# The net present value (NPV) allows you to evaluate future cash flows based on present value of money

The net present value (NPV) is the **sum of present values** of money in different future points in time. The present value (PV) determines how much** future money is worth today**. Based on the net present valuation, we can **compare** a** set of projects/ investments** with different cash-flows over time. This enables us to quantitatively assess a business' **attractiveness** using a benchmarking of NPVs.

## The closer future cash flows are to the present the more valuable your money is

The concept is also known as **time value of money **and we provide two explanations below:

- An
**intuitive**explanation: People will prefer money at present because of**risk aversion**. Would you rather have $100 today or in a year? Obviously today, because there is a risk that you may not get that $100 in a year. In addition, once you have the money, you again have a decision to either spend it right away or wait with spending it. - A
**financial**explanation: Imagine you have $100. How much is it worth in a year? If you do not leave the money in your pocket, you usually have the option to put the money in your bank account at a low and almost negligible risk. You will**earn interest**but may lose value due to inflation. However, the inflation adjusted interest rate may be 2%, in absolute terms $2. In total your $100 is worth $102 after one year. Now, you can calculate backwards: If you have a future value of $102 in a year, how much is it worth today? It is $102 divided by 1.02 which results in $100 again.

## How to calculate net present values

- PV is the present value
- FV is the future value
- i is the decimal value of the interest rate for a specific period
- n is the number of periods between present and future

**The following is the calculation**** of the above PV example** with $102 future value at an interest rate of 2%,

**Below you can find a**** slightly different version** of the above example, in which you receive $102 in **two** years instead of next year. The two year investment earns you a theoretical interest two times, which is why you discount twice.

**The Net Present Value** of those two $102 payments in one and two years is simply its sum.

## Apply NPV shortcuts to succeed in case situations

It’s **unlikely** that you will need to **calculate a complex NPV** during a case interview because the calculations tend to get overly complicated. But in some cases you can apply some shortcuts as discussed below:

**1) Perpetuity: the NPV for infinite cash flows (meaning business will generate profits for an infinite period of time)**

For infinite cash flows, there is a simplified formula:

Imagine you have to __value a company__ in a case interview. A common approach is to define the value of a company as the **sum of all its discounted future profits**. If you assume that a company will have the same profits every year for an indefinite time horizon, you just divide the future value of all profits by the respective discount rate. For instance, if you expect the company to yield $100 every year, the company is worth $2500 (at a discount rate of 4%).

To make this more pragmatic, you could assume that the company's profits will **grow every year** at a certain rate g.

Especially for short-term horizons, defining expected growth is difficult. An approximated growth rate for profits that are far into the future is often around **2%**.** ** After a while, every business or product life cycle ends up in a **competitive market environment** and simply grows at the **same rate as the overall economy**. The above example recalculated with a continuous growth rate of 2% results in a **net present value** of $5000 for the company.

Notice that the value is** twice** the value compared to the calculations without a growth. NPV calculations are very **sensitive** towards changes of inputs. Therefore, a sensitivity analysis is conducted in most cases. To do so you need to create a range of possible NPVs by using a range of possible growth and discount rates.

**2. Find the right interest rate i**

Finding the correct discounting factor for NPV calculations is the business of entire banking departments. In general, there is one basic rule: **the bigger the risk the higher the discount rate.**

The rationale behind this rule is simple: The less you can be sure about receiving future earnings, the less you value them. By **increasing the discount rate,** the **NPV** of future earnings will **shrink**. Discount rates for quite secure cash-streams vary between 1% and 3%, but for most companies, you use a discount rate between 4% - 10% and for a speculative start-up investment, the applied interest rate could reach up to 40%. In case interviews, you could ask for the discount rate directly or estimate it at 10% for most scenarios if the interviewer requests you to approximate it.

## Key takeaways

- Use
**NPVs to evaluate future cash-flows**in today’s time value of money. - By calculating
**risk-adjusted NPVs,**you can quantitatively**compare different investments**. - NPVs are used
**to value a company**based on its future profits.

__Gravestone case__, an interviewer led case which includes a company valuation

Can someone please explain better solution for problem 4?To me it depends on how the money is delt when I receive since I don't have interest rate or anything to evaluate it. Let's say I put them in a bank once that I get them.

If the interest rate that I get is lower than 4% than solution 2 is the best (after 12 years I get 143M$ with solution 1 and 146M$ with solution 2).

If the interest rate that I get is equal or higher tan 4%, then the case 1 is the best. Assuming an interest rate of 4%, I get 159M$ with solution 1 and 156M$ with solution 2.

How the NPV is used here? I would have used a compoung growth rate evaluating the interests from the bank.

The right answer for #5 is 2.9% IRR. You can verify it by plugging the IRR formula in excel.

I did not get the factor, (1/IRR - 1/(IRR*(1+IRR)^11), I though it was (1/IRR - 1/(IRR*(1+IRR)^12), because it is said in the question "for 12 years"

Dear Julia,

thanks for the answer. However, I have difficulties understanding clearly where the factor 1/(1+i) comes from. The only way for me to explain it is to find a certain arbitraryness in the definition of "coming year". This could mean that the cashflow of the first two million should not be discounted. It could also mean that I assume the cashflow to be dated to the 31.12. of a year and shoud therefore be discounted to (almost one year) in the future. Do you see what I mean. Let me give an example, assuming that today is the 1.1.2017

case1:

date cashflow discounted cashflow

1/1/2017 2.0m 2.0m (not discounted, as it occurs today)

1/1/2018 1.92m 1.75m (discounted by one full year)

1/1/2019 1.84m 1.52m (discounted by two full years)

and so on...

====================================================

case 2

31/12/2017 2.0m 1.82m (discounted by one year (or 364 days))

31/12/2018 1.92m 1.59m (discounted by two full year (minus one day))

31/12/2019 1.84m 1.38m (discounted by three full years (minus one day))

and so on...

====================================================

These two cases differ in the geometric series exactly be the additional factor of 1/(1+i), as our results do. Since I do not have a very strong BA background, I would be happy to understand why the assumption behind case 2 seem to fit the question better.

The way I see it, this difference can depend on the real business case at hand. Is the pipeline user paid for his services for the whole year in advance or at the end of the year or "continuously" (say on a weekly or montly basis)? In the latter case, the answer should be in between (like approximately discounting to the 30th of June of each year).

Anyway thanks for the lively discussion

Max

Hi Max,

Thanks a lot for your answer. The technically correct formula for calculating the NPV in case of infinite series of future cashflows with growth rate g and discount rate i is NPV = (CF/(1+i)) / (1 - (1+g)/(1+i)).

This means that the exact NPV in question 3 of our quizz is 14,3 which is similar to the approximation. This means that the best answer in the quizz should be 14m.

I hope this helps!

Best,

Julia from PrepLounge

Another small comment to infinite series of future cashflows with growth rate g and discount rate i. The correct formula (geometric series) should be

NPV= CF/(1-(1+g)/(1+i)) where CF is the cashflow in the first year, like the 2m in the pipeline case. For small enough g and i, this can be Taylor expanded to find

NPV = CF/(i-g) as stated above.

However, calculating the technically correct solution, gives 15.7mn while the approximation gives 14.3m. So the best answer in the quiz should be 16m, not 14m. I understand that the approximation will be the best in most case interviews if one is not super quick with numbers. Still, marking the best fitting answer in the quiz as wrong, just because it does not fit the approximation, is not very informative, in my humble opinion.

Discount rate is a more general term which can refer to an interest rate (financial explanation) but may also take into account the expected risk or uncertainty of projects or investments (intuitive explanation).

In the shortcut section we use the more general term discount rate because it is our aim to provide you with a basic formula for calculating the NPV.

The formula for calculating the NPV for infinite cash flows and a constant growth rate is only valid if the growth rate g is maller than the discount rate i. This should usually be the case since the growth rate and the discount rate build on the same factors, i.e. the growth rate of the economy and the expected inflation rate.

Is i the discount rate or the interest rate throughout the explanation? It states it is the interest rate in the first equation then states it is the discount rate in the shortcut equation for calculating NPV for perpetuity.

Also if: NPV = FV / (i - g)

and g is greater than i

then NPV is negative. How does that work?

Hi Wyzek,

Thank you for pointing this out here and providing a link for more details. May I know whats your background in? Our understanding is that, knowledge of rule of 72 could be expected from candidates with business background particularly finance/accounting etc. who are familiar with these terms on a more regular basis.

It's true that NPV almost never comes up. Once however, I had a case where a candidate had to apply a rule of 72 (http://www.investopedia.com/video/play/rule-of-72/). The task was to calculate the present value of revenues if their reception is delayed 6 years and WACC=12%. 72/12=6 hence 50% future value, see link