~= ln (r) is f r very small.

Philippe Penelle
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  • As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Jul 07 '22 at 13:29
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The textbook example (from Brealey, et. al. I presume) involves a perpetituity with annual compounding and compares two scenarios: (1) one payment, $C$, at the end of each year and (2) "continuous cash flows" over the year (summing to C): a good approximation to daily payments of C/365.

I derived the following equation from first principles: $$ \frac{C/n}{\left(1+\frac{r}{m}\right)^{m/n}-1}\left( 1- \left(1+\frac{r}{m}\right)^{-mT} \right) $$ This calculates the value of a $T$-year annuity, compounded $m$ times a year, with $n$ payments during the year adding up to an annual payment of $C$.

If we take the limit as $ n \rightarrow \infty$, we get the special case with continuous cash flows (after applying La Hopital's Rule): $$ \frac{C}{\ln\left(1+\frac{r}{m}\right)^{m}}\left( 1- \left(1+\frac{r}{m}\right)^{-mT} \right) $$ In the textbook, $m=1$ (annual compounding) and $T \rightarrow \infty$ (perpetuity). I suspect this gives the equation you are looking for:

$$ \frac{C}{\ln\left(1+r\right)} $$

Notice that $\ln\left(1+r\right)$ is the continuous compounded rate with an effective annual interest rate of $r$. Since the continuously compounded interest rate is less than its corresponding effective annual rate then $\ln\left(1+r\right) < r$, so continuous payments are worth more than a single end-of-year payment.

This is consistent with the textbook Example 2 since $r =18.5\%$ and so $\ln\left(1+r\right)=17\%$. For $C=\$100$, the perpetuity with continuous cash flow is valued at $100/.17$, which is compared with the standard formula for a pertituity namely $C/r = 100/.185$.

The more general formulation can be used with Example 3, where $T=20$ and $C=\$200,000$.

Finally, take the limit as $m \rightarrow \infty$, and we get the formula for continuous compounding:

$$ \frac{C/n} {e^{r/n}-1}\left(1- e^{-rT}\right) $$

And as $n \rightarrow \infty$, again using La Hopital's Rule,

$$ \frac{ A}{r}\left(1- e^{-rT}\right) $$

Glenn Anderson
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