## Present Value

**Present Value:** the value today of a set of anticipated cash flows future.

\[\begin{eqnarray}
PV &=& \frac{CF_t}{(1+r)^t}
\end{eqnarray}\]

\[\begin{eqnarray}
CF_t &=& \text{cash flow at any given time $t$}\\
r &=& \text{the higher the risk $r$, the higher the discount rate.}
\end{eqnarray}\]

**Net Present Value:** the present value of acquiring the asset – the cost of acquiring the asset (negative cash flow) at \(t\) = 0.

\[\begin{eqnarray}
NPV = \Large \sum_{t=0}^N \normalsize \frac{CF_t}{(1+r)^t} &=& CF_0 + \Large \sum_{t=1}^N \normalsize \frac{CF_t}{(1+r)^t}
\end{eqnarray}\]

Though the concept of opportunity cost is omnipresent in the study of Microeconomics, ceteris paribus, the exact financial cost of any endeavor must be met with the value(s) of the ensuing alternative(s).

Going further, if we are to look at the value of an investment, we must look at its return and compare it against other feasible investment alternatives. From a purely technical standpoint, adjusting the discount rate *r* will obviously affect the net present value.

If a disciplined investor was to consider two different investments with an equal amount of risk and forego one investment’s rate of return for another, this is the cost of capital of the investment decision, or, once again, opportunity cost.

As we cover valuation, we will see that some returns do not stack up against the company’s cost of capital, thereby increasing risk, and ultimately decreasing valuation.

Finance, as opposed to theoretical economics delves deeper into opportunity costs and quantifies these costs as real dollar figures. In the ensuing excel demo, we will show how NPV is calculated step by step.

## Example 1

Calculate Present Value where Cash Flow is constant at time *t.*

In this example, assuming the cash flow is static at time *t*, we can use Excel’s PV() function to calculate NPV; however, Excel’s PV() function **CANNOT** be used to calculate NPV when the cash flows vary across time *t*.

A | B | C | D | |
---|---|---|---|---|

1 | ||||

2 | ||||

3 | Discount Rate | 0.03 | ||

4 | ||||

5 | Year | Cash Flow | Present Value | Formula |

6 | 1 | $ 100.00 | $ 97.09 | <– =B6/(1+$B$3)^A6 |

7 | 2 | $ 100.00 | $ 94.26 | <– =B7/(1+$B$3)^A7 |

8 | 3 | $ 100.00 | $ 91.51 | <– =B8/(1+$B$3)^A8 |

9 | 4 | $ 100.00 | $ 88.85 | <– =B9/(1+$B$3)^A9 |

10 | 5 | $ 100.00 | $ 86.26 | <– =B10/(1+$B$3)^A10 |

11 | ||||

12 | NPV | $ 457.97 | <– =SUM(C6:C10) | |

13 | NPV(Excel Function) | $ 457.97 | <– =NPV(B3,B6:B10) | |

14 | PV | $ 457.97 | <– =PV(B3,5,-100) | |

15 | ||||

16 |

## Example 2

Calculate Present Value where Cash Flow changes over time *t.*

*t*= 0. From

*t*= 1 through

*t*= 5, cash flow increases by $100.00/ year. We calculate Present Value for each year starting in cell C6 , by using the formula =B6/(1+$B$3)^A6 =

*PV*= \(\frac{CF_t}{(1+r)^t}.\)

A | B | C | D | |
---|---|---|---|---|

1 | ||||

2 | ||||

3 | Discount Rate | 0.03 | ||

4 | ||||

5 | Year | Cash Flow | Present Value | Formula |

6 | 0 | $ (250.00) | $ (250.00) | <– =B6/(1+$B$3)^A6 |

7 | 1 | $ 100.00 | $ 97.09 | <– =B7/(1+$B$3)^A7 |

8 | 2 | $ 200.00 | $ 188.52 | <– =B8/(1+$B$3)^A8 |

9 | 3 | $ 300.00 | $ 274.54 | <– =B9/(1+$B$3)^A9 |

10 | 4 | $ 400.00 | $ 355.39 | <– =B10/(1+$B$3)^A10 |

11 | 5 | $ 500.00 | $ 431.30 | <– =B11/(1+$B$3)^A11 |

12 | ||||

13 | NPV | $ 1,096.85 | <– =SUM(C6:C11) | |

14 | NPV(Excel Function) | $ 1,096.85 | <– =B6+NPV(B3,B7:B11) | |

15 | ||||

16 |

## Example 3. Calculate Interal Rate of Return (IRR).

In this example, we calculate the IRR by using Excel’s built-in IRR function IRR(values, [guess]) . IRR (the internal rate of return) is the rare of return where NPV = 0 . The higher the IRR, the healthier the investment.