Think of two investments that have the same expected values but one of them is significantly more risky.
See solution.
Practice makes perfect
Let's begin by recalling the concept of an expected value. The expected value of a discrete random variable is the weighted average of the values of the variable.
E(X)=∑ [X* P(X)]
In other words, it is the sum of the products of every possible value of X and its corresponding probability P(X). Now let's think of two types of investments.
Type of Investment
Description
1-year treasury bond
We lend money to the U.S. government and receive our money back, plus interest, after 1 year. The interest that we earn depends on the treasury rate.
Investing in a start-up company
We invest money in a start-up and could potentially make a serious profit after 1 year. However, most start-ups fail within the first year, so the probability we end up with nothing is about 80 %.
Let's assume that we want to invest $100 in a 1-year treasury bond that has an interest rate of 1 %. Since treasury bonds are considered risk-free investments, the expected value is equal to the value of the investment plus an interest yield of 1 % of $100.
E(Bond)= 100+( 1 %* 100)= $101
Now, let's assume that investing in a start-up can leave us with $505 if the company is successful or $0 if the company fails. There is a 210 probability that the company will succeed and a 810 probability that the company will fail. The risk of this investment seems very big.
E(Start-up) = 2/10* 505 + 8/10* 0=$101
We can see that both investments have the exact same expected values — $101. Now let's evaluate and compare their standard deviations. The standard deviation of a probability distribution is given as the following.
σ = sqrt(∑[[X-E(X)]^2* P(X)])
Let's calculate the standard deviation for our investments. Notice that, in case of bonds, we have only one possible outcome, so the probability of obtaining this value is 1.
Investment
Standard Deviation
Simplify
Treasury Bond
σ = sqrt([101-101]^2* 1)
0
Start-up
σ = sqrt([505-101]^2* 2/10+[0-101]^2*8/10)
202
As we said before, bonds are considered a risk-free investment, so their standard deviation is equal to $0. In our example, investing in a start-up is significantly more risky with a standard deviation of $202.
Investment
Expected Value
Standard Deviation
Treasury Bond
$101
$0
Start-up
$101
$202
Therefore, when we want to make an investment, we should consider both the expected values and the standard deviations. While the expected values can be equal, the difference in risks of the investments can be seen in their standard deviations. The choice we make should be based on how much we are willing to lose.
