Chapter 7 – Risk and Return:

7.1. Investment and the rate of return

An investment is the allocation of current money and resources with the hope of achieving greater returns in the future. For instance, buying a share in a company is done with the expectation that the return generated from owning the share will justify the time, the money locked up in the investment and the risk of the investment. An investment decision should only be executed after considering the risk and return from the investment. Let’s first define what constitutes the return of an investment.

The return on an investment measures the performance of the investment over a certain period. It captures the gain and loss that the investor experienced on the investment. There are two component that makes up the total return: the income component i.e. dividends from the stock, coupon (interest) payment from the bond or rental income from properties. The second component is the capital gain (loss) from the investment. It is the increase or decrease in the value of the investment asset itself, which reflects the changes in the price of the asset during the holding period.

Realised return, as the name suggests, is the return from an asset that has been realised. It measures the performance of the asset during a given period, which is the change in cashflows, divided by the initial investment. The realised return takes into account both the capital gain and income components.

where:

• R_t                         Return over the period
• Asset Price_t       Price at the end of the period
• Asset Price_t-1    Price at the end of the previous period
• CF_t                       Dividend received over the period

Unlike realised return, expected return is the rate of return expected to be earned from the investment over a period. Expected return is about the future, while realised return is about the past. Often, the expected return is also referred to as the required rate of return, signifying the minimum rate of return investors require to justify the risk of their investment and to make the investment attractive to them.

7.2 Risk

7.2.1 Definition of risk

Webster’s dictionary defines risk in the context of investments as the potential for an asset, such as stock or commodity, to decrease in value. Risk, in a broader sense, refers to the likelihood that an investment’s actual return will fall short of its anticipated return. In the financial markets, since investors dislike risks while prefer assets with high expected returns, investors typically require higher expected returns as compensation for assuming greater levels of risk. The principle is that the more substantial the risk one takes, the greater the return they will demand to justify the possibility of incurring a loss. This dynamic is fundamental to investment strategy and affects how portfolios are constructed, as investors seek to align their risk tolerance with their financial goals.

7.2.2 Measuring expected return and risk

To determine the anticipated return and risk associated with an asset, we employ two methodologies:

Historical data average: This approach involves calculating the average return of the asset over a past period. By analysing historical performance data, investors can get a sense of the asset’s average rate of return, which is then used as a basis to estimate future returns. The expected return estimated using the historical average is calculated as:

This process simply involves adding up each return observations and divide by the number of periods. It is just a simple arithmetic average:

Probability distribution method: This technique is applicable when the potential outcomes and their respective probabilities are known. By assigning probabilities to different possible returns, one can calculate the expected return as a weighted average, where each potential return is weighted by its likelihood. This method is more nuanced and can accommodate complex scenarios where returns are contingent on certain events or conditions. It allows for a more tailored analysis based on specific forecasts or models of future events.

7.2.3. How do we measure risk?

Harry Markowitz in his seminal 1952 “Portfolio Selection” paper measure risk as the standard deviation of an investment’s returns. Standard deviation, a statistical measure, gauges the range of possible outcomes, indicating the volatility of returns.

To find standard deviation, we first need to compute variance and standard deviation can be found by taking the square root of the variance:

Intuitively, variance measures the “average” squared distance from expected value i.e. what you expect to get compared to what you might get. Since it is a squared measure, standard deviation or risk is obtained by taking the square root of the variance. Essentially, the wider the deviation from expected value — or the greater the standard deviation — the higher the risk associated with the investment.

Most investors dislike risks or have an aversion to risk; this phenomenon is termed ‘risk aversion.’ Thus, investors typically demand higher expected returns from investments that are perceived to be riskier. However, how large these compensations for risk should be?

It depends on individual risk preferences, which vary significantly among investors as some are more risk averse than others. In addition, the degree of risk aversion for a given investor can change overtime and can be affected by various factors such as personal financial situations, life stages, past investment experiences, and even broader economic conditions. Understanding this subjective element of investment risk is crucial for both individual investors crafting a personal investment strategy and for financial economists designing models that attempt to predict market behaviour.

7.2.4. Risk premium

An investment with zero risk is called a risk-free asset. An investment which has a risk element is called a risky asset, the higher the risk the higher the expected return. Risk premiums represent the additional returns that riskier investments provide over safer alternatives, such as government bonds. This premium serves as a form of compensation for investors who tolerate the increased uncertainty inherent in riskier assets compared to more secure ones.

Consider the rationale behind an investor’s decision to engage in the stock market rather than depositing funds into a bank account. The choice often stems from the expectation that, although the stock market involves more risk, it also typically offers a higher return as a reward for bearing that risk. On average, historical data shows that stocks have yielded a risk premium of approximately 4-6% over Treasury bonds. This premium incentivizes investors to allocate capital to the stock market, accepting the additional risk in anticipation of greater financial rewards.

7.3 Portfolio theory

Modern Portfolio Theory (MPT), introduced by Harry Markowitz in 1952, is a revolutionary approach to investment strategy that extends beyond the merit of individual asset selection to the dynamics of portfolio composition. It’s not solely about identifying a good investment but about creating the optimal mix of assets to balance the overall portfolio risk and return.

At the heart of MPT is the concept of diversification. The theory asserts that a portfolio composed of various investments will, on the whole, bear less risk than the sum of the risks of its individual components. This reduction in risk is achieved because the price movements of different assets are not perfectly correlated.

The key question is how do we come up with a “diverse investment”? This brings us to the concept of portfolio.

7.3.1 What is a portfolio?

A portfolio is a collection of assets. A portfolio is formed to reduce risk. Suppose you have $10,000 to invest and you decide to put $3,000 in stock A and $7,000 in stock B. Stock A earns an expected return of 7% p.a. while stock B earns an expected return of 10% p.a. What is the expected return of the portfolio?

Answer: The weight of stock A in the portfolio is 7,000/10,000 = 0.7. A makes up 70 percent of the portfolio and B therefore makes up 30 percent of the portfolio. The weight of all securities that make up the portfolio has to be 1. The expected return of the portfolio is the weighted average of the expected returns of the assets that make up the portfolio. In this case, it is 0.7 x 0.07 +0.3 x 0.1 = 0.052.

7.3.2. How does diversification reduce risk?

The volatility of the portfolio can be reduced if the assets in the portfolio do not perfectly move in the same direction. Suppose you have two assets X and Y and if X goes up 10%, Y goes down by 10% then by investing in both X and Y, you can achieve a portfolio with zero risk. On the other hand, if X goes up by 10%, Y also goes up by 10%, then putting both X and Y in the same portfolios will not eliminate any risk. How much risk you can reduce depends on the co-movement in returns between the assets in the portfolio. The correlation coefficient is a statistical measure that dictates the relationship between two variables. A correlation coefficient only fluctuates between -1 and 1. The lower the correlation coefficient, the greater the diversification benefits. If the correlation is 1, there is no diversification benefit.

Since every company is different, their correlation is not 1 and therefore the more companies you put into a portfolio, the greater the diversification benefits or risk reduction. However, there is a limit to diversification. As the number of companies in the portfolio increases, there is some risk that still remains. This risk is known as systematic risk.

Assets tend to move positively with each other since they are affected by macroeconomic factors that have pervasive impacts on the whole economy. Through diversification, the risk that we can reduce is unsystematic risk, risks that are specific to individual companies. On the other hand, market wide factors that affect all risky assets cannot be diversified away. Risk can be decomposed into two components:

Total risk = Systematic Risk + Firm Specific Risk.

Systematic risk, also known as market risk, is the type of risk that affects the entire market or economy. Examples of systematic risks are interest rate changes, inflation, recessions and wars etc.. Systematic risks cannot be diversified.

Firm specific risk, also known as unsystematic risk, is the type of risk that only affects a specific company or industry. Examples of unsystematic can include poor management decisions, production capabilities or changes in consumers demand for its products and services.

Since unsystematic risks can be eliminated, the relevant measure of risk should only be systematic risk. If investors wish to hold an undiversified portfolio and hence be exposed to unsystematic risk then that is purely the decision of the investors which the market will not compensate for.

7.3.3 How is systematic risk quantified?

In finance, we measure systematic risk using beta. Beta captures the proportionate movement in the stock returns relative to the returns on the market as a whole. For example, consider a stock with a beta of 1.5. This suggests that if the market return increases by 2% during a specific timeframe, the stock’s return is expected to increase by 3% (1.5 times the market movement) in that same timeframe. On the other hand, a stock with a beta of 0.5 implies that if the market return rises by 2%, the stock’s return would likely increase by just 1% in that period, showing less sensitivity to market movements.

Stocks with a beta greater than 1 are considered more volatile than the entire market, while those with a beta equal to 1 have volatility that mirrors the market average. Stocks with a beta less than 1 are less volatile in comparison. The market itself has a beta fixed at 1.0. Typically, a stock’s beta is found between 0 and 2, although negative betas can occur. Over time, it is observed that betas generally gravitate toward the mean value of 1.0.

7.4. The Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) developed by William Sharpe (1964) provides a description on the relationship between expected return and risk of the asset. According to the CAPM, the only relevant measure of risk is not the variance or the total risk but only the systematic risk component, which is the covariance with the market portfolio, known as beta. The higher the beta, the higher the expected return.

Expected return = Risk free rate + risk premium

The CAPM provides a useful concept for evaluating investment risks and returns. It suggests that the expected return on a security is equal to the risk-free rate plus a risk premium. The risk premium is determined by the beta of the security, reflecting how much risk the investment adds to a diversified portfolio.

Since beta is computed based the covariance with the market portfolio, we need to define what is the market portfolio. The market portfolio is a portfolio that includes all securities in the market. Since constructing such portfolio is typical costly and hence not feasible, we can get a pretty good proxy for the market portfolio. In Australia, we use the All Ordinaries Index or S&P/ASX 200 Index to proxy for the market portfolio. In the U.S, the S&P500 index or Russell 2000 are often used as the market portfolio.

Once you know the betas of the individual assets that make up the portfolio, you can calculate the beta of the portfolio as follows:

Example: Support you have 3 stocks in your portfolio A, B and C. You invest $10,000 in stock A, $15,000 in stock B and $25,000 in stock C. The beta of stock A is 0.8, the beta of stock B is 1.2 while beta of stock C is 1.5. What is the beta of the portfolio?

Answer:

Total Portfolio Value=$10,000+$15,000+$25,000=$50,000.

The weight of stock A is 10,000/50,000 = 0.2.

The weight of stock B is 15,000/50,000 = 0.3

The weight of stock C is 25,000/50,000 = 0.5.

Beta of the portfolio is (0.8×0.2)+(1.2×0.3)+(1.5×0.5) = 1.27.

References:

Investments by Zvi Bodie, Alex Kane and Alan Marcus 12 edition.