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)

Q1 Q2 Q3 Q4Assets 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

Ml n

-1-0. 1-0. 50. 8

1

I RR% 4. 17%

Scenario AnalysisState Economy Scenario (s) Probability p(s) HPR r(s)

Boom 1 25% 44%

Normal 2 50% 14%

Recession 3 25% -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

Business scenario probability End of year share price estimate

Annual dividend

Good 1 35% $ 35 $ 4.40

Normal 2 30% $ 27 $ 4

Stagnate 3 35% $ 15 $ 4

Portfolio’s

• 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

Portfolio’s

• 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 Fund Bond Fund

Scenario Probability Rate of Return Col B x Col C Rate of Return Col B x Col E

Recession 0.3 -11 -3.3 16 4.8

Normal 0.4 13 5.2 6 2.4

Boom 0.3 27 8.1 -4 -1.2

Expected or Mean Return: SUM: 10.0 SUM: 6.0

Portfolio’s      Stock Fund Bond Fund

    Deviation   Deviation  

  Rate from Column B Rate from Column B

  of Expected Squared x of Expected Squared x

Scenario Prob. Return Return Deviation Column E Return Return Deviation Column I

Recession 0.3 -11 -21 441 132.3 16 10 100 30

Normal 0.4 13 3 9 3.6 6 0 0 0

Boom 0.3 27 17 289 86.7 -4 -10 100 30

      Variance = SUM 222.6     Variance: 60

    expected Return= 10%  

Standard deviation = SQRT(Variance) 14.92

 expectedReturn=6%   Std. Dev.: 7.75

Stock Fund Bond 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?

Your answer …

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(rS) E(rB) S B SB SB SB SB SB

10 6 25 12 -1 -0.5 0.5 1 0

             Portfolio Weights             Expected Return          

wS wB = 1 - wS E(rP) = Col A x A3 + Col B x B3         Std Deviation*

0.0 1.0 6.00 12.00 12.00 12.00 12.00 12.00

0.1 0.9 6.40 8.30 9.79 12.24 13.30 11.09

0.1873 0.8127 6.75 5.07 8.45 12.76 14.43 10.8183

0.2 0.8 6.80 4.60 8.32 12.85 14.60 10.8240

0.3 0.7 7.20 0.90 7.99 13.78 15.90 11.26

0.4 0.6 7.60 2.80 8.94 14.96 17.20 12.32

0.5 0.5 8.00 6.50 10.83 16.35 18.50 13.87

0.6 0.4 8.40 10.20 13.27 17.89 19.80 15.75

0.7 0.3 8.80 13.90 16.01 19.55 21.10 17.87

0.8 0.2 9.20 17.60 18.91 21.30 22.40 20.14

0.9 0.1 9.60 21.30 21.92 23.12 23.70 22.53

1.0 0.0 10.00   25.00 25.00 25.00 25.00 25.00

Note: The weight of stocks in the minimum variance portfolio is

          wS = (B^2 - BS)/(S^2 + B^2 - 2*BS) = .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 (beta’s)

• (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 portfolio’s 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 Beta’s

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 it’s 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

company’s 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