Discounted Cash Flow Calculator

What is this?

The discounted cash flow calculator is a tool to help estimate the present value of a stream of cash flows discounted to the present.

How do I use it?

The calculator takes three inputs to produce the present value:

• Cash Flow - Latest cash flow

• Discount Rate - The rate at which future cash flows are discounted
• Cash Flow Growth Rate - The rate of growth of the cash flows

The growth rate may be specified as a simple growth rate or a multi-stage growth rate. A simple growth rate is entered as an integer and implies a constant rate of growth over the lifetime of the earnings generator. If a simple growth rate number is entered, the perpetual annuity formula is used to calculate the present value of the earnings stream (see How does it work). However, since most companies cannot be expected to show constant and simple growth, a multistage model must be used to calculate the present value of the earnings stream. You can enter a specification in the growth field to define the multistage model. It takes the form of g|y:tg where g is the growth rate for y years and tg is the terminal growth rate for the perpetual annuity calculation. You can string as many stages together as you like to model the growth. For example, the specification g1|y1:g2|y2:tg would mean grow at g1 for y1 years, then grow at g2 for y2 years with a tg terminal growth rate.

Additionally, some of the cash flow may be assumed to be plowed back into the business in order to generate the faster growth. To specify the non-plowback (payout) percentage, the specification g1|y1%p1:tg may be used for each stage where g1 is the growth rate of the stage, y1 is the length in years of the stage, and p1 is the percentage of the present value of the stage not to be reinvested to generate the growth (also known as the payout ratio). By default, none of the present value of the individual stages is assumed to be used to fund the growth (p1=dividend payout ratio=100).

How does it work?

In order to understand how the calculator works, you have to have a basic understanding of discounted cash flow valuation methodology. The best way I know of to explain this is to start with an example. Let's say that you have a company or other entity that produces annual earnings or cash flows of exactly \$1000 each year. This amount never changes but goes on forever into the future. What should one pay for this series of cash flows?

Well, one year from now we will receive our first \$1000. What should we pay today for \$1000 in a year? It depends on what we think we could have earned on our money or what we demand to earn on our money over the course of that year. This opportunity cost or required return is called the discount rate. For our example, we'll assume that we require a rate of return of 10%. This means that for the promise of receiving \$1000 a year from now, we should be willing to pay \$1000/1.1 or \$909.09. Working the other way, you can see that a 10% annual return on 909.09 results in a value of \$1000 in one year.

OK, that gets us the present value of one years earnings given our 10% discount rate. But, remember that our example company pays out \$1000 each year forever. In the second year, we will again receive our \$1000 but this time we will have to wait two years to get it. This means that we have use two years worth of discounting to arrive at the present value of the second years earnings today. This value is \$1000/(1.1*1.1) or \$1000/(1.1^2) or \$826.45.

Following this same line of thinking, you can see that the third years cash flow can be valued at \$1000/(1.1*1.1.*1.1) or \$1000/(1.1^3) or \$751.315. By now you can see that the present value of each years cash flow can be written as \$1000/(1.1^n) where n is the year number. You may also notice that as the years increase, the present value each years cash flow decreases because the denominator is rising each year. In fact, if you go out far enough, the denominator becomes so large that the values beyond a certain point do not contribute enough to the total to be meaningful. The present value of all of the cash flows is the sum of the present values of each of the individual cash flows and because of this convergence, the sum of this infinite series of cash flows can be reduced to a simple equation:

pv = c / k where
c = coupon
k = discount rate

Using this formula, we can calculate the present value of an infinite series of annual payments of \$1000 with a discount rate of 10% as \$1000/.1 or \$10000. We could have achieved the same result if we had actually finished calculating and summing the present value of each years cash flows: pv = \$909.09+\$826.45+\$751.315+... (try it out on a spreadsheet if you aren't convinced) but this is quite a bit quicker.

What if we have growing cash flows? The same technique is applicable. Let's now assume that in addition to our first criteria, that our cash flows are growing by 5% per year. At the end of the first year we will receive \$1000*1.05 or \$1050 whose present value we can calculate as (\$1000*1.05)/1.1 or \$954.545. The value of the second year cash flow will be (\$1000*1.05*1.05)/(1.1*1.1) or (\$1000*(1.05^2))/(1.1^2) or \$911.157. You should be able to see the pattern forming here. The present value of each years cash flow is (\$1000*(1.05^n))/(1.1^n) where n is the future year number. Just like our non-growth example, this series also converges and can be reduced to a simple equation:

pv = c / (k - g) where
c = coupon
k = discount rate
g = growth rate

This is the formula for a constant growth perpetual annuity (our original formula was just a special case of this same formula with g=0). Again, using this reduction, we can calculate the present value of our example cash flows of \$1000 growing at 5% a year discounted at 10% as \$1050/(.1-.05) or \$21000. Notice that in this case, the coupon input to the formula was \$1050 instead of \$1000. This is because the perpetual annuity formula expects the year one coupon as its input. Since our coupon is growing at 5% a year, it will be \$1000*1.05 or \$1050 at the end of a year. The calculator takes care of this first year growth for you - it takes as input the current or year zero cash flow.

You may be asking yourself at this point, given the simple nature of the equations we came up with: Why do we need a calculator? Well, for the examples we used, you don't, you could have done them in your head. But, the examples we used were very simple and not too realistic. For one thing, you can see that the perpetual annuity formula doesn't work if the growth rate is greater than the discount rate although many companies may exhibit short term growth rates much higher than useful discount rates. Another problem is that growth rates are not constant. Companies go through phases in their life cycle in which they will have different rates of growth.

The solution to these problems is to use what is known as a multi-stage discounted cash flow valuation. Each 'stage' is a period of years at a certain growth rate which can be different for each stage. These explicit growth rates are accompanied by a terminal growth rate that specifies the perpetual growth going forward after the specified growth periods. For example, we might specify that the growth will be 15% annually for 5 years, 10% for the next 7 years, and 5% after that into the future (this would be specified as 15|5:10|7:5 in the calculators growth input). Since the growth rates and timeframes can be different for each stage, there is no easy formula to lead us to the final result. This is where the calculator comes in.

The calculator calculates the present value of each years cash flows by stepping through the growth definitions. This is just like we did in our earlier examples above except that the calculator has to keep track of what growth rate it is currently using from year to year. When it is through calculating the cash flows for each year of the specific growth, the calculator takes the final cash flow from that process and grows it by the terminal growth rate. The perpetual annuity formula can then be used with this as its coupon to value the terminal value. Since the result of the perpetual annuity calculation is some distance into the future (remember that this process starts at the end of the specific growth periods), the result must be discounted to the present by dividing by (1+k)^N where k is the discount rate and N is the total number of years of the specific growth definition. Finally, the total present value is calculated by adding the present value of the terminal growth to the sum of the present values of all of the years of specific growth.