Measuring portfolio risk


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Measuring portfolio risk

  2. 2. Again these two concepts namely covariance and correlation coefficient are importantCovariance is a measure that combines the variance (volatility) of a stock’s return with the tendency of those returns to move up or down at the same time other stocks move up and downFor instance the covariance between two stocks X and Y tells us whether the returns of the two stocks tend to RISE and FALL together and how large those movements tend to be SAN LIO 2
  3. 3. FORMULACOV(XY)= ∑(PRX-ERX)(PRY-ERY)PWHEREPRX= Possible return of stock XERX=Expected return of stock XPRY= Possible return of stock YERY=Expected return of stock YP= Probability of the category accordingly SAN LIO 3
  4. 4. EXAMPLEYou have been provided with two stocks with the following outcomes as follows possible returnProbability X Y0.10 6% 14%0.20 8% 12%0.40 10% 10%0.20 12% 8%0.10 14% 6% SAN LIO 4
  5. 5. The expected returns of the two stocks X and Y are 10% and 10% respectivelyThe standard deviation for the two stocks is provided as follows for X an d Y respectively 2.2% and 2.2%REQUIREDDetermine the covariance between the two stocksSOLUTION SAN LIO 5
  6. 6. = (6-10)(14-10)(0.10)+(8-10)(12-10)(0.20)+(10-10)(10- 10)(0.40)+(12-10)(8-10)(0.20)+(14-10)(6-10)(0.10)=-1.6+-0.80+0+-0.8+ -1.6= -4.80 This negative sign is an indication that the rates of return on stock X and Y tend to move in opposite directions. Is this consistent with the figures provided? We can also plot X against Y on a graph and observe SAN LIO 6
  7. 7. EXAMPLE TWO (ALL TO DO) Possible return A BProbability0.10 6% 4%0.20 8% 6%0.40 10% 8%0.20 12% 15%0.10 14% 22% SAN LIO 7
  8. 8. Assume that both A and B have expected return of 10%REQUIREDCalculate the Covariance (ALL TO DO) SAN LIO 8
  9. 9. SOLUTION = +10.80Meaning these assets tend to move together as indicated by the +ve signNOTE if either stock has zero standard deviation, meaning it is RISKLESS, then all its deviations (PR-ER) will be zero and the covariance will also be zero SAN LIO 9
  10. 10. CORRELATION COEFFICIENTThis is calculated as the covariance of two assets divided by their standard deviations thusFORMULACORRELATION C= COV(XY) σXσYEXAMPLECalculate the coefficient of correlation between X and Y in our previous exampleSOLUTION SAN LIO 10
  11. 11. = -4.80 2.2*2.2= -4.80 = -1.0 4.84MEANINGSince the sign of correlation of coefficient is the same as the sign for covariance, i.e. positive sign means the variables move together and negative sign means the variables move in opposite directions, SAN LIO 11
  12. 12. and that if they are close to zero, the variables are independent of each other;Then we can observe that stock X and Y are perfectly negatively correlatedEXAMPLE TWO FOR ALLIf we are given the standard deviation of A as 2.2% AND of B as 5.3%REQUIREDCalculate the coefficient of correlation of the two assets SAN LIO 12
  13. 13. SOLUTIONCOLL C= 10.8 = 10.8 2.2*5.3 11.66 = 0.92MEANINGThere is a strong positive relationship between the two assets and therefore these assets will tend to bear similar risks SAN LIO 13
  14. 14. REMEMBER AGAIN COVARIANCE- is the measure that combines the variance or the volatility of a stock’s return with the tendency of those returns to move up or down at the same time other stocks move up or down CORRELATION COEFFICIENT- Is used to measure the degree of co-movement between two variables (stocks) . The correlation coefficient standardizes the covariance by dividing it by a product term, which facilitates comparisons by putting things to a similar scale. NOTE THAT it is difficult to interpret the magnitude of the covariance term. SAN LIO 14
  15. 15. CAPITAL ASSET PRICING MODELCapital Asset Pricing Model basically helps us determine the relationship between RISK and required rates of RETURN on ASSETS when held in a well diversified portfolio.The attitude of CAPM is the SECURITY MARKET LINESEE THE LINE SAN LIO 15
  16. 16. SECURITY MARKET LINE ER SML Assets here are under-priced RFR Assets here are over-pricedCOV SAN LIO 16
  17. 17.  NOTE that covariance is the relevant risk measure as discussed earlier. We shall at this stage introduce the asset Beta (β) Beta is a standardized measure of risk because it relates this covariance to the variance of the market portfolio. Consequently, the market portfolio has a beta of 1 The SML tells us that an individual stock’s required rate of return is equal to the RFR PLUS A PREMIUM for bearing risk (the risk premium) SAN LIO 17
  18. 18. The CAPITAL MARKET LINE which takes the same shape specifies a linear relationship between EXPECTED RETURN and RISK, with the slope of the CML being equal to the expected return on the market portfolio of risky stocks MINUS the risk-free rate (called the market risk premium) , all divided by the standard deviation of returns on the market portfolio SAN LIO 18
  19. 19. CAPM ASSUMPTIONSASSUMPTIONS- Builds on Markowitz portfolio modelAll investors are Markowitz efficient investors- risk-return utility functionInvestors can borrow and lend any amount of money at risk-free-rate of returnAll investors have homogeneous expectations i.e. they estimate intended probability distribution for future rates of return SAN LIO 19
  20. 20.  All investors have the same one-period time horizon e.g. one month, six or one year All investors are infinitely divisible i.e. it is possible to buy or sell fractional shares of any asset or portfolio There are no taxes or transaction costs involved in buying or selling assets e.g. churches There is no inflation or any change in interest rates or inflation fully anticipated SAN LIO 20
  21. 21. Capital markets are in equilibrium i.e. we begin with all investments properly priced in line with their risk levels SAN LIO 21
  22. 22. THE BETA COEFFICIENTThis is seen as a standardized measure of systematic risk because it relates the covariance to the variance of the market portfolioThe market portfolio has a beta of 1 (one)Betas are standardized around one. b = 1 ... Average risk investment b > 1 ... Above Average risk investment b < 1 ... Below Average risk investment = 0 ... Riskless investment SAN LIO 22
  23. 23. FACTORS AFFECTING BETALine of businessAmount of financial leverage undertaken by the firmDividend payoutLiquidityFirm sizeRate of growth of the firm SAN LIO 23
  24. 24. EXPECTED RETURN OF RISKY ASSETSDetermined by the RFR plus a risk premium for the individual assetThe risk premium is determined by the systematic risk of the asset (BETA) and the prevailing MARKET RISK PREMIUMS (Rm- RFR)EXAMPLE SAN LIO 24
  25. 25.  Assume the betas of the following stocks have been computed (DONE USING REGRESSION LINE) STOCK BETA A 0.70 B 1.00 C 1.15 D 1.40 E -0.30Require: calculate expected rates of return assuming an economy’s RFR of 5% and return on market portfolio (Rm) to be 9% SAN LIO 25
  26. 26. SOLUTIONThe market risk premium = 9%-5%=4%THUSE(R1)= RFR +β(Rm-RFR)E(RA)= 0.05+ 0.7(0.09-0.05) = 0.078= 7.8%CALCULATE THE EXPECTED RETURN FOR THE OTHER ASSETS SAN LIO 26
  27. 27.  B= 9.0% C= 9.6% D= 10.6% E(RE)= 0.05+(-0.30)(0.09-0.05) = 0.05-0.012 = 0.038= 3.8%MEANING These are the required rates of return that these stocks should provide based on their systematic risks and the prevailing SML (Security Market Line-relates E(R1) and CV) SAN LIO 27
  28. 28. NOTEAt equilibrium all assets and all portfolios of assets should plot on the SMLMeans all assets should be priced so that their estimated rates of returns which in effect are the actual holding period rates of return that you anticipate, are in harmony with their levels of systematic riskSecurities with an estimated rate of return above the SML are considered underpriced SAN LIO 28
  29. 29. Because this means the estimated return is above its required rate of return based on its systematic riskAssets with estimated rates of return that plot below the SML are considered overpriced because it implies your estimated rate of return is below what you should require based on the asset’s systematic risk. SAN LIO 29
  30. 30. THE ARBITRAGE PRICING THEORYNote that CAPM is a single-factor model since it specifies risk as a function of only one factor-the security’s beta coefficientFor example consider a situation where the personal tax rates on capital gains are lower than those on dividends, investors will value capital gains more than dividendsThus if two stocks had the same market risk, the stock paying the higher dividend would have the higher required rate of returnWhy? Due to the prevailing dividend policy SAN LIO 30
  31. 31. In this particular case, required returns would be a function of TWO factors namely Market risk Dividend policyAdditionally, many factors may be required to determine the equilibrium risk/return relationship rather than just one or twoStephen Ross tries to address this problem by introducing the approach called the ARBITRAGE PRICING THEORY SAN LIO 31
  32. 32.  This approach can include any number of risk factors meaning the required return could be a function of several factorsEXAMPLELets assume that all stocks returns depend on three factors; inflation, industrial production and aggregate degree of risk aversion.Lets further assume that the risk-free rate is 8%; the required rate of return is 13% on a portfolio with unit sensitivity (β=1) to inflation and ZERO sensitivities SAN LIO 32
  33. 33. (β=0) to industrial production and degree of risk aversion; the required return is 10% on a portfolio with unit sensitivity to industrial production and ZERO sensitivities to inflation and degree of risk aversion; the required return is 6% on a portfolio (the risk-bearing portfolio) with unit sensitivity to the degree of risk aversion and ZERO sensitivities to inflation and industrial production.Finally lets assume that the stock has factor sensitivities (betas) of 0.9 to the inflation portfolio, 1.2 to the industrial production portfolio and -0.7 to SAN LIO 33
  34. 34. risk –bearing portfolio.REQUIREDCalculate the stocks required rate of return using the APT approachSOLUTIONFORMULARR= ∑RFR + (SRR-RFR)SβWHERE SAN LIO 34
  35. 35.  RR= Requires Rate of return RFR= Risk Free Rate SRR= Subjective Required Rate of Return Sβ= Subjective BetaTHUSRR= 8%+(13%-8%)0.9+ (10%-8%)1.2 +(6%-8%)-0.7= 8% + 4.5+2.4+1.4=16.3% Means investors will not buy the stock if it earns them LESS than 16.3% SAN LIO 35
  36. 36. NOTEThis approach is build on very complex mathematical and statistical theories and its practical use has been limitedUsage may increase in the future however and thus the need to be aware of the approach accordingly SAN LIO 36