Selecting a portfolio in financeBen Sloman1 Ye Liu2 Jonathan Tawn2
1Maths, University of Oxford; 2Maths & Stats, Lancaster University
Introduction
In finance investors try to make as much money aspossible, whilst keeping risk low.
This trade off between expected return and risk drivesmost investment theory.
Portfolios reduce the amount of risk involved.
Investments are correlated, so statistics are used to find thebest portfolios.
Aims of the project
To assess the appropriateness of fitting a multivariatenormal distribution to real stock returns data.
To use optimisation techniques to produce the bestportfolio for different investment strategies.
To assess the performance of these optimal portfolios.
To analyse the model used and challenge the underlyingassumptions made.
Data
Data obtained from Yahoo! Finance UK(http://uk.finance.yahoo.com/)
Selected 20 stocks from the FTSE 100 which were activelytraded and from a variey of sectors.
Adjusted closing prices between 3rd January 2003 and 12thJuly 2010.
Stocks chosen: AAL, BA, BARC, BAY, BP, BSY, BTA,GSK, HSBA, IMT, IPR, LLOY, MKS, RBS, SBRY, STAN,TSCO, UU, VOD, XTA.
Simulation
The stock returns are modelled as being multivariatenormally distributed.
A random vector X is distributed as a multivariate normalif every linear combination has a univariate normaldistribution
X = (X1, ...,Xk)X ∼ MVNk(µ,Σ)
µ = (E (X1),...,E (Xk))
Σ =
σ1σ1 σ1σ2 . . . σ1σkσ2σ1 σ2σ2 . . . σ2σk
... ... . . . ...σkσ1 σkσ2 . . . σkσk
Letting w be the vector of fixed portfolio weightings thenthe portfolio returns R = wTX, with:
R ∼ N(µTw,wTΣw)
Assuming the underlying process was known I ran 1000simulations for the standardised share price of the 1
Nportfolio, which is where weightings are evenly dividedbetween all stocks.
0 500 1000 1500
02
46
810
Time (days)
Sta
ndar
dise
d P
rice
2.5% and 97.5%25% and 75%50%Actual Standardised Price
Figure 1: Simulated standardised prices for 1
N portfolio
The median of the simulations appears to give a trendlinefor the historic prices, as expected, since the distributionparameters are calculated from this data.
The large fall in the historic price is due to the financialcrisis, which severely reduced confidence in the market in2008. Such extreme movements tend not to be modelledwell by the multivariate normal distribution.
Optimisation
Using a constrained optimisation program on R, I foundthe best portfolio for different investment strategies:
Strategy 1: Minimise Var(R) given E (R)>µ0Optimisation constraint: µTw>µ0
Strategy 2: Maximise E (R) - λVar(R)
The risk coefficent, λ = Risk PremiumVariance
Both strategies allow for different levels of cautiondepending on the value of µ0 or λ.
In fig 2 the portfolio weightings are shown for the optimalportfolios arising from strategy 1 for 10 values of µ0
0.0018 0.0016 0.0014 0.0012 8e−04
XTA
UU
TSCO
SBRY
MKS
IPR
IMT
HSBA
GSK
BSY
BP
BA
Value of µ0
Por
tfolio
Wei
ghtin
g
0.0
0.2
0.4
0.6
0.8
1.0
Figure 2: Optimal portfolios for minimising Var(R) given E (R) > µ0
When a higher level of E (R) is required the portfolioheavily favours a small number of high return stocks.
However, when minimising Var(R) is of higher priority theportfolio is more diversified.
Due to fixed costs involved traders will only alter theirportfolio if there are sufficient margins to be made. Toreflect this weights are multiples of 0.01.
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●
●
●
●
●
●
●
●
0e+00 1e−04 2e−04 3e−04 4e−04 5e−04 6e−04
0.00
060.
0010
0.00
140.
0018
Variance
Exp
ecta
tion
● Strategy 1Strategy 2
Figure 3: Scatterplot of performance of optimal portfolios
Fig 3 is a scatterplot of the Var(R) against the E (R) ofoptimal portfolios for 10 values of µ0 and λ.
They all lie on a parabola, which is known as the efficientfrontier.
Portfolios to the upper left of the frontier are impossible,whilst those to the lower right are inefficient, as Var(R)could be reduced for the given level of E (R) or E (R)increased for the given level of Var(R).
Strategy 1 approaches the frontier horizontally from µ0 onthe E (R) axis, whereas strategy 2 locates the point wherethe frontier has gradient λ.
Performance
In order to assess how well these portfolios would performin real life I divided the data into in and out samples, on aroughly 2
3 to 13 split.
The in sample was used to calibrate the optimal portfolios.
The performance of these optimal portfolios was thentested in the out sample.
As the market suffered as a whole during the out sample,because of the credit crunch, it is fair to compareperformance against the 1
Nportfolio.
Performance (continued)
0 500 1000 1500
12
34
5
Time (days)
Sta
ndar
dise
d P
rice
In Sample Out Sample
Optimal portfolio1/N Portfolio
Figure 4: Line plot showing the performances of the 1
N portfolio and the portfolio for
maximising E (R) - 2 Var(R)
In fig 4 you can see the optimal portfolio far outperformsthe 1
Nportfolio in the in sample but is marginally worse in
the out sample, suggesting that the 1
Nportfolio would have
been a better investment.
Portfolio In Sample Out SampleE (R) SD(R) IR VaR5 MDD E (R) SD(R) IR VaR5 MDD
Min Var(R) s.t. 0.17 0.170 1.838 0.093 -2.688 24.6 0.003 3.671 0.001 -6.048 79.1E (R)>µ0 for 0.15 0.150 1.397 0.108 -2.091 23.6 0.017 2.452 -0.007 -3.725 64.8following µ0 0.13 0.130 1.131 0.115 -1.713 20.7 0.017 2.029 -0.008 -3.000 59.6
Max 0.5 0.181 2.193 0.083 -3.111 34.8 0.014 4.473 0.003 -7.322 85.2E (R)− λVar(R) 1 0.168 1.728 0.097 -2.562 25.1 -0.003 3.389 -0.001 -5.421 76.5for following λ 2 0.158 1.509 0.105 -2.190 25.1 -0.015 2.714 -0.005 -4.482 68.1
1/N Portfolio 0.072 0.922 0.078 -1.390 18.9 -0.016 2.096 -0.008 -3.127 58.7
Table 1: Comparison of performance of portfolios in in and out samples.E (R), SD(R), VaR5 and MDD are given as percentages rather thandecimals, as the values in figures 2 and 3 are displayed.
Sharpe’s Information Ratio (IR): IR =E (R)
SD(R)
Value at Risk 5%(VaR5): 5th percentile of returns data. A measure of the amount
that could be lost each day
Maximum drawdown (MDD): The greatest decline in share price as a percentage. A
measure of how sustained losses can be.
From table 1 we can see that although the optimalportfolios tend to have a higher E (R) than the 1
Nportfolio
in the out sample, they also tend to be more volatile andare more likely to sustain large losses.
Improvements
Fig 5 shows that whilst modelling stock returns as beingnormally distributed is reasonable, fitting a student tdistribution might be more appropriate
−0.2 −0.1 0.0 0.1 0.2
05
1015
20
Value
Den
sity
Empirical for AALN(µ,σ2)
Figure 5: Comparison of an empirical density function for a stock with the normal pdf
for the parameters taken from the stock’s return data
Rather than using a fixed mean an exponentially weightedrolling window would be more accurate, as recent datawould have a greater influence on future predictions.
The sample covariance matrix used leads to extremeportfolio values. This could be improved by shrinking thecovariance matrix or adding more portfolio weightingconstraints.
Conclusion
The multivariate normal is a reasonable model for anintroduction to portfolio optimistion but more advancedoptimisation techniques have to be used before the portfoliosproduced are competitive in real life.
References
Markowitz, H. (1952)
Portfolio Selection.
In Journal of Finance, 1952,
Jagannathan, R. and Ma, T. (2002)
Risk reduction in large portfolios: Why imposing the wrong constraints helps.
Ledoit,O. and Wolf,M. (2003)
Honey, I shrunk the sample covariance matrix
http://www.stor-i.lancs.ac.uk/ [email protected]
