## The basics of Net Present Value

You’ve probably heard of the Stanford Marshmallow experiment. Researchers offered children a piece of their favorite candy with two options: to eat it right now, or to wait for fifteen minutes and receive an extra piece of candy. Supposedly, the children who were able to delay gratification grew up to be more successful than the ones who ate their candy right away.

Why am I telling you this? Because I want to talk about Net Present Value, which is a tool that helps you decide which option is best: to have one marshmallow today, or to have two tomorrow. Here’s a nice video introduction.

Now let’s make you the subject of a similar experiment, but no marshmallows involved: would you rather receive \$100 today, or wait seven years and get \$150 instead? You can solve this question using the NPV method. But before we start calculating, you need to answer another question first: what would you do with the \$100 if you choose to take it right now?

#### Punishment

For many people, the only option available for investing this newly won \$100 is to put it in a savings account. That would increase its value by 1,50% annually. But if you’re a wealthy person, you might hand the \$100 over to your hedge fund manager who is able to return 8% on it. Or if you own a profitable company, you might add this \$100 to your working capital and receive 15% ROCE (Return On Capital Employed).

Why is this important? Because the more return you can get on your money, the more leverage you have to earn additional money. You are punished if you have to wait for your money. The punishment is that you cannot use the money until you’ve received it. The amount of the punishment is the opportunity cost of not being able to invest right away. That’s why \$100 tomorrow is worth less than \$100 today. It’s why you get a discount if you pay cash. It’s why you have to pay interest on a car loan.

We’ve learned two things:
1 – future money is worth less than today’s money and
2 – how much less depends on your available alternative investment opportunities

#### Interest

By now, I can hear you guys thinking: but where’s the math? Give me the math! So let’s get on with it. What would \$100 earn me if I invest it at 5% return over 7 years? We could calculate this with the formula for compound interest:

In this formula, FV is the future value, PV is the present value, the interest rate is i and n is the number of years. So for our example:

So the future value of \$100, seven years from now, at an interest rate of 5%, is \$140,71. I asked you at the start of this post if it would be better to receive \$100 today, or to get \$150 after seven years. Since \$100 today equals \$140,71 in seven years, it is clearly a better option to choose to receive \$150 in seven years, assuming that you cannot do better than an annual return of 5%.

#### Opportunity

But you’re still not telling me how to calculate Net Present Value! Yes, you’re right. Fortunately, that is really easy. Present value is simply the inverse of future value, so we re-arrange the formula for compound interest like this:

The difference is that when we’re calculating present value, i doesn’t represent interest. It’s called the discount rate. The discount rate is nothing more than a way of saying: I want my money to do better than X. If you’re not very ambitious and you just want your investments to keep pace with inflation, you set X as the inflation rate. For investments that beat inflation, the future value will be higher than the present value. For investments that can’t beat inflation, the future value will be lower than the present value. Or X could represent an opportunity cost, like an alternative investment.

If we apply this formula for calculating present value to our example:

This means that the present value of receiving \$150, seven years from now, at a discount rate of 5%, is \$106,60. A present value of \$106,60 is better than receiving \$100 immediately (which has a present value of \$100, duh), so this confirms that it’s better to receive \$150 seven years from now, than to get \$100 immediately.

#### Vending Machine

So now we know how to calculate future value using the compound interest formula, and we know how to calculate present value using discount rate. What about Net Present Value? This is simply the sum of all present values for a series of future cash flows. This sounds pretty abstract, but it’s really quite easy if I give you an example.

Let’s say you have the opportunity to buy a vending machine. It will cost \$2500, and you expect a profit of \$750/year. After 10 years in operation, it will be worn out and useless. Your cash flow stream looks like this:

Over 10 years, you make \$7500 profit in operations, from an initial investment of \$2500. So if you’re not discounting for time, you could say that the net profit from this investment is \$5000. If we want to know the Net Present Value of this investment, the first thing we have to do is discount all cash flows for the different years to their present value. I’m not going to repeat the formula, you’ve seen how it works. I add this formula in a row below the cash flow values, so you can see their discounted present values. I’m using a discount factor of 5%.

You can see that the present value of \$750 after one year is \$714,29. The present value of receiving \$750 ten years from now, at a discount rate of 5%, is only \$460,43. The Net Present Value is the sum of all present values over the 10-year life of this investment. You can see that it amounts to \$3291,30. It means that doing this project should be worth exactly the same to you as receiving \$3291.30 today.

#### Comparison

Right now, you might be wondering what the point of all this is. Well, NPV really comes to life when you use it to compare different investment opportunities. In the three tables below, I’ve listed the cash flow streams for three different projects. All three of them have the same total cash flow: +\$100. The first one requires an immediate investment of \$100, and then pays \$20 per year for ten years. The second one requires no investment, but pays only \$10 per year. The third scenario requires you to invest for three years before you start to see any profits, but the profits are much higher. Quickly, which one do you think is the best investment?

Now, let’s calculate the NPV’s of these projects:

Scenario 2 is by far the best, with an NPV of \$77,22. Scenario 3 is the worst, with an NPV of \$16,87. Was this the result that you expected? This example illustrates how dramatic the effect of the distribution of cash flow in time can be. There is one lesson that you learn very quickly when you’re doing NPV calculations: you have to be very careful to give up current capital in return for future gains. The numbers show no mercy for your investment plans. The compound effect of discounting profits in the future is huge, and it gets worse if your discount rate is higher.

#### Apartment

I’ll give you one more example. Let’s say you buy an apartment for \$200.000, including all transaction costs, a new paint job etc. Your rent it out for \$750/month, which gives you \$9.000 per year in revenue. You have \$1.000 in costs per year, so your profit is \$8.000 per year. After 20 years, you sell it for \$275.000. Should you make this investment? Let’s do the math:

The Net Present Value of this investment is very close to zero. At a discount rate of 5%, it’s a breakeven operation, and basically a waste of time.

Of course, this model I’ve created is very simple. A real NPV-model will take into account many more effects like taxes, operational costs, insurance, depreciations etc. But the basics remain the same: sum up all the cash flow effects in a given year, do this for every year in the project life, discount the cash flow in each year and sum them up over the project life.