Risk-Return Trade Off

Risk-Return Trade-Off Definition

The risk-return trade-off definition is fairly straightforward. The reward you get for a decision you make is the increase or decrease in happiness, safety, security, popularity, and so on. In financial markets, the reward is the return on your investment, or how much your investment increases or decreases over time.

The other factor in decision-making is risk. In financial markets, risk refers to the fact that investors do not know with certainty what the outcome of their investment decision will be. There is some probability, or chance, that a certain outcome, which can be good or bad, will occur, but that probability is almost always less than 100%. Thus, when making financial decisions, people need to be aware of the risk involved as well as the outcomes that are possible.

If you need to decide between investment A, which is very risky, and investment B, which is much less risky, which investment do you think should cost more? Well, if you were to choose investment A, and you know the chance that it might go down in value is a lot greater than that for investment B, you would probably want to be compensated for taking on more risk. This compensation is the return on your investment. Because lower-priced assets generally have higher returns, the riskier investment A would have a lower price than the less risky investment B.

This example shows why prices are lower and returns are higher for riskier assets. Risk and return are positively correlated. That means that the higher return an investor wants, the greater risk they must accept. This is the risk-return trade-off that is inherent in all financial decisions.

Risk-return Trade-Off Formula

The risk-return trade off formula shows how risk and return are positively correlated. Let's take a look.

The expected value of an investment is the probability-weighted average of the possible outcomes. If there are two possible outcomes A and B, and two probabilities P_A and P_B, then the expected value is:

\(\hbox{Expected Value} = P_A \times \hbox{Value of Outcome A} + P_B \times \hbox{Value of Outcome B}\)

In a similar way, instead of using probabilities for weights, we can use the share of a portfolio that is in stocks and the share that is in a risk-free asset as weights. Investors consider short-term US government Treasury bills risk-free because there is very little chance the US government will default on its debt before the Treasury bills mature. Thus, the interest rate on these Treasury bills is considered the risk-free rate.

Let's denote the following:

\(R_p = \hbox{Expected Return on the Portfolio}\)

\(R_m = \hbox{Expected Return on the Stock Market}\)

\(R_f = \hbox{Risk-free Rate}\)

\(b = \hbox{Share of Portfolio in Stocks}\)

\((1 - b) = \hbox{Share of Portfolio in Risk-free Treasury Bills}\)

Then the risk-return trade off formula is:

\(R_p = b \times R_m + (1 - b) \times R_f\)

The less risk-averse an investor is, the more they will invest in stocks, and the higher b will be, and vice versa.

If b = 1, the entire portfolio will be in stocks, so the expected return on the portfolio will be equal to the expected return on stocks. If b = 0, the entire portfolio will be in risk-free Treasury bills, and the expected return on the portfolio will be the risk-free rate.

By rearranging the above equation, we have:

Equation 1:

\(R_p = R_f + b \times (R_m - R_f)\)

So how does an investor know what the value of b should be?

The standard deviation, which is a measure of variability, is used to determine the riskiness of an investment. Since the standard deviation of short-term US Treasury bills is virtually nil, the only measure of risk in a well-diversified portfolio is the standard deviation of the market.

Let's denote:

\(S_p = \hbox{Standard Deviation of Portfolio}\)

\(S_m = \hbox{Standard Deviation of Market}\)

Thus:

Equation 2:

\(S_p = b \times S_m\)

That is, the standard deviation of the portfolio is equal to the share of the portfolio in the stock market multiplied by the standard deviation of the stock market.

By rearranging this equation, we have:

Equation 3:

\(b = \frac{S_p} {S_m}\)

That is, the share of the portfolio in the stock market should be equal to the standard deviation of the portfolio divided by the standard deviation of the stock market. But there is a problem here. If we don't know the value of b, how can we know the value of S_p? We have one equation and two unknowns. To solve this conundrum, we first plug the formula for b into Equation 1 as such:

Equation 4:

\(R_p = R_f + \frac{S_p} {S_m} \times (R_m - R_f)\)

We now have an equation that shows that as S_p (risk) increases, so too does R_p (return). This is known as the investor's budget line. But how does the investor know how much risk (S_p) to take on? The answer is they have to compare the budget line with their level of risk aversion, represented by indifference curves.

Indifference curves are upward-sloping because risk is not desirable. That is, the greater the risk, the greater the return must be. A risk-averse investor will have steeper indifference curves, meaning that they need a large increase in returns in exchange for a small increase in risk. A less risk-averse investor will have flatter indifference curves, indicating they need a much smaller increase in returns in exchange for a small increase in risk.

When we plot the budget line and an investor's indifference curves on a graph, the investor chooses the level of risk, S_p, where the budget line is tangent to the highest possible indifference curve, as shown in Figure 2 below. This is denoted by S*.

Fig. 2 - Budget Line, One Investor

Thus, now that S* is known, the investor can calculate b as:

Equation 5:

\(b = \frac{S*} {S_m}\)

The investor can then calculate R*, the expected return on the portfolio as:

Equation 6:

\(R* = R_f + \frac{S*} {S_m} \times (R_m - R_f)\)

The investor can also then calculate how much the expected return on the portfolio should increase for a given increase in risk, or conversely, how much more risk the investor needs to accept to obtain a higher expected return on the portfolio.

The slope of the budget line,

\(\frac{(R_m - R_f)} {S_m}\)

is called the price of risk, because it shows how much more risk an investor must accept to receive a higher expected return.

Risk-Return Trade-Off Diagram

Figure 3 below is the risk-return trade-off diagram. It shows the optimal combination of risk and return for an investor given their risk preferences. The green curves, called indifference curves, show how much more risk an investor is willing to accept for a given increase in returns. Each point on a particular curve gives the investor the same amount of satisfaction. The higher up the curve, the higher the expected return for each given level of risk.

There are an infinite number of these curves but, for simplicity, we only show three curves here to make a point. The blue line, the budget line, shows the maximum expected return possible for each given level of risk. The optimal portfolio is where this maximum expected return possible exactly matches the expected return required by the investor for each given level of risk. In other words, it is where the budget line is tangent to the highest possible indifference curve, which is IC₂. At this point, the optimal level of risk is S*, and the expected return on the portfolio is R*. The satisfaction received from being on curve IC₃ is not possible because it is higher than the budget line.

Fig. 3 - Risk-Return Trade-Off Diagram

Risk-Return Trade-Off Example

Let's take a look at a risk-return trade-off example.

Suppose we have two investors A and B. Investor A is very risk-averse. Therefore, his indifference curve, IC_A, is very steep, meaning for any increase in risk, he requires a large increase in returns. Investor B is much less risk-averse. Therefore, his indifference curve, IC_B, is much less steep, meaning for any increase in risk, he requires only a small increase in return.

As can be seen in Figure 4 below, Investor A's expected return is very close to the risk-free rate, R_f, meaning his portfolio is mostly in short-term US Treasury bills. Meanwhile, Investor B's expected return is much closer to the expected return on the market, meaning his portfolio is mostly in stocks.

Notice that these two investors are invested in their optimal portfolios. This means that they cannot increase their expected returns without becoming less risk-averse, which would mean their indifference curves would need to become less steep, allowing the budget line to be tangent to the highest possible indifference curve at a higher expected return.

Look at it this way. Suppose, instead of these two indifference curves representing two different investors, they instead represented the same investor. If this investor is at risk level A, his optimal portfolio is at the point S_A, R_A, where curve IC_A is tangent to the budget line. Now, if he wants to increase his expected return, he is going to have to accept more risk, which means he will have to become less risk-averse. If he increases his risk tolerance to risk level B, his indifference curve will be less steep, and his optimal portfolio will be at point S_B, R_B, where curve IC_B is tangent to the budget line.

The bottom line is, regardless of an investor's current risk tolerance, if they are invested in the optimal portfolio, the only way to increase expected returns is to become less risk-averse. They have to trade a higher level of risk for a higher expected return.

Fig. 4 - Budget Line, Two Investors

Risk-Return Trade-Off Importance

The importance of the risk-return trade-off cannot be understated. It is very important for investors to understand that if they are invested in a well-diversified portfolio and want a higher expected return, they are going to have to accept a higher level of risk. If, however, they are not invested in a well-diversified portfolio, they can reduce their risk without reducing their expected returns by investing in a more diversified portfolio.

Once the diversifiable risk has been eliminated by investing in a good number of companies whose risks cancel each other out, all that is left is non-diversifiable risk, which is the risk inherent in the overall market. At that point, the only way to reduce risk in a portfolio is to invest less in stocks and more in short-term Treasury bills. Similarly, the only way to increase expected returns is to invest more in stocks and less in short-term Treasury bills.

The same goes for a single stock. If an investor wants a higher expected return, they are going to have to invest in a risky stock. If they don't want to take much risk, they can invest in a low-risk stock, but must understand their expected returns will be lower as well.

Risk-Return Trade-Off Implications

The implications of the risk-return trade-off are fairly straightforward but critical to understand. The fact that people don't like risk has an enormous effect on asset prices and average expected returns. The avoidance of risk and uncertainty causes people to pay higher prices for less-risky assets of any kind and lower prices for more-risky assets. However, because asset prices and expected returns are inversely related, this means that more-risky assets will have higher expected returns than less-risky assets. In other words, risk and expected returns are positively correlated.