**How to Value Known Cash Flows**

Investors are often faced with the problem of knowing the fair value of an investment that is expected to deliver future cash flows. Fortunately, there is some standard mathematics that can be applied.

In regards to estimating the value of an investment, Warren Buffett, in his 1991 letter to Berkshire Hathaway share owners said:

*In The Theory of Investment Value, written over 50 years ago, John Burr Williams set forth the equation for value, which we condense here: The value of any stock, bond or business today is determined by the cash inflows and outflows – discounted at an appropriate interest rate – that can be expected to occur during the remaining life of the asset. Note that the formula is the same for stocks as for bonds. Even so, there is an important, and difficult to deal with, difference between the two: A bond has a coupon and maturity date that define future cash flows; but in the case of equities, the investment analyst must himself estimate the future “coupons.”*

So it’s all very simple in theory:

*The value of any stock, bond or business today is determined by the cash inflows and outflows – discounted at an appropriate interest rate – that can be expected to occur during the remaining life of the asset.*

Therefore, to place a value on any financial investment we need to do two things each of which is simple to describe but is hard to do:

1. We must know or estimate the future net positive cash flows, and their timing, that will emerge from the investment, and

2. We must discount these cash flows at an appropriate interest rate to calculate their “present value” as of today.

In the case of the safest bonds or of money on deposit at a bank, the cash flows and their timing are known with (almost total) certainty and so step 1 is essentially done for us and we are left with step 2.

This article explains how we can calculate the present value of any *known* future cash flow. The summation of the present value of the individual cash flows that an investment will produce is the value of the investment.

Perhaps the most basic concept of borrowing, lending and investing is the following:

A dollar to be received in the future is less valuable than a dollar in hand today.

To illustrate, imagine that risk-free bank accounts pay 3% annually. What then it the value of a guarantee to receive $1.00 in ten years?

The answer is that the $1.00 to be received in ten years has a value today of only 74.4 cents. This is the case because 74.4 cents deposited at 3.0% annual interest in a simple bank account will grow to that same $1.00 in ten years.

In the language of finance, the present value (i.e. today’s value) of a guaranteed $1.00 to be received in ten years when the risk-free interest rate is 3.0% is 74.4 cents. Alternatively stated, the discounted value today of this $1.00 to be received in ten years is 74.4 cents.

The formula to confirm this on your calculator is 1 divided by (1 plus the interest rate) raised to the power of 10. In this case $1.00/(1.03)^^{10}

This simple formula can be used to calculate the present value of any cash flow when you know, with certainty, the amount to be received and when it will be received.

In theory, the risk-free interest rate should compensate you for two things:

1. A percentage rate to compensate you for the lost opportunity to spend the money now as opposed to in the future. This is known as the compensation for the time value of money.

2. An additional percentage rate to compensate you for expected inflation, which is the expected decline in the purchasing value of money.

In practice, the risk-free interest rate can simply be observed as the interest rate available on risk-free investments. For individuals the interest available from banks on guaranteed investment certificates can usually be considered the risk free interest rate.

In practice the two components of the risk free interest rates (compensation for delaying the ability to spend the money and compensation for expected inflation) are not separated. Also, in practice, the available risk free investments may not really offer a “fair” compensation. This appears to be the case today as governments appear to have worked to push interest rates down to unnaturally low levels.

The process of finding the present value of know cash flows is illustrated in the following table. This is an example of a ten year $1000 bond that pays 2.5% ($25.00) at the end of each year for ten years and then returns the $1000 face value. The present (today’s) value of each individual cash flow is calculated in the table.

Year |
Cash received |
Market Interest Rate |
Present Value |

1 | $25.00 | 1.30% | 24.68 |

2 | 25 | 1.65% | 24.19 |

3 | 25 | 1.65% | 23.8 |

4 | 25 | 1.70% | 23.37 |

5 | 25 | 2.00% | 22.64 |

6 | 25 | 2.05% | 22.13 |

7 | 25 | 2.10% | 21.62 |

8 | 25 | 2.17% | 21.06 |

9 | 25 | 2.23% | 20.49 |

10 | 1,025.00 | 2.30% | 816.52 |

Total |
$1,250.00 |
$1,020.51 |

The market interest rates shown in this table are the actual interest rates applicable to guaranteed investment certificates (GICs) at the Royal Bank of Canada at this time.

The present value of the $25.00 to be received at the end of the first year is $24.68. That’s because $24.68 deposited for one year at the market rate of 1.30% will grow to $25.00 in one year. Similarly, $22.64 is the present value today of the $25.00 cash flow to be received at the end of year five. That’s because $22.64 deposited for five years in a compounded GIC year at the market rate of 2.00% compounded will grow to $25.00 in five years.

This bond will ultimately pay out $1250.00 in total cash flows over its life. But at the market interest rates of today it is worth $1020.51.

If the unlikely event that one were to come across this bond offered in the market at $1000, then it would be a bargain as it would provide a return slightly higher than the going market rate.

The next example deals with the valuation of a risk-free perpetual preferred share.Imagine that there exists a perpetual risk-free preferred share that will pay out $1.00 per year in perpetuity.

The formula to value such in investment is simply the annual cash flow divided by the applicable interest rate.

It can be very difficult to determine the risk-free rate for such an investment because risk-free perpetual investments are quite rare.

If the applicable interest rate in this case is 4.00% then the investment is worth $1.00 / 0.04 of $25.00. It’s worth $25.00 because this $25.00 if deposited in an account that paid 4.00% in perpetuity would return the same $1.00 per year.

It’s interesting to observe that the total cash that is to be paid out from now until forever is infinite. Nevertheless, the value of the perpetual investment is far from infinite due to the fact that interest rates reduce the value today of a dollar to be received in the future. In theory, the value of a perpetual investment would approach infinity as extremely long-term interest rates approached zero. Today, short-term interest rates are about zero, But long-term interest rates are still well above zero.

A final example is to calculate the value of an investment that starts paying out $1.00 per year but where that $1.00 is expected to grow each year.

Imagine a common share that currently pays out $1.00 per year and where that dividend is expected to increase by 2% forever.

The formula to calculate the value of such a perpetual, and growing annual cash flow is: The annual payment divided by (the applicable interest rate minus the growth rate).

Unfortunately, the reality is that we will not find such an investment that is risk free. But, assuming we could, and assuming that the very long-term risk free interest rate was 4.00% then the value of the cash flows described here would be: $1.00 / (0.0400-0.0200) = $50.00

Compared to the value of the constant cash flow of $1.00 per year forever, the value of a dollar growing at just 2% per year is twice as high in this example. This illustrates the amazing impact that growth has on valuation.

**END**

Shawn Allen, CFA, CMA. MBA, P.Eng.

November 22, 2014