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Risk and Return, Diversification and Portfolio Theory, CML and SML The way to the CAPM Week 3- Session 3 FINC5000

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Holding Period Return HPR= (P(end) – P(begin))/P(begin) +cash dividend/P(begin) Q1Q2Q3Q4 Assets under Management start of Quarter 1.0 Mln.1.2 Mln.2 Mln.0.8 Mln. HPR % 10%25%-20%25% Total Assets before inflows 1.1 Mln.1.5 Mln.1.6 Mln.1.0 Mln. Net inflow $ Mln. 0.1 Mln.0.5 Mln.- 0.8 Mln.0 Mln. Assets under Management end of Quarter 1.2 Mln.2.0 Mln.0.8 Mln.1.0 Mln.

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Returns? Arithmetic Return: (10%+25%-20%+25%)/4=10% Geometric Return: ((1.10)(1.25)(0.8)(1.25))^(1/4) – 1= 8.29% $ weighted return: Net cash flow: -1 -0.1 -0.5 +0.8 +1.0 IRR%= 4.17%

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IRR% in Excel

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Scenario Analysis State EconomyScenario (s)Probability p(s)HPR r(s) Boom125%44% Normal250%14% Recession325%-16%

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Class Assignment: Return and Risk? E(r)= Σp(s)r(s)= 25%*44%+50%*14%+25%* -16%= 14% Var(r)=Σp(s) (r(s) – E(r))^ 2 = 25%(44-14) 2 +50%(14-14) 2 +25%(-16-14) 2 = 450 STDEV(r)= (450)^ (0.5) = 21.21% Calculate HPR for below stock for each of the three scenarios, and calculate HPR and STDEV of HPR…is the stock is now selling at $ 23.50 BusinessscenarioprobabilityEnd of year share price estimate Annual dividend Good135%$ 35$ 4.40 Normal230%$ 27$ 4 Stagnate335%$ 15$ 4

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Portfolios Assume a portfolio of riskless and risky assets: We invest y in the risky asset and (1-y) in the risk free asset Rf=7% E(Rp)=15% STDEV(Rp)=22% If y=1 what is your expected return? (P) If y=0 what is your expected return? (F) You may choose any combination of y and (1-y)… your reward/risk will be in between… Draw the CAL (Capital Allocation Line) Slope: (E(Rp)-Rf)/STDEV(Rp)= (15%-7%)/22%=0.36

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Portfolios Assume a portfolio of two risky assets lets say Bonds and Stocks: How to understand how returns and risk on these assets interact? Assume; Stock FundBond Fund ScenarioProbabilityRate of ReturnCol B x Col CRate of ReturnCol B x Col E Recession0.3-11-3.3164.8 Normal0.4135.262.4 Boom0.3278.1-4-1.2 Expected or Mean Return:SUM:10.0SUM:6.0

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Portfolios Stock Fund Bond Fund Deviation RatefromColumn BRatefromColumn B ofExpectedSquaredxofExpectedSquaredx ScenarioProb.Return DeviationColumn EReturn DeviationColumn I Recession0.3-11-21441132.3161010030 Normal0.413393.66000 Boom0.3271728986.7-4-1010030 Variance = SUM222.6 Variance:60 expected Return= 10% Standard deviation = SQRT(Variance)14.92 expected Return= 6% Std. Dev.:7.75 Stock FundBond Fund E(r)= 10%E(r)= 6% Risk (STDEV)= 14.92%Risk (STDEV)= 7.75%

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Assume You invest 60% in stocks and 40% in bonds What is the E(Rp) and STDEV(Rp) of this portfolio?

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Your answer …

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How can the STDEV(Rp)

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Historical data…(do it!) For 2 risky assets in the same portfolio: R(p)= wb*rb+ws*rs E(Rp)=wbE(rb) + wsE(rs) Var(Rp)= (wb*σ(b)) 2 +(ws*σ(s)) 2 +2*wb*ws*Cov(rb,rs) If Cov(rb,rs)= ρ(bs) *σ(b)*σ(s) then replace in above equation and get: Var(Rp)== (wb*σ(b)) 2 +(ws*σ(s)) 2 +2*wb*ws*σ(b)*σ(s)*ρ(bs) And the STDEV(Rp) = (Var(Rp))^ 0.5 IF : E(rb)=6% E(rs)=10% σ(b)=12% σ(s)=25% ρ(bs)=0 wb=0.5 and ws=0.5 Calculate: E(Rp) and STDEV(Rp) Calculate: E(Rp) and STDEV(Rp) if we change wb=0.75 and ws=0.25

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Your answer… E(Rp)= 50%*6%+50%*10%= 8% Var(Rp)=(0.5*12)^2+(0.5*25)^2+2*0.5*12*0.5*25*0=192.25 STDEV(Rp)=(192.25)^0.5= 13.87% If wb=0.75 and ws=0.25 E(Rp)=7% Var(Rp)=(0.75*12)^ 2 +(0.25*25)^ 2 +2*(0.75*12)*(0.25*25)*0= 120 STDEV(Rp)= (120)^ 0.5 = 10.96% So you started say with bonds and your return was 6% with risk 12% (stdev) and you added stocks to your portfolio and it REDUCED your risk! to 10.96% at an even higher return… SEE HERE THE POWER OF DIVERSIFICATION!

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Searching for lowest risk portfolio… Input data E(r S )E(r B ) S B SB 1062512-0.50.510 Portfolio Weights Expected Return wSwS w B = 1 - w S E(r P ) = Col A x A3 + Col B x B3 Std Deviation* 0.01.06.0012.00 0.10.96.408.309.7912.2413.3011.09 0.18730.81276.755.078.4512.7614.4310.8183 0.20.86.804.608.3212.8514.6010.8240 0.30.77.200.907.9913.7815.9011.26 0.40.67.602.808.9414.9617.2012.32 0.5 8.006.5010.8316.3518.5013.87 0.60.48.4010.2013.2717.8919.8015.75 0.70.38.8013.9016.0119.5521.1017.87 0.80.29.2017.6018.9121.3022.4020.14 0.90.19.6021.3021.9223.1223.7022.53 1.00.010.00 25.00 Note: The weight of stocks in the minimum variance portfolio is w S = ( B ^2 - B S )/( S ^2 + B ^2 - 2* B S ) =.1873 * The formula for portfolio standard deviation is: SQRT[ (Col A*$C$3)^2 + (Col B*$D$3)^2 + 2*$E$3*Col A*$C$3*Col B*$D$3 ]

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Portfolio at different correlations…

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Capital Market Line (CML) The Rf connected to the Optimal Risky Portfolio Complete Portfolio: Choice of investor on the CML depends on risk averseness Minimum Variance Portfolio the point most North West on the Efficient Frontier…

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Individual securities… Move in tandem with the market (systematic risk) but correct for different risk levels (betas) (E(Rm) – Rf)/1 = (E(R(Dell)) – Rf) /Beta(Dell) Thus E(R(Dell))= Rf + Beta(Dell)*(E(Rm) – Rf) The general expression of the CAPM! Note we are assuming that all investors are fully diversified in portfolios and that therefore they only need to be compensated for systematic risk!

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Security Market Line (SML) Relationship between Risk (Beta) and return of an individual Asset… From this picture we see Rf=6% Beta=1.2 and assume the return on the Market is expected to be 14%...then The SML predict: E(r) = 6%+1.2(14% - 6%)= 15.6% if you believe instead that this stock will provide 17% return then the implied alpha (surprise) would be 1.4% (see picture)

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Understanding Regression statistics of Betas

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Homework: Calculating Beta Use monthly (at least 5 year data up to 31 March 2012) Use monthly (5 year data) up to 31 March 2007) Perform an OLS Regression for both periods Show your results/output Estimate the Beta for your Company Interpret Beta and its reliability Interpret your Regression outputs (Stdev(beta), R(sq), t score, p score)

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Collecting/Interpreting data Please use 5 years of Monthly returns (at least 60 returns) Multiple R: the correlation coefficient between the excess return on the companys returns and the S&P500 (the market) was 0.7205 The adjusted R square: correlation coefficient squared and adjusted for degrees of freedom; telling us that 47.54% of the variation in excess returns in the company is explained by the variation in the excess return on the market… Standard Error: In about two third of the observed periods the excess return was between +/- 3.56% indicating some volatility From ANOVA use: SS/Df (degrees of freedom) indicate the variance of excess returns; STDEV= Var^0.5 per period Intercept close to 0 ; Beta=1.369 estimated the slightly negative alpha indicates that the returns of the stock were slightly below the SML in this period however the t-statistic and p-value indicate that the alpha estimate is not very reliable; the beta estimate is much better at t= 3.446 (significantly different from 0) and p close to 0! The 95% interval shows a very disappointing large area in which the true beta may be (between 0.5 and 2.2)…this area is too large and therefore the estimate is not very reliable at 1.369….

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Homework: The Portfolio Create a portfolio of at least 3 stocks Based on historical data calculate return and risk (stdev(return)) and show your calculations Show that a portfolio of these stocks may create better reward/risk ratio than investing in the individual stocks Given the data what is the Minimum Variance Portfolio? (estimate/calculate wA, wB, and wC) Draw the efficient frontier of the 3 stocks

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