# 4.3. FCFF for multiple periods and a restricted residual value

While discussing about business valuations, there is often a common denominator: that businesses do not last through eternity and that investors do not really care much about the cash flows that the business may generate past a certain number of years. While that is not true for large and public-listed companies, for small and medium sized businesses it is a recurring critic to valuations. Restricting the number of years of the residual value is an answer to this issue. This approach follows the same logic of the FCFF for multiple periods and a terminal or residual value; the difference is in the constraint of the residual value.

- \(Business\ value=NPVFCFF\ +Residual\ value\)

- \(NPV\ FCFF=\frac{{FCFF}_t}{{(1+WACC)}^t}+\frac{{FCFF}_{t+1}}{{(1+WACC)}^{t+1}}+\ldots+\frac{{FCFF}_n}{{(1+WACC)}^n}\)

- \(Residual\ value=\frac{{FCFF}_{n+1}}{\left(WACC-g\right)}\frac{1}{\left(1+WACC\right)^n}-\frac{{FCFF}_{n+x}}{\left(WACC-g\right)}\frac{1}{\left(1+WACC\right)^{n+x}}\)

The residual value equation comprises two perpetuities. The second perpetuity acts as a cut to the first perpetuity. Hence, the business value only comprises the years of forecast and the years between the two perpetuities.

Restricting the residual value embodies a disbelief that the company will generate results into perpetuity or that relevant value will only be created within a given timeframe. Although it is not common to find this approach at theoretical and academic levels, in practice it may be relevant for businesses whose activity or industry have significant volatility and/or where there a high rate of business mortality.

###### Example – 5-year NPV of FCFF and a 10-year residual value

Pascal has a startup project and he was able to estimate the FCFF for the five first years of business. At the sixth year, he expects the company to be within a stable position and that it will only grow a 0,5% real rate. These are the FCFF of the three first years that he has estimated:

Year | 1 | 2 | 3 | 4 | 5 |

FCFF | -150.000 EUR | -10.000 EUR | 35.000 EUR | 60.000 EUR | 70.000 EUR |

Pascal thinks that the business will be in the maturity stage for 10 years. After those 10 years, he believes that technogic improvements will bring the industry to other standards and the company will enter a declining stage and that the FCFF of such period is not relevant.

The WACC of Pascal’s company is 8,5%. The estimated average inflation rate is of 2,5%.

Year | 1 | 2 | 3 | 4 | 5 |

FCFF | -125.000 | -10.000 | 45.000 | 60.000 | 70.000 |

WACC | 0,085 | 0,085 | 0,085 | 0,085 | 0,085 |

Discount factor | 1,085 | 1,177225 | 1,277289 | 1,385859 | 1,503657 |

Accumulated FCFF | -125.000,00 | -135.000,00 | -90.000,00 | -30.000,00 | 40.000,00 |

Discounted FCFF | -115.207,37 | -8.494,55 | 35.230,86 | 43.294,46 | 46.553,18 |

Accumulated discounted FCFF | -115.207,37 | -123.701,93 | -88.471,06 | -45.176,60 | 1.376,57 |

The NPV of the five-year forecast is 1.376,57 EUR.

To compute the residual value, you need an estimate on the FCFF_{6} and the FCFF_{15}. The growth rate is:

- \(g=\left(1+inf\right)\left(1+real.g\right)-1\)
- \(g=\left(1+0,025\right)\left(1+0,005\right)-1=0,030125\)

Now that you have growth rate, estimating the FCFF_{6} and FCFF_{15} is easy:

- \({FCFF}_6=70.000\times(1+0,030125)=72.108,75\)
- \({FCFF}_{15}=70.000\times{(1+0,030125)}^{10}=94.188,37\)
- \(Residual\ value=\frac{72.108,75}{(0,085-0,030125)}\frac{1}{\left(1+0,085\right)^5}-\frac{94.188,37}{\left(0,085-0,030125\right)}\frac{1}{\left(1+0,085\right)^{10}}=369.039,37\)

The business value is:

- \(Business\ value=1.376,57+369.039,37=370.415,94\)

Next Section: 4.4. Business value, company value and equity value