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INTRODUCTION
This project aims to construct an active international equity portfolio based on trade date Feb 8 00. We perform the key stages in the investment management process including investment analysis, formulating risk-return expectations, ¡®blending in¡¯ investment heuristics, portfolio optimisation and performance evaluation (both in and out of sample).
PRELIMINERY CONSIDERATIONS
Choice of Indices
We select the FTSE All-World index (FTAW) and 48 constituent country indices for three reasons
First, the FTAW index is a well-known and widely used benchmark index, so it can be readily accepted as a valid data origin.
Second, The FTAW covers a broad range of stock markets [both in terms of geographical area and the risk-return spectrum], including developed markets, advanced emerging markets and emerging markets. Further there are standardized indices for each of the 48 constituent countries.
Third, and related, we can obtain (from DATASTREAM) dollar denominated total return indices for all countries allowing us to conduct the analysis as US investors. As discussed by Solnik (5) , there are in fact two possibilities with respect to managing currency exposure. The first, which is the approach we take implicitly take here, is to take on currency exposure so that total portfolio risk includes currency and stock risk. The second is to hedge it using derivative securities such as forward contracts. In this case, the returns available to international investors are equivalent to local currency returns. The latter may appear more sensible; however, Solnik () finds that currency risk is actually a relatively small component of the total risk of an international equity investment, and further observes currency risk reduction effects in a portfolio diversified across many currencies [which is the case here]. It must also be highlighted that hedging involves transaction costs and requires periodic rebalancing. Hence, we decide not to hedge.
Choice of risk free rate
The risk free rate assumption, for all investment decisions, is 1 year US dollar LIBOR, at Feb 8 00. At this date, the annualized risk free rate (rf) is .4%, so the monthly risk free rate is
Choice of 15 Countries
At the first step, we excluded all the Latin America markets to avoid the political and economic risk. Brazil, Venezuela, Peru, Colombia, Chile, Mexico and Argentina were excluded, from a commonsense viewpoint, because of the continuously widening negative influence from Argentine financial crisis over that area.
We decide to select the set of 15 countries with lowest average correlation to maximize international diversification benefits. To achieve this, we construct a correlation matrix from the FTAW 48 constituent countries¡¯ total return indices, with monthly frequency, for the 5-year period from 8 Feb 17 to Feb 8 00. However, based on this 5-year criteria, we find sufficient data in DATASTREAM for only countries Russia, Czech Republic, Egypt, Poland, Hungary, South Africa and Morocco have to be deleted from the list.
From the average correlation shown in Table 1, we chose the 15 countries China, India, Indonesia, Israel, Malaysia, Pakistan, Philippines, Taiwan, Thailand, Turkey, Austria, Belgium, Denmark, Finland, Ireland.
INVESTMENT STRATEGY
Tools for portfolio analysis
The two optimization techniques used in the project involve the Markowitz (MM) and Single Index (SIM) models.
Markowitz E(Rp)=¡ÆXiE(R) sp = ¡Æ¡ÆXiXjsi,j Single Index E(Rp)= ¦Áp+¦ÂpE(Ri) sp=¦Âpsm +se,p
MM is a standard approach to portfolio optimisation, but requires the full variance-covariance matrix to compute portfolio risk. The SIM bypasses the operational difficulties associated with this by assuming a simplified covariance structure between securities; the optimal portfolio is then computed by a ranking procedure conditional on the required inputs. The SIM is based on the standard OLS assumptions (1) E(ei)=0, () COV (ei,Rm) = 0 & () COV (ei,ej) = 0. The SIM & MM portfolio variances can be compared from the formula
MMsp=SIMsp + ¡Æ¡ÆXiXj COV (ei, ej).
From assumption () above, SIMsp will over- or under-estimate the true variance [MMsp].
A key and pertinent difference between the models is information requirements. For the MM, total information requirements are (N+N)/ (N expected returns, N variances, (N-N)/ unique covariance) making a somewhat impractical total of 15 for all 15 countries. However, the SIM, due to the simplifying assumption about the covariance structure, requires only N+ (N alpha, N beta, N r, Rm, sm) making a more manageable total of 47. For this reason, we decide to use the SIM, at the first stage, to select the 5 constituent countries for the portfolio followed by MM to blend in the required ¡®heuristics¡¯.
Single Index for preliminary country selection
Our natural starting point for the portfolio analysis is the optimal portfolio selection procedure from EGBG (00) Ch. based on the input data from the single index model. The procedure enables optimal portfolio weights to be established, and we consider the solution case with no short sales. It is based on a particular ranking criteria, and EGBG (00) note that it can be proved that the particular criteria lead to an optimal portfolio (conditional on the single index model representing the covariance structure between assets). The ranking is purely driven by the well-known Treynor ratio capturing the ratio of excess return over the risk free to beta.
Country
Israel 0.00688 0.5108 0.0040 0.007406 0.004714 0.01446
Italy 0.00454 0.8648 0.005 0.00756 0.0011 0.00841
Turkey 0.0076 .1070 0.01846 0.0164 0.008 0.007746
Pakistan 0.001 0.014 0.0061 0.001618 0.04741 0.00567
Denmark 0.00051 0.7005 0.005508 0.00505 0.001515 0.00500
Belgium 0.00148 0.5470 0.004185 0.0018 0.00004 0.008
Ireland -0.0005 0.86456 0.0074 0.001740 0.00040 0.0001
Thailand -0.011776 .1718 -0.00105 -0.00055 0.08665 -0.001406
Austria -0.007 0.677 0.00014 -0.001854 0.00411 -0.00
Taiwan -0.008457 1.075465 -0.00150 -0.00515 0.00816 -0.00471
Malaysia -0.0110 1.15881 -0.00671 -0.00874 0.018884 -0.0071
India -0.00505 0.55871 -0.006 -0.00465 0.00686 -0.007618
Indonesia -0.0114 1.6647 -0.01004 -0.0107 0.001 -0.00771
Philippine 0.00688 0.5108 -0.015548 -0.017551 0.0114 -0.0174
China 0.00454 0.8648 0.015475 0.0147 0.017 -0.44518
Table . Data required to determine optimal portfolio with rf=0.0000
Table illustrates the first part of the procedure. The Treynor ratio was computed for each of the 15 countries, and they were then ranked from highest (most desirable) to lowest (least desirable). We now require a cut-off point, and to determine the C, we compute Ci for each necessary country, where
(1)
The Ci is gradually built up in table ; it assumes that we keep adding one more country to the optimal portfolio until we get C. We seek a unique C such that all countries ranked higher satisfy Ci ¡Ý C, while countries ranked lower satisfy Ci ¡Ü C. We find C = 0.004656. Having obtained the set of 5 countries, it remains to determine the optimal portfolio weights. Each Zi value from () below is proportional to the optimal weight Xi, which is obtained by scaling each Zi using () below. Based on the five countries; we obtain -
() Z1=1.065188, Z1=1.0174, Z=0.150, Z4=0.008664, Z5=0.1604
() X1=0.4087, X=0.41, X=0.0876, X4=0.00504, X5=0.06486
Country
Israel 0.804457 55.654 0.804457 55.654 0.00166
Italy .070 54.4568 .0147 10.0844 0.00888
Turkey 1.14811 148.11 4.0467 458.15804 0.00458
Pakistan 0.01711 .67464 4.068 461.8868 0.00451
Denmark 1.6117 4.05847 5.60511 786.046766 0.004656
Belgium 0.5667 14.800467 6.8718 5.847 0.00458
Ireland 0.774 66.561 7.045 10.05 0.004041
Thailand -0.1581 164.746587 6.751 1466.50110 0.0056
Austria -0.486510 166.07705 6.0644 16.0715 0.00048
Taiwan -0.564568 117.801 5.74187 1750.857454 0.0065
Malaysia -0.507760 71.1117 5.411 181.78771 0.0018
India -0.4767 45.64847 4.886476 1867.61617 0.0011
Indonesia -0.7118 .55077 4.1758 15.86864 0.001741
Philippine -.06576 168.0657 .10748 18.17566 0.0008
China -0.0115 0.047516 .0866 18.78 0.00081
Table . Calculations for determining cut-off rate with ¦�m=0.00
Given the procedure; these are clearly the optimal weights. However, we are also required to formulate and apply ¡®heuristic¡¯ rules in the portfolio choice problem. There are numerous possibilities such as upper or lower limits, portfolio tilting or even risk controls. A key issue is, of course, how to incorporate such portfolio constraints into an analytical solution.
A second approach is therefore to consider the quadratic program from EGBG(0) (pp0) based on the Kuhn-Tucker conditions plus an upper limit weight constraint
(K1)
(K)
(K) and
(K4) (N=15)
An attempt was made to compute the solution via ¡®Excel Solver¡¯. A total of 15 ¡®target cells¡¯ are required. These represent (K1) rearranged such that each is set equal to zero. Then, in the solver format, Z and M need be varied to obtain solution values that meet (K) to (K4). Unfortunately, Excel Solver function allows only one target cell, so the solution was not obtainable.
Markowitz to build in ¡®heuristics¡¯
Therefore, we opt for the Markowitz approach via ¡®Excel Solver¡¯, which will enable us to determine portfolio weights based on the following ¡®heuristic¡¯ rules (1) No countries weight should be more than 5%, Xi ¡Ü 0.5, () No countries weight should be less than 5%, Xi ¡Ý 0.05. The rationale for the constraints is to limit single country exposure, balance the portfolio and control risk. They allow us to modify the portfolio so that it is not solely a product of historical analysis.
The portfolio strategy is based on the maximization the objective Sharpe ratio
,
where Rp is the expected return of the portfolio, and Rf is the risk free rate, sp is the total portfolio risk conditional on varying the 5 country weights under the constraints
(1) Fully invested åXi = 1;
() No short selling Xi ¡Ý 0 for all i;
() No countries weight should be more than 5%, Xi ¡Ü 0.5;
(4) No countries weight should be less than 5%, Xi ¡Ý 0.05.
We uses the Excel Solver function to solve the problem, the procedure is as follows
Firstly, we report the summary statistics (Table 4) that the Markowitz Model requires. They includes N expected returns, N variance terms & (N-N)/ unique covariance terms [N=5].
Country Israel Italy Turkey Pakistan Denmark
Expected Returns 0.0040 0.005 0.01846 0.0061 0.005508
Variance 0.0055 0.00501 0.041150 0.056 0.00666
Covariance Matrix
Country Israel Italy Turkey Pakistan Denmark
Israel 0.0055
Italy 0.0014 0.00501
Turkey 0.00788 0.00554 0.041150
Pakistan 0.0018 0.00047 0.0066 0.056
Denmark 0.00074 0.0016 0.005 -0.00074 0.00666
Table 4. Summary Statistics for Markowitz Approach
Secondly, we input those data into the Excel spread sheet (Table 5) and define B, B10, B11, B1, and B1 as the adjustable cells. Then we added the following constrains into solver function (1) B+ B10+B11+B1+ B1 =1, ()$B$ $B$1 ¡Ü 0.5, and () $B$ $B$1 ¡Ý 0.05. After that, we set B0 as the target cell to be maximized by varying the 5 adjustable cells representing the constraints. B0 is equal to the (B17-B1)/B18. B17and B18 will vary based on the weights of the 5 countries in our portfolio. We set 0% as default weights for each country, as solver requires initial guesses.
7 A B C
8 Country (Xi) Weight
Israel (X1) 0.
10 Italy (X) 0.
11 Turkey (X) 0.
1 Pakistan (X4) 0.
1 Denmark (X5) 0.
14 X1+X+X+X4+X5 1.00
15
16 Monthly Annually
17 Expected Return (Rp) 0.0065 0.117018
18 Total Risk (sp) 0.07408 0.5665
1 Interest Rate (Rf) 0.0000 0.0400
0 Objective Sharpe Ratio 0.0801 0.6158
Table 5. Excel Solver function Inputs
The solver output (Table 6) shows that the maximum achievable Sharpe ratio (by varying the weights of the 5 countries under those constraints) is 0.14514 (monthly). Thus, the weights of those 5 countries in our portfolio are Israel (X1) = 0.5, Italy (X) = 0.5, Turkey (X) = 0.085, Pakistan (X4) = 0.05, and Denmark (X5) = 0.167005.
Target Cell (Max)
Cell Name Original Value Final Value
$B$0 Objective Sharpe Ratio 0.07714 0.14514
Adjustable Cells
Cell Name Original Value Final Value Reduced Gradient
$B$ Israel (X1) 0. 0.5 0.010401
$B$10 Italy (X) 0. 0.5 0.0075
$B$11 Turkey (X) 0. 0.085 0
$B$1 Pakistan (X4) 0. 0.05 -0.0786
$B$1 Denmark (X5) 0. 0.167005 0
Constraints
Cell Name Cell Value Formula Status Slack
$B$14 X1+X+X+X4+X5 1.00 $B$14=1 Binding 0
$B$ Israel (X1) 0.5 $B$=0.05 Not Binding 0
$B$10 Italy (X) 0.5 $B$10=0.05 Not Binding 0
$B$11 Turkey (X) 0.085 $B$11=0.05 Not Binding 0.67005
$B$1 Pakistan (X4) 0.05 $B$1=0.05 Binding 0.
$B$1 Denmark (X5) 0.167005 $B$1=0.05 Not Binding 0.185
$B$ Israel (X1) 0.5 $B$=0.5 Binding 0.
$B$10 Italy (X) 0.5 $B$10=0.5 Binding 0.
$B$11 Turkey (X) 0.085 $B$11=0.5 Not Binding 0.05
$B$1 Pakistan (X4) 0.05 $B$1=0.5 Not Binding 0
$B$1 Denmark (X5) 0.167005 $B$1=0.5 Not Binding 0.117005
Table 6. Microsoft Excel .0 Solver Answer and Sensitivity Report
From the optimal weight we obtained from Markowitz Approach we can calculate out the following basic summary of our portfolio (Table 7).
Monthly Annually
Expected Return (Rp) 0.001 0.116148
Total Risk (sp) 0.05774 0.0005
Interest Rate (Rf) 0.0000 0.0400
Porfolio Beta 0.7701 0.7701
Objective Sharpe ratio 0.14514 0.458768
Treynor ratio portfolio 0.008 0.11485
Table 7. Summary for the optimal portfolio with rf = .4% p.a.
Computation of Efficiency Frontier
We begin to trace out the portfolio efficient frontier by the suggested approach in EGBG (P10-11) by varying the risk free rate. We select 5% as the interest rate for obtain the optimal portfolio, and solve it through solver function, and then we obtain the optimal portfolio with respect to 5% risk less rate. The weights of the 5 countries are Israel (X1) = 0.5, Italy (X) = 0.5, Turkey (X) = 0.1805, Pakistan (X4) = 0.05, and Denmark (X5) = 0.06065. Its annualised expected return is 0.14464, the annualised total risk is 0.4585, and the variance is 0.060465.
From the basic statistics for those portfolios shown in Table 8, we can conclude that the variance of a new portfolio consisting of &frac1; of portfolio (rf =.4%) and &frac1; of portfolio (rf = 5%) is
s = (&frac1;)(0.0005) + (&frac1;)(0.4585 ) + (&frac1;)(&frac1;) s1 = (0.14)
Therefore, s1 = 0.04771.
Country (Xi) P1 (rf = .4%) P (rf =5%) &frac1;P1+ &frac1;P
Israel (X1) 0.5 0.5 0.5
Italy (X) 0.5 0.5 0.5
Turkey (X) 0.085 0.1805 0.16015
Pakistan (X4) 0.05 0.05 0.05
Denmark (X5) 0.167005 0.06065 0.1185
W1+W+W+W4+W5 1.00 1.00 1.00
Summary Annually Annually Annually
Expected Return 0.116148 0.14464 0.157
Total Risk 0.0005 0.4585 0.14
Table 8. basic statistics for portfolio (P1, P, and &frac1;P1+ &frac1;P)
After we calculated out the expected returns, variance and covariance, then we could trace out the efficient frontier from varying the weights of P1 and P.
The efficient frontier is showed in the Figure 1. In our case, the risk free rate, rf, is .4%, the optimal portfolio is the tangent point (0.0005, 0.116148), and the minimum variance portfolio is the starting point (0.16461, 0.088857).
Figure 1. Efficient Frontier with the minimum risk portfolio and optimal portfolio.
PERFORMANCE ANALYSIS
As highlighted by EBGB(0) performance evaluation is an integral aspect of any investment decision making process. In this section, the ex-post performance of the Single index and Markowitz portfolios are analyzed over the sample estimation and subsequent one-year period. We aim to discover whether the selected portfolios have achieved sufficient return conditional on various performance measures. The required input data, based on annualized mean return and standard deviation, is summarized in table . For the in-sample period, the risk free assumption rf=0.054777, which represent the annualized mean monthly rate. For the out of sample period, the risk free assumption is rf=0.00568, which represents the annualized mean weekly rate.
We observe that, in the in-sample period, the ¡®Single Index 5¡¯ (SI5) and ¡®Markowitz 5¡¯ (M5) portfolios exhibited higher return and standard deviation than the FTSE All World (FTAW), while both portfolio betas were less than 1. For the out of sample period, the general observation is that all risky equity portfolios under-performed the risk free. From these preliminary observations, we can move to more definitive conclusions by utilizing a range of analytical techniques.
In Sample -
MARKOWITZ 5 0.116148 0.0005 0.7701
SINGLE INDEX 5 0.1555 0.158 0.81
FTSE ALL-WORLD 0.06085 0.165876 1
Out of Sample -
MARKOWITZ 5 -0.04885 0.171641 0.6084
SINGLE INDEX 5 -0.066 0.18188 0.6401
FTSE ALL-WORLD -0.16580 0.168 1
Table . Performance measurement input data
The Sharpe Ratio
As noted by EBGB(0), in the return & standard deviation space, all combinations of a risky portfolio and risk free asset lie along the straight line connecting the risky portfolio and risk free asset. The Sharpe ratio is the slope of this line and represents the ratio of excess return (over risk free) divided by return standard deviation (total risk). More formally -
Under this measure, the greater the Sharpe ratio, the better the portfolio performance. Further, since the ratio is based on total risk, it can be used to rank actively managed equity portfolios. In this respect, we note that our five asset portfolios represent active selections from the set of 4 countries that comprise the FTAW. The respective Sharpe ratios are reported in table 10.
In Sample ranking Out of sample ranking
Rank Portfolio Sharpe ratio Rank Portfolio Sharpe ratio
1 SINGLE INDEX 5 0.860 1 MARKOWITZ 5 -0.41046
MARKOWITZ 5 0.06540 SINGLE INDEX 5 -0.481175
FTSE ALL-WORLD 0.066 FTSE ALL-WORLD -1.08105
Table 10. Sharpe ratio portfolio performance ranking
From the table, we observe that, over both periods, both the SI5 and M5 portfolios outperformed the FTAW, but neither SI5 nor M5 dominated. For the in-sample period, the respective factors of market outperformance, for the SI5 and M5 portfolios, were around 8.7 and 8.7 times the FTAW Sharpe ratio. For the out of sample period, the negative Sharpe ratios indicate underperformance with respect to the risk free. The respective factors of market outperformance, for the M5 and SI5 portfolios, were around 0.4 and 0.4 times the negative FTAW Sharpe ratio.
Differential return when risk is measured by beta
Again, this is appropriate for actively managed equity portfolios. As discussed by EGBG(0), the essential intuition is to compare the return on an active portfolio to a passive strategy of combining the risk free and FTAW to obtain a portfolio of identical beta. The differential return, known as the Jensen alpha measure, represents the (beta) risk adjusted abnormal return. More formally -
The respective ratios are reported in table 11, and we note that the performance ranking is identical to the Sharpe ratio.
In Sample ranking Out of sample ranking
Rank Portfolio Alpha Rank Portfolio Alpha
1 SINGLE INDEX 5 0.065758 1 MARKOWITZ 5 0.076646
MARKOWITZ 5 0.05651 SINGLE INDEX 5 0.06458
Table 11 Performance ranking by the Jensen differential performance index
Decomposition of performance
The alpha measure, or return to selectivity, can be further decomposed into the return to diversifiable risk and net selectivity via the technique of Fama decomposition.
Markowitz approach Single Index Approach
Performance analysis (annual basis) in sample Performance analysis (annual basis) in sample
Fama Decomposition Fama Decomposition
Return Beta Return Beta
Selected Portfolio (P1) 0.116148 0.7701 Selected Portfolio (P1) 0.1555 0.81
Benchmark (P) 0.0565 0.7701 Benchmark (P) 0.05767 0.81
Portfolio alpha (P1-P) 0.05651 Portfolio alpha (P1-P) 0.065758
Benchmark (P) 0.0610 1.0658 Benchmark (P) 0.0666 1.78
Divers. Risk (P-P) 0.00474 Divers. Risk (P-P) 0.0085
Selectivity (P1-P) 0.0540 Selectivity (P1-P) 0.06864
Markowitz approach Single Index Approach
Performance analysis (annual basis) Out of sample Performance analysis (annual basis) out of sample
Fama Decomposition Fama Decomposition
Return Beta Return Beta
Selected Portfolio (P1) -0.04885 0.608 Selected Portfolio (P1) -0.066 0.6401
Benchmark (P) -0.1651 0.6084 Benchmark (P) -0.1181 0.6401
Portfolio alpha (P1-P) 0.076646 Portfolio alpha (P1-P) 0.06458
Benchmark (P) -0.1867 0.8740 Benchmark (P) -0.10 0.687
Divers. Risk (P-P) -0.06061 Divers. Risk (P-P) -0.06781
Selectivity (P1-P) 0.1607 Selectivity (P1-P) 0.1176
Table 1 Performance analysis by Fama decomposition
As discussed by EBGB(0), this is achieved by comparing the active portfolio return to the return on a passive portfolio with the same total risk, i.e. the total risk of the passive portfolio (which is all systematic) is set equal to the total risk of the active portfolio. This allows a second benchmark to be computed, which represents the ¡®required¡¯ alpha from an active fund conditional on its total risk. Any more than this represents a pure selection gain or ¡®net selectivity¡¯.
For the in sample SI5 portfolio, we compute the beta and return of the required naïve passive portfolio with the same total risk.
The Fama decomposition for all four portfolios are reported in table 1 and are illustrated for SI5 and M5 in fig 1. For the SI5 in sample portfolio, we observe that only around 4% of the alpha was required to achieve the benchmark return for diversifiable risk. Similarly for the M5 in sample portfolio, around 6% of the alpha represented a net selection gain.
Figure . Performance Analysis by Fama Decomposition
For the out of sample period, the positive alpha measures for the SI5 and M5 portfolios must be put into the overall context of a negative market risk premium of nearly ¨C4%. Figure illustrates the Fama decomposition for the out of sample M5. The passive benchmark P illustrates that RP is the ¡®benchmark¡¯ return required from an active portfolio to ¡®compete¡¯ with a naïve portfolio of equivalent total risk to P1, leading to negative return to diversifiable risk. In this scenario, even fund managers with negative alpha can legitimately claim selection skills if they exceed RP. We note that positive alpha required positive net selectivity in excess of the magnitude of return to diversifiable risk; this was comfortably achieved by M5. However, the overall M5 return was around 7% below the risk free. We observe similar decomposition effects for the SI5 out of sample portfolio, where the return was nearly % below the risk free.
Performance summary
Both SI5 and M5 portfolios were market outperformers with the SI5 featuring as the dominant in sample portfolio. In this respect, we note that the SI5 represented an unconstrained solution, while the M5 portfolio was optimized subject to several constraints resulting in a performance trade off. However, these ¡®heuristics¡¯ proved beneficial for the out of sample, M5 has % more return than SI5. Illustrating that limiting country exposure and balancing out a portfolio can act as prudent modifications in supplementing a purely historically driven analysis. Finally, in the out of sample period, we must acknowledge that both portfolios underperformed the risk free.
CONCLUSION
We need not look very far for the source of the out of sample. At the current time, there appears to be a world-wide lack of confidence in global equities driven by a range of factors including the general fall out from the dot-com bubble, a number of high profile accounting scandals, the plague of political instability in respect of Iraq sending shockwaves across global stock markets. We note that US and UK stock markets have fallen heavily for the last years or so, and maybe, therefore, we should have considered a bond or even equity put option portfolio given the stock market environment. Indeed, we were only able to find one rising stock market ¨C Pakistan (by 1%) - for the last year, but the portfolio weight amounted to only 5%. If only we had a crystal ball!
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