How to calculate intrinsic value of a company with discount free cash flow analysis

One way to value a company is add up all the cash it will produce in the future. This is similar to how someone value a house by adding up the rents the house can collect. After collecting the future cash flow, a small adjustment is needed. The cash generated in the future needs to be adjusted to get its equivalent value in today’s value or present value. The reason is a dollar that’s available today is worth more than a dollar that might be available sometime in the the future. The future dollar has risk that it might not be there. There is also opportunity cost of that dollar. A dollar today can produce work and start earning risk free interest right away. A person can invest today’s dollar in the risk free 10 years US bonds. The cash in the future therefore has to be discounted to account for the risk of not being there and the lost interest by not having the cash today.

Using a discount rate of 10% means that the value of having that cash today is worth 10%. A lower discount rates can be used.

Present value of all future free Cash flow

A company will produce cash flow for many years. The ability to predict beyond 5 or ten years is limited. This method will predict the cash flow up to 10 years assuming a constant growth rate. After year ten, the growth rate will change to a lower rate that is closer to the rate at which the economy grows, the US GDP. At that point, we assume it will generate cash forever and grow at that terminal growth rate and calculate that as the perpetuity cash flow. The present value of all that cash flow can be summed up to arrive at a valuation by dividing it by the total outstanding shares.

Toal cash flow = cash flow present value year1 + cash flow present value year 2 + … case flow present value year 10 + present value perpetuity cash flow

Deriving the formula for present value.

In this section we will derive the formula to calculate the present value. In following example, we have $100 and earning 10%. The cash can earn the following stream going in the future. We will analyze this for 2 years to see what the future value is after year 2.

Year 0 = $100

Year 1 = Year 0 + (gain for Year 0) = Year 0 + (Year 0 x .10) = Year 0 (1 + .10) = Year 0 x 1.10.

Year 2 = Year 1 * (gain for Year 1) = Year 1 * (Year 1 * 1.0) = Year 1 (1 + .10) = (Year 0 x 1.10) x 1.10 = Year 0 x 1.10^2

$100 year 0—> $110 year 1 –> $121 year 2

The value at year 2 can be simplified to $100 x 1.10 ^ 2

In the above example, after year 2, the value is $121. So if we had $121 in the future and we want to see what was the present value that produced $121 at a 10% rate. We can go backwards by reversing the x 1.10. The inverse of multiplying by 1.10 is divide by 1.10.

121 year 2 -> 121 / 1.10 year 1 -> (121 /1.10)/1.10 -> year 0.

121 -> 110 -> 100

We see that the future value of $121 has a present value of $100 with a discount rate of 10%.

From that exercise we can generalize an equation to obtain the present value given a future value and a discount rate or risk free return rate.

\[PresenValue = \frac {ValueInFuture} {(1 + DiscounRate)^n} \]

Let’s calculate the $100 present value in 1 year with a discount rate of 10%

$100/(1 + .10) = $100/1.10 = $90.91

Calculating the perpetuity cash flow value

It’s unlikely that a company can maintain its present growth rate after ten years and it is also difficult to arrive at a growth rate prediction so far into the future. If a company is healthy into year 10, the company will likely still grow. We can assume that it can grow at the rate of the economy and that it can maintain that rate forever. The perpetuity or terminal cash flow value is the expected cash flow after year 10 onward forever. The formula is:

\[perpetuity = \frac {FreeCashFlowYear10 \times (1 + g)}{ (DiscountRate -g)}\]

g is the expected growth rate after year 10 or the terminal growth rate.

The formula is obtained by mapping the sum of cash flow to the geometric series:

\[ total = {a + ar + ar^2 + ar^3 + … + ar^n} = \frac {a} {(1-r)} \]

With the requirement that r < 1

The proof is obtained by forming a difference that will subtract out the middle terms.

\[s = a + ar + ar^2 + ar^3 + … ar^{(n-1)}\] \[rs = ar + ar^2 + ar^3 + .. ar^n\]

Then

\[s – rs = a – ar^n \]

because the middle terms are common and subtracted out.

Factor the common terms:

\[s(1 -r) = a(1-r^n)\]

Solve for s:

\[s = \frac{a(1-r^n)}{(1-r)}\]

If we take the limit as n approach infinity AND r is less than 1, the r^n term becomes so small that it becomes 0.

\[ s = \frac {a(1 – 0)} {(1-r)} = \frac {a} {(1-r)}\]

The present value of perpetual cash flow is expressed as:

\[perpetuity = \frac{C}{(1+d)} + \frac{C(1+g)}{(1+d)^2} + \frac{C(1+g)^2}{(1+d)^3} …\]

C is the cash flow at year 11 = cash flow at year 10

We can map the perpetuity sum to the geometric series with the following substitution. Recall that the geometric series is:

\[ total = {a + ar + ar^2 + ar^3 + … + ar^n} = \frac {a} {(1-r)} \]

Substitute:

\[a = \frac {C}{(1+d)}\] and \[r = \frac {(1+g)}{(1+d)}\]

Then the perpetuity converge to

\[perpetuity = \frac{a}{(1-r)} \] \[= \frac {\frac {C}{(1+d)}} {1 – \frac {(1+g)}{(1+d)}} \]

Let’s simplify the denominator:

\[1 – \frac {1+ g}{1+d} = \frac {1+ d}{1+d} – \frac {1+ g}{1+d} = \frac {(d – g)}{(1+d)} \]

Then putting that back to the expression:

\[s = \frac {\frac {C}{(1+d)}} {\frac {(d-g)}{(1+d)} } = \frac {C}{(1+d)} {\frac {(1+d)}{(d-g)} } = \frac {C}{(d-g)} \]

With C being the free cash flow at year 11 is,

\[ C = FreeCashFlowYear10 \times (1+g) \]

The perpetual cash flow is

\[ Perpetuity = \frac {FreeCashFlowYear10 \times (1+g)} {(d-g)} \]

The perpetuity needs to adjust the cash flow at year 10 to present value so we divide by (1 – discount_rate)^10.

\[PresentValuePerpetuity = \frac {perpetuity} { (1 – DiscountRate)^{10}} \]

d = discount_rate

Intrinsic value

Adding up all the cash flow the company can expect to generate and divide by the total shares outstanding gives an estimate of the present day share price.

intrinsic value = (total_cash_flow to year 10 + present value perpetuity )/total shares outstanding

\[intrinsic value = \frac {TotalFreeCashFlowEstimateFor10 + PresentValuePerpetuity}{TotalShares} \]

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