Transcript Risk and Return - Webster in china

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
Q4
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.
Assets under
Management
start of Quarter
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
Mln
-1
-0.1
-0.5
0.8
1
IRR%
4.17%
Scenario Analysis
State Economy
Scenario
Boom
(s)
Probability p(s)
HPR
1
25%
44%
Normal
2
50%
14%
Recession
3
25%
-16%
r(s)
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
Scenario
Probability
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:
Rate of Return
Bond Fund
Col B x Col C
SUM:
10.0
Rate of Return
Col B x Col E
SUM:
6.0
Portfolio’s
Stock Fund
Bond Fund
Deviation
Rate
from
of
Expected
Prob.
Return
Return
Recession
0.3
-11
-21
Normal
0.4
13
Boom
0.3
27
Scenario
Deviation
Column B
Rate
from
Column B
Squared
x
of
Expected
Squared
x
Deviation
Column E
Return
Return
Deviation
Column I
441
132.3
16
10
100
30
3
9
3.6
6
0
0
0
17
289
86.7
-4
-10
100
30
Variance = SUM
222.6
Variance:
60
expected
Return=
14.92 6%
Std. Dev.:
7.75
expected
Return= 10%
Standard deviation
= SQRT(Variance)
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)
sS
sB
rSB
rSB
rSB
rSB
rSB
10
6
25
12
-1
-0.5
0.5
1
0
Portfolio Weights
Expected Return
wS
wB = 1 - wS
0.0
1.0
E(rP) = Col A x A3 + Col B x B3
6.00
12.00
12.00
12.00
12.00
Std Deviation*
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 = (sB^2 - sBsSr)/(sS^2 + sB^2 - 2*sBsSr) = .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…
12.00
10.00
Series1
8.00
Series2
6.00
Series3
Series4
4.00
Series5
2.00
0.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
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