Session 2 :
Portfolio Theory (Reminder)
Asset Management (ieb Programme)
Dr Dirk Nitzsche (E-mail : d.nitzsche@city.ac.uk)
Content – A Reminder
• Principal of diversification (i.e. widely applied in hedging and risk
management in the insurance industry and asset management)
• Mean variance approach to portfolio theory (e.g. Markowitz approach)
• Combining risky assets
• Combining (a bundle of) risky assets and a risk free rate – capital allocation line (also
known as transformation line).
• Capital market line (best capital allocation line).
• Security market line
• Optimum weights and standard errors
• Some practical issues related to the implementation of portfolio theory
Two Questions
• Question 1 : Asset Allocation
• How much money are you going to put into risky assets and how much into the risk
free security? (What proportion of your ‘own’ money should you put into a bundle
of risky stocks?)
• Allowing you to borrow and lend (from a bank). How does this alter your choice of
‘weights’ and the amount you actually choose to borrow or lend?
• Depends on your attitude towards risk
• Question 2 : Portfolio Theory
• How do you allocate the money which you invest into risky assets amongst different
risky securities?
• Different weights give rise to different ‘risk-return combinations and this leads to the
‘efficient frontier’
We will address Q2 first!
Introduction
• Two questions : How should we allocate our wealth (say $1,000)?
• Between risky assets and a risk free investments (Q2).
• Money allocated to risky assets : How should we allocate the money we want
to invest in risky assets?
• Correlations/covariance play a key role
• Diversification, portfolio theory
• Also used by insurance, structuring, hedging, risk management, …
• Assume investors are risk averse!
Investments : Risk – Return Profile
ERi
Asset B
Asset A
Asset C
Risk
Risk averse investor would prefer asset-B to asset-A or
risk averse investor would prefer asset-C to asset-A.
Portfolio Theory (Question 2)
• Portfolio theory works out the ‘best combination’ of stocks to hold in your
portfolio of risky assets.
• What proportions of your $100 (which you invest in risky assets – Question
2) should you put into different stocks – say two? For instance if weights
are 25%, 75% than that means $25 and $75 respectively.
• Objective : We assume the investor is trying to mix or combine stocks to
get the best risk return outcome.
• Rem. : Investors like returns and dislike risk.
Diversification : Mathematical
Concepts
Analysing Financial Data : Prices and Returns
• Financial Data : Most financial data is collected and made available in
form of prices
• Financial Analysis : Conducted on returns
Expected Return, Variance, Standard
Deviation, Correlation
• Expected return :
• measure of central tendency (i.e. mean,
average)
• Risk :
• measure of fluctuation/variability (i.e.
variance, std dev)
• Covariance/Correlation
• Correlation = ρ = Cov(X, Y)/[σXσY]
• Covariance
= (1/n) [(X1-Xmean)(Y1-Ymean) + … + (Xn-Xmean)(Yn-Ymean)]
Portfolio : Expected Return and Variance
• Formula for 2-asset case :
• Expected portf. return : ERp = wA ERA + wB ERB
• Var. of portf. return : Var(Rp) = wA2 Var(RA) + wB2 Var(RB) + 2wAwBCov(RA,RB)
• Using matrix notation :
• Expected portfolio return : ERp = w’ERi
• Variance of portfolio return : Var(Rp) = w’Σw
where w is (n x1) vector of weights
ERi is (n x 1) vector of expected return of individual assets
Σ is (n x n) variance covariance matrix
Diversification : Focusing ONLY on
Risk (e.g. variance)
Power of Diversification : Random Selection
of Stocks
Variance
1 2 ... 20 40
No. of shares in portfolio
Power of Diversification
• As the number of assets (n) in the portfolio increase, the portfolio SD
(total riskiness) falls
• Assumptions (for simplification) :
• All assets have the same variance : σi 2 = σ2
• All assets have the same covariance : σij = ρσ2
• Invest equally in each asset (i.e. 1/n)
• General formula for calculating the portfolio variance :
σ2p = Σwi2 σi2 + ΣΣ wiwj σij
Power of Diversification
• Starting with the general formula for portfolio variance :
σ2p = Σwi2 σi2 + ΣΣ wiwj σij
• Include the (three) assumptions we get
σ2p = (1/n) σ2 + ((n-1)/n) ρσ2
• If n is large (1/n) is small and ((n-1)/n) is close to 1.
• Hence σ2p ≈ ρσ2
Self Study Exercise
• Suppose you have n assets. All assets have the same standard
deviation (σ = 25) and the same correlation (ρ = 0.35)
• Questions :
• Calculate the portfolio standard deviation for an equally weighted portfolio of
size n = 2, 20 and infinity.
• How many assets do you have to include in your equally weighted portfolio so
that the portfolio standard deviation does not exceed 16?
Excel Spreadsheet : Sigma = 25, Corr = 0.35
Risk Reduction - Portfolio Theory
30
25
20
15
10
0
0 10 20 30 40 50 60
Diversification : Considering Risk
and Expected Return
Example (Two Risky Assets) : Summary
Statistics
Equity 1 Equity 2
Mean 8.75% 21.25%
SD 10.83% 19.80%
Correlation -0.9549
Covariance -204.688
Example (Two Risky Assets) : Portfolio Risk
and Portfolio Expected Return
Share of wealth Portfolio
in
w1 w2 ERp σp
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% 19.80%
Example (Two Risky Assets) : Efficient Frontier
25
0, 1
20
0.5, 0.5
Expected return (%)
15
10 1, 0
0.75, 0.25
5
0
0 5 10 15 20 25
Standard deviation
Efficient Frontier : Different Correlation
Assumptions (2 Risky Assets)
Y
ρ = -0.5
Expected return
ρ = -1 ρ = +1
B A
ρ = 0.5
Z
ρ=0
C
X
Std. dev.
Minimum Variance Portfolio (2 Risky Asset)
• Two asset case : wA + wB = 1 or wB = 1 – wA
Var(Rp) = wA2 σA2 + wB2 σB2 + 2wAwB ρσAσB
Var(Rp) = wA2 σA2 + (1-wA)2 σB2 + 2wA(1-wA) ρσAσB
• To minimise the portfolio variance : Differentiating with respect to wA
∂σp2/∂wA = 2wAσA2 – 2(1-wA)σB2 + 2(1-2wA)ρσAσB = 0
• Solving the equation :
wA = [σB2 – ρσAσB] / [σA2 + σB2 – 2ρσAσB]
= (σB2 – σAB) / (σA2 + σB2 – 2σAB)
Minimum Variance Portfolio (2 Risky Asset)
ERp
0% Asset 1.
Minimum Variance
100% Asset 2
portfolio
100% Asset 1,
0% Asset 2
σp
Efficient Frontier : N-assets
ERp
U A
ERp* = 5% x
x x
L x
ERp** = 4% x
B
x x P1 x
x x x
x
x x
P1 x
x
x C
σp** σp* σp
Summary : Portfolio Theory
Expected Return
ER2
ER1
Formula for Varp
→ function of weights
→ different weights give different
values for Varp
σp
Asset Allocation : Risky Assets
and Risk Free Investment
Example : One ‘Bundle’ of Risky Assets Plus
Risk Free Rate
Suppose the ‘one-bundle’ of risky assets consists of 3
securities, i.e. 0.45 asset-A, 0.3 asset-B and 0.25 asset-C
Returns
T-Bill Equity
(safe) (Risky)
Mean 10% 22.5%
SD 0% 24.87%
Introducing Borrowing and Lending at the
Risk Free Rate
• Stage 2 of the investment process :
• You are now allowed to borrow and lend at the risk free rate r while still
investing in any SINGLE ‘risky bundle’ on the efficient frontier.
• For each SINGLE risky bundle, this gives a new set of risk-return combinations
known as the ‘capital allocation line’.
• Rather remarkable the risk-return combinations you are faced with is a
straight line (for each single bundle of risky assets) – capital allocation line
• You can be anywhere on this line.
New Portfolio of Risky Assets and the Risk
Free Investment
• ‘New’ Expected Portfolio Return : ERN = (1-x) rf + x ERp
• ‘New’ Portfolio Standard Deviation : σ2N = x2 σ2p or σ N = x σp
where x = proportion invested in the portfolio of risky assets
ERp = expected return on the portfolio containing only risky assets
σp = standard deviation of the portfolio of risky assets
ERN = expected ‘new’ portfolio return (including the risk free asset)
σN = standard deviation of ‘new’ portfolio
Example : ‘New’ Portfolio ER and SD (One
Bundle of Risky Assets Plus Risk Free Rate)
Share of wealth Portfolio
in
(1-x) x ERN σN
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%
Example : Capital Allocation Line
35
30
0.5 lending +
25 0.5 in 1 risky bundle
20
15 No borrowing/
no lending
10
-0.5 borrowing +
5 All lending 1.5 in 1 risky bundle
0
0 5 10 15 20 25 30 35 40
Standard deviation (Risk)
Capital Allocation Line : Borrowing and
Lending
Expected Return
Borrowing/
leverage
Z
Lending
X all wealth in risky asset
L
Q
r
all wealth in
risk-free asset
σR
Standard Deviation, σN
Choosing the Most Efficient Portfolio
Expected Return
Portfolio B’ Portfolio B
rf
Portfolio A
σp
Capital Market Line – ‘Best’ Capital Allocation
Line
Capital allocation line 3
Expected Return
– best possible one
Portfolio M
Capital allocation line 2
Capital allocation line 1
rf Portfolio B
Portfolio A
σp
The Capital Market Line (CML)
Expected Return CML
Market Portfolio
Risk Premium / Equity Premium
(ERi – rf)
rf
σm Std. dev., σi
Portfolio Choice – Capital Market Line and
Indifference Curves
IB Z’
Expected Return
Capital Market Line
K
IA
M Y
ERm
ERm - r
A
r
α L
σm σ
Summary : Market / Optimum Portfolio
• How do you find the optimum
portfolio?
Expected Return
Market (or optimum)
Portfolio
• Requirements :
ER, Variances, Covariances of individual
assets (returns), risk free rate
• Stage 1 :
• Find efficient frontier using constrained
rf optimisation
• Stage 2 :
• Find best capital allocation line (
bundle of risky assets = optimum
portfolio)
Std. dev.
Britton-Jones (1999) ‘The Sampling
Error in Estimates of Mean-Variance
Efficient Portfolio Weights’, JoF
The Paper : Idea, Data, Methodology
• Idea :
• Are the optimum weights from an international portfolio statistically significantly
different from ZERO?
• Data :
• Returns are measured in US Dollars and fully hedged
• 11 countries : US, UK, Germany, Japan, …
• Monthly data 1977 – 1996 (plus two subperiods : 1977-1986 and 1987-1996)
• Methodology
• Regression analysis
• Non-negative restrictions on weights not possible
Optimum Weights and t-statistics
1977-1986 (10 years) 1987-1996 (10 years) 1977-1996 (20 years)
weights t-stats weights t-stats Weights t-stats
Australia 21.6 0.66
Austria 22.5 0.74
Belgium 66 1.21
Canada -68.9 -1.10
Denmark 68.8 1.78
France -22.8 -0.48
Germany -58.6 -1.13
Italy -15.3 -0.52
Japan -24.5 -0.87
UK 3.5 0.07
US 107.9 1.53
Optimum Weights and t-statistics
1977-1986 (10 years) 1987-1996 (10 years) 1977-1996 (20 years)
weights t-stats weights t-stats Weights t-stats
Australia 21.6 0.66 12.8 0.54
Austria 22.5 0.74 3.0 0.12
Belgium 66 1.21 29.0 0.83
Canada -68.9 -1.10 -45.2 -1.16
Denmark 68.8 1.78 14.2 0.47
France -22.8 -0.48 1.2 0.04
Germany -58.6 -1.13 -18.2 -0.51
Italy -15.3 -0.52 5.9 0.29
Japan -24.5 -0.87 5.6 0.24
UK 3.5 0.07 32.5 1.01
US 107.9 1.53 59.3 1.26
Optimum Weights and t-statistics
1977-1986 (10 years) 1987-1996 (10 years) 1977-1996 (20 years)
weights t-stats weights t-stats Weights t-stats
Australia 6.8 0.20
Austria -9.7 -0.22
Belgium 7.1 0.15
Canada -32.7 -0.64
Denmark -29.6 -0.65
France -0.7 -0.02
Germany 9.4 0.19
Italy 22.2 0.79
Japan 57.7 1.43
UK 42.5 0.99
US 27.0 0.41
Optimum Weights and t-statistics
1977-1986 (10 years) 1987-1996 (10 years) 1977-1996 (20 years)
weights t-stats weights t-stats Weights t-stats
Australia 6.8 0.20 21.6 0.66
Austria -9.7 -0.22 22.5 0.74
Belgium 7.1 0.15 66 1.21
Canada -32.7 -0.64 -68.9 -1.10
Denmark -29.6 -0.65 68.8 1.78
France -0.7 -0.02 -22.8 -0.48
Germany 9.4 0.19 -58.6 -1.13
Italy 22.2 0.79 -15.3 -0.52
Japan 57.7 1.43 -24.5 -0.87
UK 42.5 0.99 3.5 0.07
US 27.0 0.41 107.9 1.53
Optimum Weights and t-statistics
1977-1986 (10 years) 1987-1996 (10 years) 1977-1996 (20 years)
weights t-stats weights t-stats Weights t-stats
Australia 6.8 0.20 21.6 0.66 12.8 0.54
Austria -9.7 -0.22 22.5 0.74 3.0 0.12
Belgium 7.1 0.15 66 1.21 29.0 0.83
Canada -32.7 -0.64 -68.9 -1.10 -45.2 -1.16
Denmark -29.6 -0.65 68.8 1.78 14.2 0.47
France -0.7 -0.02 -22.8 -0.48 1.2 0.04
Germany 9.4 0.19 -58.6 -1.13 -18.2 -0.51
Italy 22.2 0.79 -15.3 -0.52 5.9 0.29
Japan 57.7 1.43 -24.5 -0.87 5.6 0.24
UK 42.5 0.99 3.5 0.07 32.5 1.01
US 27.0 0.41 107.9 1.53 59.3 1.26
Forecast Error (ERP, σP) : Error in Weights
Mathematical optimum
ERp = (50%, 50%)
A = (100% US, 0%)
x
x
x Portfolio
x
x x xx = (90% US, 10% EU)
C = (0% US, 100%)
σp
2 assets, say European stock index and S&P500
Practical Considerations
Portfolio Theory and Practical Issues
• All investors do not have the same views about expected returns and
covariances. However, we can still use methodology to work our
optimal proportions / weights for each individual investor.
• The optimal weights will change as forecasts of returns and
covariances change
• Weights might be negative which implies short selling, possible on a
large scale (if this is impractical you can calculate weights where all
the weights are forces to be positive).
• The methods can be easily adopted to include transaction costs of
buying and selling and investing ‘new’ flows of money.
Portfolio Theory and Practical Issues
• To overcome the sensitivity problems :
… limit the analysis to only a number (say 10 or maybe only 5) countries
… choose the weights to minimise portfolio variance (weights are independent of
‘badly measured expected returns).
… do not allow any short selling of risky asserts (only positive weights).
• Set up a portfolio you are ‘comfortably’ with. (Maybe rebalance from time
to time).
… choose ‘new weights’ which do not deviate from ‘index tracking weights’ by more
than x% (say 2%)
… choose ‘new weights’ which do not deviate from the existing weights by more than
x% (say 2%)
use optimisation
No Short Selling Allowed (i.e. wi > 0)
E(Rp) Unconstraint efficient frontier
(short selling allowed)
Constraint efficient frontier
(with no short selling allowed)
- always lies within unconstraint
efficient frontier or on it
- deviates more at high levels of ER and σ
σp
Summary
• Power of diversification – Portfolio risk is covariance
• Mean variance approach (developed by H. Markowitz)
• Efficient frontier
• Capital allocation line and capital market line
• Market (or optimum) portfolio
• Practical issues to make portfolio theory work
• Britton-Jones (1999) - Standard errors of the optimum weights and
the issue of home country biasness.
References
• Cuthbertson, K. and Nitzsche, D. (2008) ‘Investments’, J. Wiley (2nd
edition)
• Chapters 10 and 11
• Britton-Jones, M. (1999) ‘The Sampling Error in Estimates of Mean-
Variance Efficient Portfolio Weights’, Journal of Finance, Vol. 52, No.
2, pp. 637-659
End of Lecture
D.Nitzsche@city.ac.uk
Asset Management
