S09 Portfolio Theory
S09 Portfolio Theory
S09 Portfolio Theory
2 n
Var,σ p = pi (Rp - Rp ) 2
n
Cov(R1 ,R2 ) = ∑ pi (R1 - R1 )(R2 - R2 )
1
Cov(R1 ,R2 )
Coefficient of Correlation, ρ =
σ1σ 2
2 2 2 2 2
Variance of the portfolio, σ p = w1 σ1 + w 2 σ 2 + 2w1w 2Cov(R1 ,R2 )
Return, %
30 x Security 2
ρ = +1.00
ρ = -1.00
-1.00 < ρ < 1.00
15 x Security 1
10 20
Standard Deviation, %
Portfolio Theory 31
RAJIV SRIVASTAVA
ISO-UTILITY CURVE
• Iso-utility means that two of more investments despite different risks
and returns can provide identical utility.
• The locus of all those investments with different returns and risks that
provide same utility is called iso-utility curve.
– For example consider an investor with risk aversion coefficient, A = 1.00 the utility
of an investment with 20% return and 20% risk is U = R1 – 0.005σ12= 20 – 0.005 x 202
= 18.
– If another investment offers 25% return then the same investor would be
indifferent if the risk associated with this investment is 37.42% as this would provide
the same utility of 18.00.
U = R1 – 0.005σ12= R2 – 0.005σ22=25 – 0.005 x 20 x 20
= 25 – 0.005 σ22 or σ22 = 1400 equivalent to σ2 = 37.42%
Return, %
Increasing utility
U=5
U=4 B
O Efficient
R0 U=3 Frontier
U=2
A
U=1
σ0 Standard Deviation, %
RAJIV
Portfolio Theory 36
SRIVASTAVA
ISO-UTILITY CURVE - EXAMPLE
• Iso-utility means that two or more investments despite having
different risks and returns can still provide identical utility.
• The locus of all those investments with different returns and risks that
provide same utility is called iso-utility (indifference) curve.
– For example consider an investor with risk aversion coefficient, A = 1.00 the utility
of an investment with 20% return and 20% risk is U = R1 – 0.005σ12= 20 – 0.005 x 202
= 18.
– If another investment offers 25% return then the same investor would be
indifferent if the risk associated with this investment is 37.42% as this would provide
the same utility of 18.00.
U = R1 – 0.005σ12= R2 – 0.005σ22=25 – 0.005 x 20 x 20
= 25 – 0.005 σ22 or σ22 = 1400 equivalent to σ2 = 37.42%
Portfolio Theory 38
RAJIV SRIVASTAVA
PROPERTIES OF UTILITY CURVE AND RISK
AVERSION
Properties of utility curve and its relationship with risk aversion are:
– Higher the coefficient of risk aversion more risk averse the investor
is. More risk averse investor would require greater increase in return
for the same increase in risk.
– Greater the risk aversion higher is the value of coefficient, A
– Higher the coefficient of risk aversion steeper is the utility curve.
– For the same investment i.e. asset with same return and risk an
investor with higher coefficient of aversion would have lesser utility
value than the investor with lower coefficient of aversion.
U=1
• No two utility curves would ever
intersect each other. They Standard Deviation, %
remain parallel to one another.
Return, %
15 - 5 = 10%
y<1
Excess Return RF = 5%
F
= = =0.50
σs 20
(Refer Point O)
U=2
Rs− RF
y∗=
0.01 x A x σs 2 RF
F
U=1
off.
• CAL with risk free asset and Portfolio 15 x Security 1
A has higher slope (Sharpe ratio)
than the CAL with Portfolio B. RF=7
• Therefore CAL with portfolio A is 10 20
preferred over CAL with portfolio B.
𝑅𝑜 = 𝑤1 𝑅1 + 𝑤2 𝑅2
𝜎𝑜 2 =w1 2 𝜎1 2 +w2 2 𝜎2 2 +2w1 𝑤2 ρσ1 𝜎2 ; where w1 +w2 =1
For maximum slope of the capital allocation line we differentiate the CAL equation
with respect to w1 and equate it to zero for the value of w1 The proportion of fund
in Portfolio 1, w1* that yields the maximum slope capital allocation line is given by:
(𝑅 −R )𝜎 2 − (𝑅 −R )𝜌𝜎 𝜎
∗ 1 𝑓 2 2 𝑓 1 2
𝑤1 =
(𝑅1 −R𝑓 )𝜎2 2 + (𝑅2 −R𝑓 )𝜎1 2 − (𝑅1 + 𝑅2 −2R𝑓 )𝜌𝜎1 𝜎2
and 𝑤2∗ =1−𝑤1∗
Portfolio Theory RAJIV SRIVASTAVA 54
OPTIMISING PORTFOLIO - EXAMPLE
Given two securities 1 and 2 with expected returns of 16% and 24% respectively,
and standard deviations of 20% and 30% respectively. T-bill serving as proxy for risk
free asset offer a return of 8%. The coefficient of correlation between Sec 1 and Sec
2 is 0.5.
What would be the return and risk of the optimum portfolio, and the Sharpe ratio?
SOLUTION
Given R1 = 16, R2 = 24 RF = 8, σ1 = 20, σ2 = 30, and ρ = 0.50
(𝑅1 − 𝑅𝑓 )𝜎2 2 −(𝑅2 − 𝑅𝑓 )𝜌𝜎1 𝜎2
𝑤1∗ =
(𝑅1 − 𝑅𝑓 )𝜎2 2 +(𝑅2 − 𝑅𝑓 )𝜎1 2 −(𝑅1 +𝑅2 −2 𝑅𝑓 )𝜌𝜎1 𝜎2
3 5
𝑅𝑜 = 𝑤1 𝑅1 + 𝑤2 𝑅2 = 𝑥 16 + 𝑥 24 = 21%
8 8
2 2 2 2 2
𝜎𝑜 = w1 𝜎1 +w2 𝜎2 +2w1 𝑤2 ρσ1 𝜎2 ;
9 25 35
= 𝑥 400 + 𝑥 900 + 2 𝑥 𝑥 300 = 548.44 𝑜𝑟 𝜎0
64 64 88
= 23.42%
Utility Curves
Portfolio C Feasible Portfolios
Portfolio B
RC
R1 x Security 1
RF =7
σC σ1 σ0 σ1
ASSET Allocation
Risk Free Asset 1 – 0.5925 = 0.6075 ≡ 60.75%
Security 1 0.5925 x 3/8 =0.2222 ≡ 22.22%
Security 2 0.5925 x 5/8 =0.3703 ≡ 37.03%
Portfolio Theory 61
RAJIV SRIVASTAVA
CAPITAL ALLOCATION LINE (CAL)
The slope of the CAL is dependent
the excess return and the
additional risk attached with S. The
slope of the CAL is
Excess Return
Slope of the CAL =
Additional Risk
Rs - R f 15 - 5
= = = 0.50
σs 20
Portfolio Return Rp = R f + Slope x σ p
Rs - R f σp
= Rf + x σ p = R f + (Rs - R f )
σs σs
and Portfolio Risk σP = y x σ S
RAJIV
Portfolio Theory 62
SRIVASTAVA
CAL AND ISO-UTILITY CURVE
• Capital Allocation Line is the locus of feasible alternatives for
investment.
• The risk and return combinations not satisfying CAL are not feasible
portfolios.
• All desired portfolios necessarily must lie on CAL. Simultaneously
portfolios must maximise utility for each investor.
• Neither return nor risk can be viewed in isolation. A combination of
risk and return forms a composite asset that provides some utility.
• Combining two would give feasible investment that maximises
utility.
2 2
U = R f + (Rs - R f )y - 0.005Aσ s y
• Utility would be maximised for y* as given by
Rs - R f
y* = 2
0.01 x A x σ s
U=5 Y
U=4 B Z
O
R0 U=3
Efficient
U=2 A
Frontier
F U=1
σ0 Standard Deviation, %
1 j =1 i =1
• Assume that the portfolio a) has n securities b) is equal weighted i.e. wi = 1/n,
c) each of the security has same variance of σ, and d) each of the
correlation coefficient is same and equals ρ giving average covariance as
AvCov(Ri,Rj).
• Segregating the variance term (variance of security i with itself, Cov(Ri,Ri)= σi2
from rest of the securities (i ≠ j) the variance of the portfolio is
n n n
1 1 2 1
p 2
=
n
∑ n
i + ∑∑ n 2
Cov( Ri , R j )
1 j =1 i =1
j ≠i
same risk.
15 x Security 1
• A rationale investor therefore would
reject all the portfolios on segment
from Sec 1 to V, since for the same
risk investor has a portfolio in the 10 20
Standard Deviation, %
segment V to Sec 2, directly above it
Portfolio Theory RAJIV SRIVASTAVA 74