# PV NPV

PV ( Present Value)

Present Value **PV**: Assuming that $1,000 tomorrow will have more value to you than $1000 in 5 years we need to find a way to compute what the actual value of $1,000 in 5 years is to us now. The formula is

PV = FV ( Future Value ) / ( 1 + Interest ) ^ n

To make the calculation simple let’s test the value in 3 years with a 4% interest

PV = $1000 / ( 1 + 0.04 ) ^ 3 = $1000 / 1.04 ^3 = $1000 / 1.12 = $892

We can calculate the PV of 1000 a year from now, add to the PV of $1000 two years from now and add it up to the value of $1000 three years from now – A more elegant formula is PV=β FV / (1+r)^t for t=1 to n

Example we have to choose between $1000 now and $200 a year for 6 years starting now:

Option 1 Today | $ 1,000 | ||||||

Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

Option 2 Cash Flow | $ 200 | $ 200 | $ 200 | $ 200 | $ 200 | $ 200 | |

Discount Rate | 12.0% | ||||||

Discounted Cash Flow | $ 200 | $ 179 | $ 159 | $ 142 | $ 127 | $ 113 | |

PV | $ 921 |

# ( Net Present Value )

The **net present value** is simply the present value of all future cash flows, discounted back to the present time at the appropriate discount rate, deducting the cost to acquire those cash flows. In other words, **NPV **is simply **the value now minus the cost now**.

Net present value (NPV) is determined by calculating the costs (negative cash flows) and benefits (positive cash flows) for each period of an investment, discounted back to the present time at the appropriate discount rate. The NPV is the sum of all the discounted future cash flows.

The NPV function in which the first payment is made at the **end** of the first period. Therefore, this value is included as the first value1 argument to the NPV function.

*NPV= β **{** Cash Flow / (1+**r**)^**t**}** *

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*The NPV function in which the first payment is made at the start of the first period.*

*NPV= β **{** Cash Flow / (1+**r**)^**t**}** – Initial Investment*

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πππ = ππ (π΅ππππππ‘π ) β ππ(πΆππ π‘π )

πππ = ππ(π΄ππ πππππππ‘ πππ β ππππ€π )

There are 3 categories NPV will fall into:

Positive NPV. If NPV is **positive** then it means youβre paying less than what the asset is worth and signifies a project that has net positive benefits after accounting for the time value of money

Negative NPV. If NPV is **negative** then it means that youβre paying more than what the asset is worth and indicates the project will result in a loss for the investor

Zero NPV. If NPV is **zero** then it means youβre paying exactly what the asset is worth at that assumed rate of return.

We are proposed to invest $1000 and receive $500 for 3 years. The Discounted Rate is 5% – Is this a good deal ?

Example Below After the 1^{st} year we calculate the discounted cash flow

Year | Cash Flow | Disc Cash flow | Rate | 5% |

0 | $ (1,000) | $ (1,000) | ||

1 | $ 500 | $ 476 | =E5/(1+5%)^D5 | |

2 | $ 500 | $ 454 | =E6/(1+5%)^D6 | |

3 | $ 500 | $ 432 | ||

NPV | Sum=$500 | $ 362 | ||

# Payback Time

A project requires an initial investment of $3,000 upfront. It will produce an annual income of $500 every year for 6 years. What is this projectβs payback period without considering discounting?

Year 1 = $500 Total = $500

Year 2 = $500 Total = $1000

Year 3 = $500 Total = $1500

Year 4 = $500 Total = $2000 Payback Period is 2 years

A project requires an initial investment of $2,000 upfront. It will produce an annual income of $1,000 every year for 3 years. Assume a discount rate of 10%. What is the above projectβs discounted payback period?

Discounted year 1 CF = 1000/1.1=909

Discounted year 2 CF = 1000/1.1^2=826

Discounted year 3 CF = 1000/1.1^3=751

2000 – 909 – 826 = 264

264/ 751 = 0.352

2+0.352=$2.352

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