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

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

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.

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%

IRR% in Excel

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

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

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

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

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%

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

How can the STDEV(Rp)<STDEV(Rb) ? Answer: diversification… Stocks and bonds do not move in tandem but in opposite directions… Cov(R(s),R(b))=Σp(s)*(r(s)-E(r(s)))(r(b)-E(r(b)))= -114… Correlation Coefficient=ρ(sb)= Cov(R(s),R(b))/STDEV(Rs)*STDEV(Rb)= -114/(14.92*7.75)= -.99

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

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!

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 ]

Portfolio at different correlations…

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…

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!

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)

Understanding Regression statistics of Betas

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)

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….

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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