If you are an investor, you probably know that risk and return are two sides of the same coin. The higher the potential return of an investment, the higher the risk involved. But how do you measure risk and return? And how do you decide which investments are worth taking?
One way to answer these questions is by using probability. Probability is the mathematical way of expressing how likely something is to happen. For example, if you flip a fair coin, the probability of getting heads is 0.5, or 50%. If you roll a fair six-sided die, the probability of getting a 1 is 1/6, or about 16.7%.
Probability can help you estimate the risk and return of different investments by calculating the expected value and the standard deviation of their outcomes. The expected value is the average outcome that you would get if you repeated the investment many times. The standard deviation is a measure of how much the outcomes vary from the expected value.
For example, suppose you have two investment options: A and B. Option A pays $100 with a probability of 0.8, and $0 with a probability of 0.2. Option B pays $200 with a probability of 0.4, and $0 with a probability of 0.6.
The expected value of option A is:
E(A) = (0.8 x $100) + (0.2 x $0) = $80
The expected value of option B is:
E(B) = (0.4 x $200) + (0.6 x $0) = $80
Both options have the same expected value, but they have different risks. To measure the risk, we can calculate the standard deviation of each option.
The standard deviation of option A is:
SD(A) = sqrt[(0.8 x ($100 – $80)^2) + (0.2 x ($0 – $80)^2)] = $40
The standard deviation of option B is:
SD(B) = sqrt[(0.4 x ($200 – $80)^2) + (0.6 x ($0 – $80)^2)] = $80
Option B has a higher standard deviation than option A, which means it has more variability and uncertainty in its outcomes. Option A is more stable and predictable than option B.
So, which option should you choose? It depends on your risk preference and your goal. If you are risk-averse, you might prefer option A, because it has less chance of losing money and more chance of getting a moderate return. If you are risk-seeking, you might prefer option B, because it has more chance of getting a high return and more excitement.
Probability can also help you compare different investments with different expected values and standard deviations. One way to do this is by using the coefficient of variation (CV), which is the ratio of the standard deviation to the expected value.
CV = SD / E
The coefficient of variation tells you how much risk you are taking per unit of return. The lower the CV, the better the investment.
For example, suppose you have two more investment options: C and D. Option C pays $150 with a probability of 0.7, and $50 with a probability of 0.3. Option D pays $300 with a probability of 0.3, and $100 with a probability of 0.7.
The expected value of option C is:
E(C) = (0.7 x $150) + (0.3 x $50) = $120
The expected value of option D is:
E(D) = (0.3 x $300) + (0.7 x $100) = $160
Option D has a higher expected value than option C, but it also has a higher risk.
The standard deviation of option C is:
SD(C) = sqrt[(0.7 x ($150 – $120)^2) + (0.3 x ($50 – $120)^2)] = $35
The standard deviation of option D is:
SD(D) = sqrt[(0.3 x ($300 – $160)^2) + (0.7 x ($100 – $160)^2)] = $70
The coefficient of variation of option C is:
CV(C) = SD(C) / E(C) = 35 / 120 = 0.29
The coefficient of variation of option D is:
CV(D) = SD(D) / E(D) = 70 / 160 = 0.44