# Quantitative Methods # 2 : Required Rate Of Returns And Means

There is a small but important distinction between a required nominal rate of return and a required real rate of return. What is it? Inflation.

When evaluating a company, a project or even just a series of cash flows, you will be given a “required” rate, often also named a discount rate. That is the rate that is used to discount all flows (see the time value for more information). However, that rate will sometimes be specifically displayed as a notional rate or a real rate. I’d say that in general you can assume that you are given the notional rate and that it should be used as is.

However, in the case where inflation is mentioned in the question, you should be very careful to see if the required rate is a “real” or “after inflation” rate. If it is, here is how you would obtain the real rate:

Required return rate (notional) = ((1+required return rate (real) (1+Inflation rate premium) – 1

You would also need to compensate the investor for the risk associated with the business itself. Remember that an investor will require a better return when investing in a risky startup company than by investing in a “risk-free” 90day tbill. It wouldn’t make sense otherwise would it? Here is the updated formula:

Required return rate (notional) = ((1+required return rate (real) (1+Inflation rate premium) (1+Risk Premium) ) – 1

The approximation would be:

E(r) = Risk Free Rate + Inflation premium + Risk premium

Means

What is a mean? Quite simple you would think and you would be right. There are a few different means though and it’s important to make the distinction:

Let’s say an investor made these returns over 4 years:

0%
-5%
5%
10%

The traditional mean (named arithmetic mean in CFA material) would be:

Arithmetic mean= 0-5+5+10/4 = 2,5%

There is one problem with that mean however. If an investor returned 2.5% every year, they would not get the same return as the investor that we just described. Why? You can try out the example but in this case:

Investor that gets 0, -5, 5 and 10%: overall return of 9.725%
Investor that gets 2,5% every year = 10.38%

For that reason, using the geometric mean is more precise in many cases:

Geometric mean = { (1+R1) x …. x …(1+RN)}^(1/n)-1

In our example:

R= {(1+0) x (1-0,05) x (1+0,05) x (1+0,1) }^ (1/4)-1
R = 2,3473%

What does it mean? It means that an investor that gets 2.3473% every year would get the same return as the investor that had irregular returns.

Quantitative Methods # 2 Required rate of returns and Means
There is a small but important distinction between a required nominal rate of return and a required real rate of return. What is it? Inflation.
When evaluating a company, a project or even just a series of cash flows, you will be given a “required” rate, often also named a discount rate. That is the rate that is used to discount all flows (see the time value (yyyyyyy) for more information). However, that rate will sometimes be specificied as a notional rate or a real rate. I’d say that in general you can assume that you are given the notional rate and that it should be used as is.
However, in the case where inflation is mentioned in the question, you should be very careful to see if the required rate is a “real” or “after inflation” rate. If it is, here is how you would obtain the real rate:
Required return rate (notional) = ((1+required return rate (real) (1+Inflation rate premium) – 1
Then, you would also need to compensate the investor for the risk associated with the business itself. Remember that an investor will require a better return when investing in a risky startup company than by investing in a “risk-free” 90day tbill. It wouldn’t make sense otherwise would it? Here is the updated formula:
Required return rate (notional) = ((1+required return rate (real) (1+Inflation rate premium) (1+Risk Premium) ) – 1
The approximation would be:
E(r) = Risk Free Rate + Inflation premium + Risk premium
Means
What is a mean? Quite simple you would think and you would be right. There are a few different means though and it’s important to make the distinction:
Let’s say an investor made these returns over 4 years:
0%
-5%
5%
10%
The traditional mean (named arithmetic mean in CFA material) would be:
Arithmetic mean= 0-5+5+10/4 = 2,5%
There is one problem with that mean however. If an investor returned 2,5% every year, he would not get the same return as the investor that we just described. Why? You can try out the example but in this case:
Investor that gets 0, -5, 5 and 10%: overall return of 9,725%
Investor that gets 2,5% every year = 10,38%
For that reason, using the geometric mean is more precise in many cases:
Geometric mean = { (1+R1) x …. x …(1+RN)}^(1/n)-1
In our example:
R= {(1+0) x (1-0,05) x (1+0,05) x (1+0,1) }^ (1/4)-1
R = 2,3473%
What does it mean? It means that an investor that gets 2,3473% every year would get the same return as the investor that had irregular returns.

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