Risk and Return - Coursework Example

Risk and return Name: University: Course: Date: Risk and return Risk is the probability that the actual return from an investment might be different from the expected. Just as life is composed of numerous uncertainties, investing just like driving or taking a walk down the street is an exposure of oneself to risk. The amount of risk one can accommodate (also known as risk appetite), is largely dependent on ones personality and lifestyle. Investors are either risk lovers or risk averse. Investors are risk averse if they avoid risky opportunities and only allow a minimal level of risk in their portfolio. On the contrary, risk lovers, for instance day traders embrace a lot of risk, and often feel inadequate if they are not making dozen of trades in a day (Campbell & Vicera, 2002). Thus, investing in any instrument such as bonds, stocks or shares exposes an investor to different types of risks. In order to maximize on the returns, prudent investors should be aware of the types of risks they are exposed to, estimate the amount of risk, and indentify methods of mitigating the risks. Return, risk and the Security market line The expected return of an investment is calculated as the weighted average of the likely profits of the individual assets in the portfolio, weighted by the possible profits of each of the asset class. The formula is Also represented as E(R) = w1R1 + w2Rq + ...+ wnRn In the first example, the expected return for a portfolio of two mutual funds; one invested bonds and the other in stocks. The expected return from the stock is 12% and the return from the bond is 6% while the allocation to each asset class is 50%; Expected return E(R) = (0.12)*(0.5) + (0.06)*(0.5) = 0.09 or 9% Although the expected return is not always a guaranteed rate or return for an investment, it can however be used to forecast the future value of a portfolio. Investors also use expected return as a guide to estimate actual returns. In Example 2, another portfolio of stocks A and B; with stock A having an expected return of 20% and weight of 30% in the portfolio and stock B has expected return of 70% and weight of 30%. The expected return will be; E(R) = (0.30)(0.20) + (0.70)(0.15) E(R) = 0.06 + 0.105 = 0.165 or 16.5% The expected return of the portfolio can also be analyzed using variance. Variance Variance (σ2) is gives the measure of the spreading of a set of data points around the mean value. Mathematically variance is the expectation of the average squared deviations from the mean. It is the probability-weighted average of squared divergence away from the expected mean. Variance being a measure of volatility (variability from the average) means it can be used to estimate the amount of risk an investor is willing to accommodate while purchasing a specific security (Luenberger, 1997). Example: From a recent research on company A and analyst’s report assigned the following probabilities to the following year’s expected sales. Scenario Probability Sales ($ Millions) 1 0.10 $16 2 0.30 $15 3 0.30 $14 3 0.30 $13 Using the expected returns formula E(R) = (0.30)(0.20) + (0.70)(0.15) E(R) = (0.1)*(16.0) + (0.3)*(15.0) + (0.3)*(14.0) + (0.3)*(13.0) = $14.2 million Variance is calculated by first computing the difference in each potential sales outcome out of the $14.2 million, then finding the square of the same; Scenario Probability Deviation from Expected Value Squared 1 0.1 (16.0 - 14.2) = 1.8  3.24  2  0.30  (15.0 - 14.2) = 0.8 0.64  3 0.30  (14.0 - 14.2) = - 0.2 0.04  4 0.30 (13.0 - 14.2) = - 1.2 1.44  Variance equals to the sum of the squared deviation multiplied by the weights; (0.1)*(3.24) + (0.3)*(0.64) + (0.3)*(0.04) + (0.3)*(1.44) = 0.96 Portfolio variance According to Rubinstein (2006), a portfolio’s return variance is composed of the variance of the individual assets added to the covariance between each of them. Covariance is defined as the measure of the degree to which returns of a pair of risky assets move in tandem; assets moving together have positive covariance, and assets moving inversely having a negative covariance. In order to reduce risk, the modern portfolio theory suggests that choosing asset classes bearing low negative covariance such as bonds and stocks (diversification). According to Connor, et al (2010), portfolio variance focuses on the correlation coefficient or covariance of the securities in the portfolio. It is calculated by multiplying the squared weight of each individual security by its subsequent variance and adding twice the weighted average weight multiplied by the covariance of all individual security pairs. Formulated as Portfolio Variance = w2A*σ2(RA) + w2B*σ2(RB) + 2*(wA)*(wB)*Cov(RA, RB) Where: wA and wB are portfolio weights, σ2(RA) and σ2(RB) are variances and  Cov(RA, RB) is the covariance  Example: below is a covariance matrix for two two-assets Stock Bond Stock 350 80 Bond 150 Because variance is equal to the covariance of any asset to itself, the variance equals to 350 for stock and 150 for bonds while the covariance between stocks and bonds is 80. If the portfolio weights for stocks and bonds equals to 0.5, the Portfolio variance is computed as below; Portfolio variance = w2A*σ2(RA) + w2B*σ2(RB) + 2*(wA)*(wB)*Cov(RA, RB) =(0.5)2*(350) + (0.5)2*(150) + 2*(0.5)*(0.5)*(80) = 87.5 + 37.5 + 40 = 165. Standard deviation The measure of the spread of a given set of data away from its mean is the standard deviation; computed as the square root of variance. The higher the deviation, the more spread apart the data will be. Zhong (2007) believes that an investor can measure the volatility of an investment by applying standard deviation on the annual rate of return of an investment. When used by investors as a measure of the amount of expected volatility standard deviation is known as historical volatility. Hence as a measure of historical volatility, a high standard deviation stock will be more volatile than a stable blue chip stock with a lower standard deviation (Zhong, 2007). From the Portfolio variance example above, Standard deviation (σ) = (165)1/2 = 12.85%­­­. Although in the preceding examples two-asset portfolio was used to illustrate the importance of variance, most portfolios are composed of more than two assets. Therefore, for multi-asset portfolios the variance formula becomes more complex and necessitates that all terms in a covariance matrix be included to the calculation (Luenberger, 1997). Variance and standard deviation example: The below data below represents three stocks in Newco Plc. If the expected return is 14%, calculate the variance and standard deviation of the stocks. Scenario Probability Return Expected Return Worst Case 10% 10% 0.01 Base Case 80% 14% 0.112 Best Case 10% 18% 0.018 Variance (σ2) = (0.10)(0.10 - 0.14)2 + (0.80)(0.14 - 0.14)2 + (0.10)(0.18 - 0.14)2= 0.0003 Standard deviation (σ) = 0.0179, or 1.79% Systematic and unsystematic risk According to Bodie, et al. (2008), the type of uncertainty common with the industry or company one invests in is known as unsystematic risk (also known as systematic/diversifiable/residual risk). This type of risk can be managed through diversification. For instance a sudden strike by employees will potentially affect the stocks of Newco Plc. On the other hand systematic risk is the uncertainty affecting an entire market/market segment (systematic risk is also known as un-diversifiable/market risk). An example of systematic risk (also known as volatility) is the day-to-day fluctuations in a stock’s price; that can only be mitigated through hedging. Profitability from stocks is primarily dependent on the market movement, hence volatility is critical for returns. Other examples of systematic (un-diversifiable) risk include interest rate, wars and recession as they cannot be mitigated through diversification and affect the entire market. The measure of volatility (or systematic risk) using the market as a basis of comparison is called Beta. It is also used to while comparing the stock’s market risk against that of other stocks. Investments analysts employing the capital asset pricing model (CAPM) use beta (denoted by the Greek letter 'ß'). Through regression analysis, calculates beta as the likelihood of a security’s return to react to market swings. Beta of 1 means security price will move with market price; less than 1 means security volatility will be less than markets; while beta greater than 1 means that security volatility will be more than market’s. Beta also measures passive and active risk (Bodie, et al., 2008). In the graph, representing time series of returns (points of data marked “+”) for a portfolio R(p) in the y-axis against market return R(m). The intersection of axes x and y represents the cash-equivalent of the asset as the returns are cash-adjusted. The line of best fit depicts the amount of passive (beta) risk vis-à-vis the active risk (alpha). The slope of the line is equivalent to beta. Optimal portfolio In order to arrive at an optimal portfolio, the investor’s utility (risk appetite) and the capital allocation line is mandatory. In the below graph, an investor can derive same utility on all the points of the three curves. Indifference curve - shows the combination of risk-return pairs that would be acceptable to an investor to maintain a given level of utility (Bodie, et al., 2008).. Therefore, an investor is indifferent while on curve 1 as all the points have the same utility; because point a has lower risk and lower return, while point b has a higher return and higher risk. Indifference is similar for curve 2; but comparing points b and c shows that point b has same risk but relatively higher utility than point c; hence curve 1 has a higher utility than curve 2. Thus investors faced with indifference curves 1 and 2 with a risk averse appetite, will opt for curve 1 (Zhong, 2007). References 1. Bodie, Z., Kane, A. & Marcus, A. J. (2008). Investments (7th International ed.). McGraw-Hill: Boston. 2. Campbell, J & Vicera, M. (2002). "Strategic Asset Allocation: Portfolio Choice for Long Term Investors". Clarendon Lectures in Economics. Oxford University Press: London. 3. Connor,G.,   Goldberg, L. R., Korajczyk, R. A. (2010). Portfolio Risk Analysis. Princeton University Press: New York. 4. Luenberger, D. (1997). Investment Science. Oxford University Press: London. 5. Rubinstein, M. (2006). A History of the Theory of Investments. John Wiley & Sons, Inc: Hoboken 6. Zhong, M. (2007). Estimation of Portfolio Value-at-Risk and Expected Shortfall Using Copulas, Extreme Value Theory, and Doubly Noncentral T Distribution. State University of ProQuest: New York at Albany. Read More