### What is Net Present Value?

In short, Net Present Value, or NPV is a method used to calculate the value of something. Typically used by managers for capital budgeting, NPV can also be used in a variety of other circumstances, like deciding between two schools for your MBA. The formula for NPV is relatively straight forward, it is simply the sum of all the future cash flows, discounted by the appropriate rate.

NPV = Σ (Year n Total Cash Flow)/(1+Discount Rate)^{n}

To understand NPV, you must first understand several fundamental concepts in finance, namely time value of money, rate of return and present value. We will briefly review each concept below.

#### What is Time Value of Money?

Most people intuitively understand that money today is worth more than money tomorrow (putting $100 in a savings account today gets you $101 next year, therefore $100 today > $100 next year). This concept is formally known as time value of money, it is closely related to rate of return.

#### What is Rate of Return?

Rate of Return, normally denoted by *r*, is the payoff of an investment as a percentage of the investment. Going back to our previous example, the savings account would have a rate of return of 1% ($101 – $100)/($100). Rate of return takes on many forms, a few being dividend yield, rental yield or coupon rate.

#### What is Future Value?

Future value is the value of present cash at some point in the future. As per our previous example, the future value of $100 in the next year is $101. The formula for future value is

FV = PV * (1 + r)^{n}

Where *PV* is the present value ($100), *r* is the rate of return (1%) and *n* is number of periods, normally in years.

#### What is Present Value?

Present value is the value of something today, given its value at some point in the future. The formula for present value is simply a modified version of the formula for future value

PV = FV / (1 + r)^{n}

As per the example above, the present value of $101 a year from now would be $101 / (1 + 0.01)^{1} = $100, where the future value is discounted by the discount rate (1% here). A more common application of present value is figuring out how much future cash flows is worth now, and to do that we must modify the formula above to the following

PV = CF_{1 }/ (1 + r_{1}) + CF_{2} / (1 + r_{2}) … CF_{n} / (1 + r_{n})

Where *CF _{n}* is the cash received or paid at period

*n*, and they are discounted by an appropriate discount rate

*r*. This is a fundamental formula in finance, a modified version of it is used to value bonds and dividend paying stocks.

### Calculating Net Present Value

Net Present Value is simply a special case of Present Value, where the upfront cost is included in its calculations (hence ‘net’). The method is very simple, you

- Calculate the PV from projected cashflows then
- Subtract the initial investment

Remember the initial investment does not need to be discounted (unless it is allocated over multiple years).

NPV = -CF0 + PV = -CF0 + CF_{1 }/ (1 + r_{1}) + CF_{2} / (1 + r_{2}) … CF_{n} / (1 + r_{n})

As a student, it is important to memorize this formula (it always ends up on exams). After college, no one does it by hand. Excel has nicely wrapped the formula into the function *NPV()*. Financial calculators such as the BA II Plus (used for the CFA exams) also include a NPV function.

### Examples of Net Present Value

#### Example 1 – Capital Budgeting

The most common application of Net Present Value is for capital budgeting. In an unconstrained environment (enough money to accommodate all projects), every non mutually exclusive project with a positive NPV should be accepted. In a more realistic, budget-constrained environment, projects should be prioritized based on decreasing NPV (highest NPV projects should be accepted first).

*Take 3 hypothetical projects: Project A costs $850 today and yields $300 a year for the next 5 years, Project B costs $450 today and yields $150 a year for the next 5 years and Project C costs $550 today and yields $200 a year for the next 5 years. The discount rate is 5%, there is a budget constraint of $1000 and the projects are not mutually exclusive. Which projects should be accepted?*

The easiest thing to do here is to calculate the NPV for each project on an Excel spreadsheet. If you are in school or if you are studying for a CFA exam, do the NPV calculations on a calculator instead and the answers should match.

Project A | |||||

Year | Cash Flow | Discount Rate | Discount Factor | Present Value | Net Present Value |

0 | -$850 | 0.00% | 1 | -850 | 448.82 |

1 | $300 | 5.00% | 0.9524 | 285.72 | |

2 | $300 | 10.25% | 0.907 | 272.1 | |

3 | $300 | 15.76% | 0.8638 | 259.14 | |

4 | $300 | 21.55% | 0.8227 | 246.81 | |

5 | $300 | 27.63% | 0.7835 | 235.05 |

Project B | |||||

Year | Cash Flow | Discount Rate | Discount Factor | Present Value | Net Present Value |

0 | -$450 | 0.00% | 1 | -450 | 199.41 |

1 | $150 | 5.00% | 0.9524 | 142.86 | |

2 | $150 | 10.25% | 0.907 | 136.05 | |

3 | $150 | 15.76% | 0.8638 | 129.57 | |

4 | $150 | 21.55% | 0.8227 | 123.405 | |

5 | $150 | 27.63% | 0.7835 | 117.525 |

Project C | |||||

Year | Cash Flow | Discount Rate | Discount Factor | Present Value | Net Present Value |

0 | -$550 | 0.00% | 1 | -550 | 315.88 |

1 | $200 | 5.00% | 0.9524 | 190.48 | |

2 | $200 | 10.25% | 0.907 | 181.4 | |

3 | $200 | 15.76% | 0.8638 | 172.76 | |

4 | $200 | 21.55% | 0.8227 | 164.54 | |

5 | $200 | 27.63% | 0.7835 | 156.7 |

With a budget constraint of $1000, you can either accept Project A or Projects B and C. The NPV for Project A is $448.82 and for Projects B and C is $199.41 + $315.88 = $515.29. As the NPV for Projects B and C is higher than Project A, Projects B and C should be accepted.

#### Example 2 – School Selection

Another application of Net Present Value is for decisions on education/schools. In this case, your future wages would be the cash flow, and your tuition would be the initial investment. Our example below is somewhat simple, you can add as much complexity as you want to refine the model.

*An 25 year old auditor making x a year (Present Value of future wages = $1,450,000) is looking to break into finance, and is accepted to an MBA at University A and MFE at University B. The MBA tuition is $50,000 a year for 2 years, and he will make y a year (Value of future wages at end of year 2 = $1,900,000) as an investment banker afterwards. The MFE tuition is $10,000 a year for 2 years, and he will make z a year (Value of future wages at end of year 2 = $1,850,000) as a trader afterwards. Assume 5% discount rate and the tuition is paid at the beginning of each year, what path should he choose?*

We are given the Present Value and Future Value of the expected wages from each option, we can plug that into our model as a lump sum payment. The three options is this example are

- Stay as an auditor (PV = $1,450,000)
- Complete MBA then go into investment banking (CF0 = -$50,000, CF1 = -$50,000, CF2 = $0, CF3 = $1,900,000)
- Complete MFE then go into trading (CF0 = -$10,000, CF1 = -$10,000, CF2 = $0, CF3 = $1,850,000)

Throwing these numbers into a spreadsheet gets us the following

Audit | |||||

Year | Cashflow | Discount Rate | Discount Factor | Present Value | Net Present Value |

0 | $1,450,000 | 0.00% | 1 | 1450000 | 1450000 |

Investment Banking | |||||

Year | Cashflow | Discount Rate | Discount Factor | Present Value | Net Present Value |

0 | -$50,000 | 0.00% | 1 | -50000 | 1543600 |

1 | -$50,000 | 5.00% | 0.9524 | -47620 | |

2 | $0 | 10.25% | 0.907 | 0 | |

3 | $1,900,000 | 15.76% | 0.8638 | 1641220 |

Trading | |||||

Year | Cashflow | Discount Rate | Discount Factor | Present Value | Net Present Value |

0 | -$10,000 | 0.00% | 1 | -10000 | 1578506 |

1 | -$10,000 | 5.00% | 0.9524 | -9524 | |

2 | $0 | 10.25% | 0.907 | 0 | |

3 | $1,850,000 | 15.76% | 0.8638 | 1598030 |

The NPV for trading appears to be the highest, therefore he should go with the MFE.

Statistics 101 | · · · |
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