Lecture 8 & 9 Risk & Rates of Return

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1 Lecture 8 & 9 Risk & Rates of Return We start from the basic premise that investors LIKE return and DISLIKE risk. Therefore, people will invest in risky assets only if they expect to receive higher returns. We define precisely what the term risk means as it relate to investment, we examine procedures managers use to measure risk and we discuss the relationship between risk and return. 1

2 Market Interest Rates The Determinants of Quoted Interest Rates In general, the quoted (Nominal) interest rate on a security is composed of a real risk-free rate of interest, plus several premiums that reflect inflation, the riskiness of the security and the security s liquidity. This relationship can be expressed as follows: Quoted Interest Rate = K = K* + IP + DRP + LP + MRP Where, K = The quoted, or nominal, rate of interest on a given security; K* = The real risk-free rate of interest, K-star, it is the rate that would exist on a risk-less security if zero inflation was expected; IP = Inflation premium, which is equal to the average expected inflation rate over the life of the security; 2

3 Market Interest Rates The Determinants of Quoted Interest Rates Quoted Interest Rate = K = K* + IP + DRP + LP + MRP K RF = K* + IP, and it is the quoted risk-free rate of interest on a risk-less security; DRP = Default risk premium, which reflects the possibility that the issuer will not pay interest or principal at the stated time and in the stated amount. DRP is zero for government T-Bills and it increases as the riskiness of the issuer increases; LP = Liquidity or marketability premium, which is the premium charged by investors to reflect the fact that some securities cannot be converted to cash on short notice at a reasonable price; MRP = Maturity risk premium, which is the premium charged by investors for taking up investments of longer horizon. 3

4 The Determinants of Quoted Interest Rates Numeric Examples 30-Day treasury bills are currently yielding 13.30%, you contemplate that the market is currently offering the following premiums: IP of 11%, LP of 0.5%, MRP of 1% and DRP of 3%, given the data what is the real risk-free rate of return? The 5-year bonds of FM Enterprises are yielding 7.75% per year. Treasury bonds with the same maturity are yielding 5.2% per year. The real risk-free rate has not changed in recent years and is 2.3%. The average inflation premium is 2.5% and the maturity risk premium takes the form: MRP = 0.1% (t-1), where t is the number of years to maturity. If the liquidity premium is 1%, what is the default risk premium on these bonds? An investor in treasury securities expects inflation to be 2.5% in year 1, 3.2% in year 2 and 3.6% each year thereafter. Assume that the real risk-free rate is 2.75%, and that this rate will remain constant over time. 3-Year treasury securities yield 6.25% while 5-Year treasury securities yield 6.8%. What is the difference in the maturity risk premiums (MRPs) on the two securities? 4

5 Statistical Flashback Revision An event s probability is defined as a chance that the event will occur. For example, a weather forecaster might state, There is a 40 percent chance of rain today and a 60% chance that it will not rain; A probability distribution is a listing of all possible outcomes or events with a probability (Chance of occurrence) assigned to each outcome; Outcome Standard deviation is a statistical measure of the variability of a set of observations; Variance is the square of the standard deviation. Probability Rain 0.40 = 40% Not rain 0.60 = 60% Total 1.00 = 100% 5

6 Expected Rate of Return Definition & Computation The rate of return expected to be realized from an investment is known as its expected rate of return; From a calculation perspective, it is the weighted average of the probability distribution of possible outcomes. If we multiply each possible outcome by its probability of occurrence and then sum these products, we have a weighted average of outcomes. The weights are the probabilities and the weighted average is the expected rate of return. Symbolically, Expected Rate of Return = P 1 K 1 + P 2 K 2 + P 3 K P n K n Where, K i P i n = The ith possible outcome; = Probability of the ith possible outcome; = The number of possible outcomes. 6

7 Expected Rate of Return Example Given the following probability distributions, calculate the expected rates of return of ABC and XYZ: Demand for the Product Probability of the Demand Occurring ABC s Rate of Return XYZ s Rate of Return Strong % 20% Normal % 15% Weak % 10% Based upon the data given and the answer computed, which company do you think is a better investment option? 7

8 Risk Definition in the Financial Context Dictionary defines risk as a hazard, a peril or exposure to loss or injury. Thus, risk refers to the chance that some unfavorable event will occur; Although there is a difference in the specific definitions of risk and uncertainty, in most financial literature the two terms are used interchangeably; As far as its financial interpretation goes, risk means the uncertainty of future outcomes. In case of investments, it is the uncertainty or the variation associated with the quantum and the timing of future cash flows expected from a particular asset; While computing the returns for ABC and XYZ, we pointed that although same rates of return are expected of both the companies the range of returns specified in their respective probability distributions differs significantly. So which one do you think is more risky? Why? 8

9 Risk Measuring Stand-Alone Risk Before diving into the complexities of portfolio management, it is imperative that we know how to calculate the risk associated with investing in a single asset; We defined risk as uncertainty or variation associated with the expected returns or values of any asset. Statistically speaking, the variability in the context of probability distribution is captured by measuring standard deviation; Therefore in order to understand the risk associated with any particular asset, we need to compute the standard deviation of the probability distribution of its returns. 9

10 Risk Measuring Stand-Alone Risk In order to calculate standard deviation, we need to undertake the following steps: Step 1 : Calculate the expected rate of return; Step 2 : Subtract the expected rate of return from each possible outcome to obtain a set of deviations about the expected value; Step 3 : Square each deviation and then multiply the result by the probability of occurrence for its related outcome; Step 4 : Sum the products obtained in step 3 in order to obtain the variance; Step 5 : Find the square root of variance to obtain the standard deviation. Reverting back to the example of ABC and XYZ, factually conclude which is the most risky of the two investment options? 10

11 Measuring Stand-Alone Risk Another Measure Under the premise that RETURN is GOOD and RISK is BAD, if a choice is to be made between two investment options: With same expected rates of return but different standard deviations, most people would choose the one with the lower standard deviation and therefore the lower risk; With the same risk but different expected returns, investors would generally prefer the investment with the higher expected return. But, how do we choose between two investment options if one has a higher expected return and the other one has a lower standard deviation? To help answer this question, we use another measure of risk, the coefficient of variation (CV), which is the standard deviation divided by the expected return. 11

12 Measuring Stand-Alone Risk Another Measure The coefficient of variation (CV) provides a more meaningful basis for comparison when the expected returns and the risk of the investment options are not the same by showing the risk per unit of return. Mathematically: Coefficien t of Variation CV Standard Deviation Expected Return Based upon your calculation of the expected returns and standard deviations of XYZ and ABC, compute their coefficients of variation. 12

13 Markowitz Portfolio Theory An Introduction In reality, most financial assets are held as part of portfolios whereby investments are diversified within the same asset class and across various asset classes; Given its significance, Harry Markowitz presented a basic portfolio model in an attempt to gauge the expected rate of return of the portfolio and the accompanying risk; Before diving into the depths of the model, it is imperative that we keep in mind the underlying assumptions: Investors consider each investment alternative as being represented by a probability distribution of expected returns over some holding period; Investors aim at maximizing one-period s expected utility; Investors estimate the risk of the portfolio on the basis of variability of expected returns; Investors base their decisions solely on expected return and risk; For a given level of risk, investors prefer higher returns to lower returns. Similarly, for a given level of expected return, investors prefer less risk to more risk. 13

14 Markowitz Portfolio Theory Expected Portfolio Returns The expected return on a portfolio (K P ) is simply the weighted average of the expected returns on the individual assets in the portfolio, with the weights being the fraction of the total portfolio size invested in each asset: Portfolio Return = K P = W 1 K 1 + W 2 K 2 + W 3 K W n K n Where, W i K i n = Fraction of the portfolio s value invested in the asset; = Expected return on the individual asset; = The number of assets in the portfolio. 14

15 Markowitz Portfolio Theory Expected Portfolio Returns (Examples) Suppose you are the money manager of a PKR 4 million investment fund. Calculate the fund return if it consists of 4 assets with the following investments and expected returns: Asset Investment (In PKR) Expected Return (K) A 400,000 15% B 600,000 38% C 1,000,000 12% D 2,000,000 16% What will be the return on the portfolio if equal proportions are invested in asset A, B, C and D with individual expected returns of 12%, 11.5%, 10% and 9.5% respectively? 15

16 Statistical Flashback Revision The tendency of two variables to move together is called correlation; A measure which is used to asses the degree of relationship between two variables is known as correlation coefficient (r); Theoretically, correlation coefficient can only assume values in the range of -1 to +1. A value of -1 indicates a perfectly negative relationship with the two variables moving in opposite directions while a value of +1 indicates a perfectly positive relationship with the two variables moving together in the same direction. Correlatio n Coefficient r Covariance of Variables A & B (Std. Deviation of A)(Std.Deviation of B) 16

17 Markowitz Portfolio Theory Portfolio Risk As we just saw, the expected returns on a portfolio is simply the weighted average of the expected returns on the individual assets in the portfolio; However, unlike returns, the riskiness of a portfolio (σ P )is generally not the weighted average of the standard deviation of the individual assets in the portfolio. The portfolio s risk in most cases will be smaller than the weighted average of the individual assets riskiness; In fact, it is theoretically possible to combine assets of varying risks (As measured by their individual standard deviation) and combine them to form a portfolio that is completely riskless, with σ P = 0. 17

18 Markowitz Portfolio Theory Combination of Two Assets (Perfectly Negative Relationship) 45% 40% Asset (A) 45% 40% Asset (B) 45% 40% Combination (AB) 35% 35% 35% 30% 30% 30% 25% 25% 25% 20% 20% 20% 15% 15% 15% 10% 10% 10% 5% 5% 5% 0% 0% 0% -5% -5% -5% -10% -10% -10% -15% % %

19 Markowitz Portfolio Theory Combination of Two Assets (Perfectly Positive Relationship) 45% 40% Asset (A) 45% 40% Asset (B) 45% 40% Combination (AB) 35% 35% 35% 30% 30% 30% 25% 25% 25% 20% 20% 20% 15% 15% 15% 10% 10% 10% 5% 5% 5% 0% 0% 0% -5% -5% -5% -10% -10% -10% -15% -15% -15%

20 Markowitz Portfolio Theory Combination of Two Assets (Conclusions Drawn) In case of perfectly negative correlation (r = -1), assets tend to move in opposite directions and it becomes theoretically possible to form a portfolio with zero risk (σ P = 0). Thus, diversification takes its full effect when assets with perfectly negative correlation are combined to form a portfolio; However, in case of perfectly positive correlation (r = +1), assets tend to move together and a portfolio consisting of such assets would be exactly as risky as the individual assets. Thus, diversification does nothing to reduce risk if the portfolio consists of perfectly positively correlated assets; In between the two extremes, combining assets into a portfolio reduces risk but does not eliminate it completely. 20

21 Markowitz Portfolio Theory Calculating the Risk of a Two-Asset Portfolio Now that we have discussed of how combination of assets in a portfolio affects its risk, we can consider the formula for computing the variance for a portfolio of assets. In case of a two-asset portfolio, variance would be: σ P 2 = W 1 2 σ W 2 2 σ W 1 W 2 Cov 12 σ 2 P = W 2 1 σ W 2 2 σ W 1 W 2 σ 1 σ 2 Cor 12 Where, σ P W i σ i = Standard deviation of the portfolio; = Weights of the asset in the portfolio; = Standard deviation of the individual assets; Cov 12 = Covariance between the two assets; Cor 12 = Correlation between the two assets; 21

22 Markowitz Portfolio Theory Calculating the Risk of a Two-Asset Portfolio (Examples) You are considering two assets with the following characteristics: K 1 = 15% σ 1 = 10% W 1 = 50% K 2 = 20% σ 2 = 20% W 2 = 50% Compute the standard deviation of the portfolio when the correlation coefficient amongst the assets is assumed to be Given: K 1 = 10% σ 1 = 3% K 2 = 15% σ 2 = 5% Compute the standard deviation of a two-asset portfolio in which asset 1 has a weight of 60 percent and the correlation coefficient amongst the assets is

23 Markowitz Portfolio Theory Calculating the Risk of a Two-Asset Portfolio (Examples) The following are yearly rates of returns for Madison Corp. and for General Electric during a six-year period. Year Madison Corp. General Electric Given a coefficient of variation of between the two assets, compute the standard deviation of a portfolio in which both the above-mentioned assets have equal weights. 23

Archana Khetan 05/09/ MAFA (CA Final) - Portfolio Management

Archana Khetan 05/09/ MAFA (CA Final) - Portfolio Management Archana Khetan 05/09/2010 +91-9930812722 Archana090@hotmail.com MAFA (CA Final) - Portfolio Management 1 Portfolio Management Portfolio is a collection of assets. By investing in a portfolio or combination