# Risk Return and Portfolio Theory by MikeJenny

VIEWS: 4 PAGES: 39

• pg 1
```									Money, Banking & Finance
Lecture 3
Risk, Return and Portfolio Theory
Aims
• Explain the principles of portfolio diversification
• Demonstrate the construction of the efficient
frontier
• Show the trade-off between risk and return
• Derive the Capital Market Line (CML)
• Show the calculation of the optimal portfolio
choice based on the mean and variance of
portfolio returns.
Overview
• Investors choose a set of risky assets
(stocks) plus a risk-free asset.
• The risk-free asset is a term deposit or
government Treasury bill.
• Investors can borrow or lend as much as
they like at the risk-free rate of interest.
• Investors like return but dislike risk (risk
averse).
Preferences of Expected return
and risk
• We have seen how expected return is defined in Lecture 2.
• The investor faces a number of stocks with different
expected returns and differ from each other in terms of
risk.
• The expected return on the portfolio is the weighted mean
return of all stocks. First moment.
• Risk is measured in terms of the variance of returns or
standard deviation. Second moment.
• Investor preferences are in terms of the first and second
moments of the distribution of returns.
Investor Utility function
U  U E ( R p ), p 
U                  U
 U1  0;         
 U2  0
E ( R p )             p
dU            dU
dU  U1              
U2      0
dE ( R p )     d p
dU    dU          dE ( R p )    U1
/                            0
d p dE ( R p )    d p          
U2
Preference Function

E(Rp) Expected return   U2

U0

σp Risk
Expected return

n
R p   i Ri
i 1

R p  1R1  2 R2
E R p   1E ( R1 )  2 E ( R2 )
Risk

         E R
p  E(Rp     )  E 1  R1  E ( R1 )   2 R 2  E ( R2 ) 2
2                       2
p

 12 12  2  2  212 E R1  E ( R1 ) R2  E ( R2 ) 
2 2

E R1  E ( R1 ) R2  E ( R2 )   CovR1 , R 2 
CovR1 , R2 
 
1, 2
Var( R1 )Var( R2 )
 p  12 12  2  2  212 1, 2 1 2
2              2 2
Return and risk
• How do return and risk vary relative to each other
as the investor alters the proportion of each of the
assets in the portfolio?
• Assume that returns, risk and the covariance are
fixed and simply vary the weights in the portfolio.
• Let E(R1)=8.75% and E(R2)=21.25
• Let w1=0.75 and w2=0.25
• E(Rp)=.75x8.75+.25x21.25=11.88
• σ1=10.83, σ2=19.80, ρ1,2=-.9549
Portfolio Risk
• σ2p=(0.75)2x(10.83)2+(0.25)2x(19.80)2+2x(0
.75)x(0.25)x(-0.95)x(10.83)x(19.80)
• =13.7
• σp=√13.7=3.7
• Calculate risk and return for different
weights
Portfolio risk and return
Equity 1 Equity 2 E(Rp)    Risk
State   w1       w2
1       1        0        8.75%    10.83%
2       0.75     0.25     11.88%   3.70%
3       0.5      0.5      15%      5%
4       0        1        21.25    19.8%
Locus of risk-return points

Expected
return                      (0,1)

(.5,.5)

(.75,.25)

(1,0)           Risk=standard
deviation
Risk – return locus
• Can see that the locus of risk and returns vary
according to the proportions of the equity held in
the portfolio.
• The proportion (0.75,0.25) is the lowest risk point
with highest return.
• The other points are either higher risk and higher
return or low return and high risk.
• The locus of points vary with the correlation
coefficient and is called the efficient frontier
Choice of weights
• How does the portfolio manager choose the weights?
• That will depend on preferences of the investor.
• What happens if the number of assets grows to a large
number.
• If n is the number of assets then will need n(n-1)/2
covariances - becomes intractable
• A short-cut is the Single Index Model (SIM) where each
asset return is assumed to vary only with the return of the
whole market (FTSE100, DJ, etc).
• For ‘n’ assets the efficient frontier defines a ‘bundle’ of
risky assets.
‘n’ asset case

E R p    i E Ri 
n

i 1
n    n
   i j i j  ij
2
p
i 1 j 1
How is the efficient frontier
derived?
• The shape of the efficient frontier will depend on
the correlation between the asset returns of the two
assets.
• If the correlation is ρ = +1 then the portfolio risk is
the weighted average of the risk of the portfolio
components.
• If the correlation is ρ = -1 then the portfolio risk
can be diversified away to zero
• When ρ < +1 then not all the total risk of each
investment is non-diversifiable. Some of it can be
diversified away
Correlation of +1

 p   2 12  (1   ) 2  2  2 1, 2 (1   ) 1 2 
2

1, 2  1
 p   2 12  (1   ) 2  2  2 (1   ) 1 2 
2

     ( 1  (1   ) 2 ) 2
  1  (1   ) 2
Correlation of -1
 p   2 12  (1   ) 2  2  2 (1   ) 1 2 
2

p      1  (1   ) 2 2

min risk
 p   1  (1   ) 2  0
  1   2   2  0
2

1   2
Check

 2              2 
p 
   1  1 
       2
 1  2           1   2 

 1 2    1 2
p                   0
1   2 1   2
Correlation < +1

 p   1  (1   ) 2
Efficient frontier
E(Rp)

Ρ = -1

-1 < Ρ < +1   Ρ = +1

σp
The general case – applied to
two assets
 p   2 12  (1   ) 2  2  2 1, 2 (1   ) 1 2 
2

 p   2 12  (1   ) 2  2  2 1, 2 (1   ) 1 2 
2                           2

d p
2

 2 12  2(1   ) 2  2 1, 2 1 2  4 1, 2 1 2  0
2

d
  12  (1   ) 2  1, 2 1 2  2 1, 2 1 2  0
2

  ( 12   2 )   2  2 1, 2 1 2   1, 2 1 2
2       2

  ( 12   2  2 1, 2 1 2 )   2  1, 2 1 2
2                        2

 2 ( 2  1, 2 1 )
 2
[ 1   2  2 1, 2 1 2 ]
2
Efficient Frontier

E(Rp)

X

Y

σp
Risk-free asset
• Lets introduce a risk-free asset that pays a
rate of interest Rf.
• The rate Rf is known with certainty and has
zero variance and therefore no covariance
with the portfolio.
• Such a rate could be a short-term
government bill or commercial bank
deposit.
One bundle of risky assets
• Take one bundle of risky assets and allow the investor to
lend or borrow at the safe rate of interest. The investor can;
• Invest all his wealth in the risky bundle and undertake no
lending or borrowing.
• Invest less than his total wealth in the single risky bundle
and the rest in the risk-free asset.
• Invest more than his total wealth in the risky bundle by
borrowing at the risk-free rate and hold a levered portfolio.
• These choices are shown by the transformation line that
relates the return on the portfolio with one risk-free asset
and risk.
Transformation line

E R p   R f  (1   ) RN
 p   2 2  (1   ) 2  N   (1   ) f  N  fn
2
f
2

  (1   ) 
2
p
2    2
N

 p  (1   ) N
Linear Opportunity set
• Let the risk-free rate Rf = 10% and the return on the
bundle of assets RN = 22.5%.
• The standard deviation of the returns on the bundle
σN = 24.87%.
• The weights on the risky bundle and the risk-free
asset can be varied to produce a range of new
portfolio returns.
Portfolio Risk and Return
State    T-bill   Equity   E(Rp)    σp
(1-φ)    φ
1        1        0        10%      0%
2        0.5      0.5      16.25%   12.44%
3        0        1        22.5%    24.87%
4        -0.5     1.5      28.75%   37.31%
Transformation line
• The transformation line describes the linear risk-
return relationship for any portfolio consisting of a
combination of investment in one safe asset and
one ‘bundle’ of risky assets.
• At every point on a given transformation line the
investor holds the risky assets in the same fixed
proportions of the risky portfolio ωi.
Transformation line

E(Rp)

-0.5 borrowing + 1.5
0.5 lending + 0.5                    in risky bundle
in risky bundle
No lending all
investment in
bundle

Rf
All lending
σp
A riskless asset and a risky
portfolio
• An investor faces many bundles of risky assets (eg
from the London Stock Exchange).
• The efficient frontier defines the boundary of
efficient portfolios.
• The single risky asset is replaced by a risky
portfolio.
• We can find a dominant portfolio with the riskless
asset that will be superior to all other
combinations.
Combining risk-free and risky
portfolios
E(Rp)

C
B
A
Rf

σp
Borrowing and Lending
• The investor can lend or borrow at the risk-
free rate of interest rate.
• The risk-free rate of interest Rf represents
the rate on Treasury Bills or some other
risk-free asset.
• The efficiency boundary is redefined to
include borrowing.
Borrowing and lending frontier
C
E(Rp)

B

Rf

A
σp
Combined borrowing and
lending at different rates of
interest
• The investor can borrow at the rate of
interest Rb
• Lend at the rate of interest Rf
• The borrowing rate is greater than the risk-
free rate. Rb > Rf
• Preferences determine the proportions of
lending or borrowing,
Combining borrowing and
lending
D
E(Rp)
Q

C
B
Rb
P

Rf               A

σp
Separation Principle
• Investor makes 2 separate decisions
• Given knowledge of expected returns, variances
and covariances the investor determines the
efficient frontier. The point M is located with
reference to Rf.
• The investor determines the combination of the
risky portfolio and the safe asset (lending) or a
leveraged portfolio (borrowing).
Market portfolio and risk
reduction
Portfolio
risk

Diversifiable
risk

Non-                                        Number of
diversifiable                               securities
risk
20
Summary
• We have examine the theory of portfolio
diversification
• We have seen how the efficient frontier is
constructed.
• We have seen that portfolio diversification
reduces risk to the non-diversifiable
component.

```
To top