MODULE ON INVESTMENT SELECTION Article 3 THE RULE OF 72

This article short presents a mathematical curiosity that is very useful to calculate how long it takes for an investment to double in value given a certain rate of return. The only thing needed is to remember the number 72. Let's see.

 

If we expect a yield of 8% it will take nine years to double our money, just because 8 multiplied by 9 equals 72. That is, if we invest 100 now at an 8% return we will have 200 after 9 years. Similarly, if the return was 12%, six years would be needed to double our money, since 12 times 6 also make 72.

This simple rule assumes that the money invested is not touched until the end. That is, both the initial capital and the returns it produces are reinvested year after year until the original investment
is doubled.

The rule also works if we fix the number of years and want to know the corresponding return. For instance, what should be the return to double my money in 10 years? The answer is 7.2% (because 7.2 times 10 equal 72).

Interestingly, the rule does not have to be limited to money. For example: In how many years will the population of a country will double if population growth is 2% per year? It will take 36 years (2 multiplied by 36 make 72).

It is important to note that this is a rule of thumb that loses accuracy when yields rise above 16%. However it is still very practical when we want to do fast computations without resorting to a calculator.

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MODULE ON INVESTMENT SELECTION: HOW TO KNOW IF AN INVESTMENT IS BETTER OR WORSE THAN ANOTHER?

In our previous article (What Makes an Investment Attractive?) we concluded that a business will be more interesting the more money it produces, the quicker the money is received and when its return sufficiently compensates for the risk taken. In this piece I will explain the Net Present Value method which helps deciding whether or not an investment is attractive. This is the most important and most complex article of this module so it must be read carefully.

 

Suppose we have €100 to invest at a 12% yield over one year. This means that at the end of a year we will get €112, that is, our initial investment of €100 plus €12, which corresponds to the return on the initial investment (12% of €100).

Just as €112 is the amount you receive by investing €100 for one year with a yield of 12%, €100 is the amount you would have to invest at a 12% yield in order to get €112 at the end of a year.

If we call C the money to be received at the end of a year (€112), PV the invested amount (€100) and r the return (12%), the previous expression can be generalized as:

PV = C/(1+r)

PV (or €100 in our example) is known as the "Present Value" resulting from "discounting" C (or €112) at r% (or 12%) for one year.

As stated before, PV is the amount of money that must be invested to get the expected sum after a year given an established return. But it can also be understood as the amount in today's money that is equivalent to the amount to be received at the end of one year, taking into account the risk that is being taken.

The return r (12% in our case) is called the "discount rate". The discount rate depends on the risk being taken. We can assume that in our example, the risk taken demands a yield of at least 12%. In other words, the discount rate corresponding to the risk of the investment is 12%. A riskier investment would have required a higher discount rate and a less risky investment a lower discount rate.

Why the PV formula is important?

Let´s call A our previous investment opportunity. Imagine you are offered another investment opportunity, which we call B. B also requires an initial investment of €100 but produces €120 at the end of a year. However, B is riskier than A and thus requires a discount rate of 15%.

Applying the formula to calculate the present values ​​of A and B we obtain,

PV(A) = €112/(1+0.12) = €100

PV(B) = €120/(1+0.15) = €104.35

Note that €104.35 is greater than €100. That is, the PV of B is greater than the PV of A.

Given that the present value represents the amount in today's money that is equivalent to the money that is to be received in the future (taking into account the risk) and the present value of B is greater than the present value of A, then we can conclude B is preferable to A.

In A, €100 are invested to get the same €100 in today's money terms, so that the surplus was zero. This means that return was exactly equal to the minimum required (12%).

In B, €100 are invested to obtain €104.35 in today's money terms, so that there was a €4.35 surplus. The fact that there is a surplus implies that the investment has yielded more than the minimum required (over 15%). In fact, it is clear that the yield was 20% (the €100 invested became €120 after a year).

Then we can also affirm that B is better than A because B has produced more than the minimum required while this was not the case of A.

The surplus is the result of subtracting the initial investment from the present value. This surplus is called "Net Present Value". The Net Present Value (NPV) is useful for comparing investments when the the initial investment is not the same.

In our example, the result might have favored A if the amount to invest in B would have been higher. For example, if the initial investment for B were €110 (instead of €100) its NPV would have been:

€104.35 - €110 = -€5.65,

This is worse than a zero NPV for investment A. So now B is preferable.

Formally, the net present value is calculated as follows.

For investment A:

NPV(A) = -€100 + €112/1.12 = €0

And for investment B (when investing € 110):

NPV (B) = -€110+€120/1.15 = -€110 + €104.35 = -€5.65

The general formula is:

NPV = -C(0) + C(1)/(1+r) ,

where we differentiate between C₀ the amount invested (at time "zero") and C₁ the amount received at the end of the first year (at time "1").

The general rule is that when comparing two investment opportunities, the preferable option will be the one with that higher NPV.

But the story does not end here. An investment being “preferable” does not necessarily mean that it must be acceptable.

To explain this let´s return to case B, when €110 were invested. NPV_B was negative (-€5.65) meaning that the PV (€104.35) is less than the amount invested (€110). This means that the inverted sum failed to produce the minimum required return, or what is the same, that the yield was below 15%. In fact, if we make the computations we see that the €110 invested produced only a yield of 9.1% since 9.1% of €110 is equal to €10 (for a total received of €120).

We would have had to invest €104.35 to get the minimum demanded return of 15% since the difference between €120 and €104.35 corresponds exactly to 15% of €104.34.

Also, remember that when the initial investment was €100, NPV_B was positive. Applying the NPV formula we get:

NPV(B) = -€100 + €120/1.15 = -€100 + €104.35 = +€4.35

That is, the PV (€104.35) is now greater than the amount invested (€100). As we know, this time the yield was 20%, well above the minimum requirement of 15%.

This allows us to conclude that when the NPV is zero, the investment produces exactly the required minimum return (equal to the discount rate). When the NPV is negative, the investment produces less than the minimum required return, and when the NPV is positive the investment produces more than the minimum required.

An investment is acceptable only when it produces at least the minimum required return. That is, when its NPV is equal to or greater than zero. The investment must be rejected if the NPV is negative.

The NPV method is very useful because it allows comparison of investment alternatives with different risks and different amounts invested or received. In fact it is a method widely used by analysts when making investment decisions.

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Investment Selection: Article 1

WHAT MAKES AN INVESTMENT ATTRACTIVE?

This is the first of a number of articles that attempt to explain the significance of the basic financial computations needed to make investment decisions. The purpose is to make this topic understandable to anyone regardless of his previous knowledge on the subject. It is highly recommended for the articles to be read in the order they are posted. Let us begin.

We do business to make money. The objective is to invest a certain amount with the expectation of getting a larger amount in the future. But when considering an investment opportunity is not enough that we get more than we invested but there are two other important factors: time and risk.

It is not the same to receive €100 within a year than in 10 years. Naturally the sooner the proceeds are received the happier we will be. This is what we mean when we speak of the time factor.

The risk factor refers to the fact that the riskier a business is the higher must be its return. Otherwise, no one would invest in risky ventures. We would all limit ourselves to invest in the safest opportunities.

In conclusion. When deciding on an investment opportunity attention must be given to: 1) the amount of money we get in exchange for the sum originally invested, 2) how long it will take to receive the money and, 3) what is the risk involved.

I will illustrate with an example. Let's start with the first factor: the amount of money to receive.
Suppose we have the opportunity to invest €100 with the expectation of getting €110 in a year, i.e. a yield of 10%. And we also have the opportunity to invest the same €100 but with a 15% yield meaning €115 at the end of the year. Both alternatives have the same risk. Which alternative will be preferable? Obviously the one yielding 15%.

Then we reach our first conclusion: If the risk level is the same, an investment will be more attractive the more money it produces or, in other words, the higher its yield.

Now let´s look at the second factor: the time when we will receive the money. Returning to our initial investment opportunity: investing €100 with the expectation of receiving €110 in a year. Will we mind if the €110 is received instead at the end of two years? We surely will. The sooner we get the €110 the happier we will be.

This is because the sooner you get the money, the quicker it can be reinvested getting even more money in the end. For example, if we receive €110 in one year this amount can be invested again during the second year, say at the same 10%. This will produce an additional 10% during the second year, i.e. €11 (10% of €110). So at the end of the second year we will get the original €110 plus €11 for a total of €121. There is no doubt that this is better than receiving just €110 at the end of two years.

This brings us to our second conclusion: An investment will be more attractive the sooner the money is received.

Let's finish with the third factor: risk. Our initial investment opportunity promised a return of 10% at the end of a year for an investment of €100. At first glance it looks good. But is this attractive enough given the risk taken?

If we were investing in a bond issued by a very solid company, for example IBM, we are talking about a very attractive investment. Why? If it is known that IBM normally borrows at an interest rate of 6%, and we have the opportunity to buy a bond of the same company with a yield of 10%, there is no doubt that we would be doing a great business.

But if instead it were a bond issued by a Pakistani construction company, possibly a return of 10% would be insufficient. A Pakistani company is far riskier than IBM. We would have to ask: How solvent is the company? Will it pay its debts? Might I lose my money if the company invests in dangerous ventures? Will the Pakistani rupee lose value? How many euros will I get if the rupee is devalued?

That is, the Pakistani company will demand a return larger than 10% since we could always get 10% with IBM which is much less risky.

This brings us to our third and final conclusion: The more risky an investment is the larger must its return be.

In the next article I will explain how to implement these ideas when deciding on an investment opportunity.

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