Optimal portfolio construction in markets with no risk-free asset available

Transcription

1 Optimal portfolio construction in markets with no risk-free asset available Andreas Emmert Produced during the MSc in Finance studies at Strathclyde University in Glasgow/Scotland Portfolio Theory

2 Optimal portfolio construction in markets with no risk-free asset available Introduction In this paper we will tackle the question whether the market portfolio is the optimal portfolio in all cases, i.e. whether the market is expected to move upwards or downwards. Theory suggests that the market portfolio as the most diversified portfolio should display the smallest risk and therefore be the most secure investment for a risk-averse investor. However, the analysis is usually based on positive returns and rarely is information given about the impact of adverse outcomes in the market. It is therefore interesting to contribute some additional information to the discussion on how these negative movements in the market affect the choice of optimal portfolios. e consider the effects on utility of different investor types, applying three different utility functions to the analysis. The results are analysed by looking at utility in conjunction with risk to provide information on the optimal choice of share combinations. The findings indicate that it is not always clear that the market portfolio represents the optimal choice and one has to include the expectations of the investor to provide a meaningful suggestion for an optimal combination of assets. henever a non-market portfolio appears to be a better choice than the market portfolio, which is an efficient portfolio lying on the efficient frontier given that the assumptions underlying the single-index market model hold, we have to ask the question if arbitrage has been possible and why it has not been eliminated by the market. In addition the results show that the risk of the portfolio itself does not give a clear indication of which portfolio suits a risk-averse investor best. For this to provide information we would have to consider the risk premium, i.e. the amount of additional return to be received on the portfolio for taking on one additional unit of risk. But this indication has some flaws which we will also consider at a later stage in this paper. In the following, the methodology and data used will be described and a final conclusion of the results given.

3 Optimal portfolio construction in markets with no risk-free asset available Methodology The data used for this empirical analysis are the daily prices for all FTSE-00 shares over the period nd January to 3 st December 00. Out of these a random number of 50 shares has been chosen and randomly placed in four different portfolios of the sizes 5, 0, 0 and 50 shares. The FTSE-00 index served as a proxy for the market portfolio. A number of assumptions had to be made to facilitate the analysis: equally weighted portfolios, i.e. portfolios are weighted by number of shares not by proportion of portfolio value, so this implies that assets are not infinitely divisible and the minimum size is one share per asset no short-sales are allowed in the market no risk-free asset available, i.e. portfolios consist of risk assets/shares only perfect market conditions, i.e. no transaction costs, no taxes, investors are fully informed, rationally acting and have no market power holding period is one year, i.e. no trading is allowed during the year under consideration, therefore we look at static portfolios Portfolios were constructed by adding the individual share values according to the composition of the different portfolio sizes, where the 0-sharesportfolio consists of all shares in the 5-shares-portfolio plus an additional 5 randomly chosen shares. This procedure has been followed for all portfolio sizes. Once these portfolios have been constructed their performance is measured as an annual return in percent on the initial portfolio value. Mean return and standard deviation of the portfolio have been calculated using MINITAB 3 as the standard regression package. The risk premium of each cp. portfolio compositions in Appendix, Exhibit annual return refers to the value development over 59 trading days in 00, starting vom and ending on Detailed output data can be found in Exhibit 8 in the appendix. 3

4 Optimal portfolio construction in markets with no risk-free asset available portfolio has been calculated as annual return over standard deviation of the corresponding portfolio. For the analysis of utilities the following three functions have been applied to the analysis: [ annual (%)*00] U ( ) = ln return This function displays decreasing absolute risk aversion (DARA) and at the same time constant relative risk aversion (CRRA) which describes a typical utility function for a risk-neutral investor. 4 U ( ) = 4 * annual return (%) *00 [ annual return (%) *00] This function displays increasing absolute risk aversion (IARA) and at the same time increasing relative risk aversion (IRRA) which describes a typical utility function of a risk-averse investor. 0 U( ) = e annual return (%) *00 This function displays constant absolute risk aversion (DARA) and at the same time increasing relative risk aversion (DRRA) which describes a typical utility function of a risk-seeking investor. As the general movement in the FTSE-00 was downward sloping during 00, we have to consider the adverse effect of an upward sloping market index on the optimal portfolio choice by building a scenario. In order to provide sensible figures for the portfolio values, the betas of all portfolios against the market proxy have been estimated using MINITAB s regression function assuming the single-index market model holds. 5 Applying the obtained values to the single-index market model provides values of the individual portfolios corresponding to the different movement in the market proxy. The scenario assumes a 5% overall growth in the market in contrast to the actual 4 For the derivation and proof of these characteristics compare Exhibit 4 in appendix. 5 For detailed output data compare Exhibit 9 in the appendix. 4

5 Optimal portfolio construction in markets with no risk-free asset available -6.5% as calculated earlier. Again, utilities and risk premiums have been calculated under the assumption that the risk of the individual portfolios remains constant and results are being compared to give a final conclusion. Results of these calculations can be found in Exhibits and 3 in the appendix. Conclusion Considering the data obtained from this empirical analysis we can easily identify the portfolio with the smallest risk. 6 In this case it is not the market proxy it is the 50-shares-portfolio which is surprising in a sense that the market portfolio should be more diversified and display less risk. However, this notion refers to the average of all portfolios, i.e. as we have only considered one 50-shares-portfolio sample this particular sample can display this characteristic where on average the market portfolio will be more diversified and display less risk. Nevertheless, considering the fact that the 50-sharesportfolio also provided a better result in 00 compared with the market proxy the question arises if there was an unexploited arbitrage opportunity in the market, i.e. selling the market proxy and buying the 50-shares-portfolio, thereby making a risk-free profit. This question is easily answered and has two major flaws, one, we neglected transaction costs which might reduce the opportunity for arbitrage depending on the terms of trade for the individual investor and two, the advantage of hindsight. At the beginning of 00 it was not known that the terror attacks of September th in New York would happen and have a major impact on share returns. Therefore it is not easily concluded that an arbitrage opportunity existed as the risk of the portfolios has been estimated over the whole of 00 and not at the beginning of 00. In order to come to a conclusion here an analysis of the portfolios for a different period, i.e. 000, would have to be undertaken. The second notion arising from these results is whether expectations play a significant role for the decision of building the optimal portfolio. The data seems to suggest that depending on which expectations of future outcome the investor has, different portfolios will be chosen. Considering Exhibits 5 to 7 it becomes obvious that the minimum risk portfolio produced the smallest 6 cp. also Exhibit 5 in the appendix 5

6 Optimal portfolio construction in markets with no risk-free asset available loss for the actual data, i.e. an investor expecting a negative performance of the market would have bought the 50-shares-portfolio and not the market portfolio. For the adverse expectation of positive performance of the market proxy the market portfolio would have been the reasonable choice. 7 However, the actual risk of the portfolio doesn t seem to provide any indication for the choice of the optimal portfolio, the indicator which proves useful is the calculated risk premium, where the rule is that the higher the calculated risk premium the better the portfolio for the expectation of positive returns on the market proxy. But, as for the argument of arbitrage above, the same problem applies; we re dealing in this analysis with the advantage of hindsight and apply the data including the effects of September th, therefore the assumption of constant risk in the portfolio for the scenario of +5% might not hold true as the risk strongly depends on the fluctuations, i.e. the path to the annual return, and not the final annual return on the portfolio. In addition, other factors like change in the operating gearing or in the structure of the individual companies could lead to a different sensitivity to the market proxy and therefore also to a change in the beta of the portfolio. This, too, will influence the results. And as these impacts are not included in the scenario analysis data we cannot clearly conclude that the market portfolio is better or worse for the case of positive return expectations. In conclusion, we have to recognise that there is no clear-cut answer to the question whether the market portfolio is always the optimal choice for an investor. To answer this question different time periods would have to be considered, either a number of same length periods or at least one longer-term analysis over three to five years. Another way would be to exclude the adverse impact of the terror attacks of September th in New York. Therefore the quality of the results does not prove sufficient to provide a final answer to the question posed in this paper. 7 Although the 0-shares -portfolio produces a slightly better annual return and utility this difference is statistically insignificant recognising the quality of the regression leading to this result, i.e. R-Square statistic for the 0-shares -portfolio is 73.8%, there a difference of 0.08% in annual return is not significant and the market portfolio can be considered the better deal. 6

7 Appendix 7

8 Exhibit : Portfolio compositions Share Mean return return over year BAE Systems -0.04% -8.98% Legal&General -0.03% -3.8% Pearson -0.4% -50.5% Smith&Nephew 0.3% 33.87% Vodafone % -6.78% 5 shares portfolio Share Mean return return over year Astraze Neca -0.06% -8.% BAE Systems -0.04% -8.98% Cable&ireless -0.33% % Legal&General -0.03% -3.8% Logica -0.7% % National Grid -0.9% -9.66% Pearson -0.4% -50.5% Smith&Nephew 0.3% 33.87% Tesco -0.0% -8.7% Vodafone % -6.78% 0 shares portfolio Share Mean return return over year Allied Domecq -0.06% -7.86% Astraze Neca -0.06% -8.% BAE Systems -0.04% -8.98% British Airways -0.87% % Cable&ireless -0.33% % Innogy Holdings 0.03% -0.5% Kingfisher % -.5% Legal&General -0.03% -3.8% Lloyds TSB 0.040% 5.37% Logica -0.7% % National Grid -0.9% -9.66% Pearson -0.4% -50.5% Powergen 0.077% 9.46% Reuters Group -0.38% % Sage Group -0.07% -7.8% Six Continents % -6.7% Smith&Nephew 0.3% 33.87% Tesco -0.0% -8.7% United Utilities -0.07% -7.44% Vodafone % -6.78% 0 shares portfolio 8

9 Share Mean return return over year Abbey National % -9.6% Allied Domecq -0.06% -7.86% Assorted British Foods 0.03% 0.0% Astraze Neca -0.06% -8.% BAE Systems -0.04% -8.98% BG Group 0.046% 6.87% BHP Billiton 0.43% 35.7% BOC Group 0.036% 4.3% BP 0.05% -.% British Airways -0.87% % British Sky Broadcasting -0.% -3.56% Cable&ireless -0.33% % Cadbury Schweppes % -5.40% Canary harf Group -0.07% -8.56% Dixons Group 0.054% 4.9% Enterprise Oil % -7.8% Gallaher Group 0.056% 0.59% Hanson 0.03% 3.7% HBOS 0.096% 9.97% Hilton Group 0.036% 0.96% HSBC Holdings % -8.7% Imperial Tobacco Group 0.5% 30.08% Innogy Holdings 0.03% -0.5% Kingfisher % -.5% Land Securities -0.0% -7.00% Lattice Group 0.037% 3.3% Legal&General -0.03% -3.8% Lloyds TSB 0.040% 5.37% Logica -0.7% % Marks&Spencer 0.78% 94.09% Morrison (M) Sp. Markets 0.058% 0.85% National Grid -0.9% -9.66% Pearson -0.4% -50.5% Powergen 0.077% 9.46% Reuters Group -0.38% % Royal&Sun All. In % -3.% Sage Group -0.07% -7.8% Sainsbury -0.07% -7.8% Schroders NV -0.09% -3.85% Scottish&Southern Energy 0.008% -.6% Severn Trent 0.00% -3.36% Shell Transport & Trading % -4.03% Shire Pharmaceuticals -0.04% -8.48% Six Continents % -6.7% Smith&Nephew 0.3% 33.87% South African Brews 0.00% -0.% Standard Chartered -0.0% -4.98% Tesco -0.0% -8.7% United Utilities -0.07% -7.44% Vodafone % -6.78% 50 shares portfolio 9

10 Exhibit : Portfolio characteristics and calculated utilities (actual data) Portfolio Mean Return Portfolio Value after year Portfolio Value at beginning of year annual return Standard deviation (portfolio) Risk premium Utility (logarithmic) Utility (quadratic) Utility (exponential) 5 shares portfolio -0.8%,854,7-3.63%.93% -6.40% not defined not defined 0 shares portfolio -0.9% 6,600 9,6-3.40%.59% -9.74% not defined -4.4 not defined 0 shares portfolio %,638 5,4-4.54%.40% -7.58% not defined not defined 50 shares portfolio % 8,0 33,99-5.3%.8% -.93% not defined not defined FTSE % 5,7 6, -6.5%.35% -.94% not defined not defined Exhibit 3: Portfolio characteristics and utilities (scenario data) Portfolio Beta against market proxy esimated regression constant R-Square statistic from regression Value after year Value at beginning of year annual return risk premium Utility (logarithmic) Utility (quadratic) Utility (exponential) 5 shares portfolio % 3,099,7 4.7% 7.40% shares portfolio %,07 9,6 5.08% 9.48% shares portfolio % 7,587 5,4 4.04% 0.06% shares portfolio % 37,467 33,99.5% 0.57% FTSE % 7,56 6, 5.00%.09%

11 Exhibit 4: Utility functions characteristics (proofs/derivation) Logarithmic utility function: U( ) = ln U'( ) = U''( ) = U''( ) A( ) = U'( ) A'( ) = (with > 0) = = A'() < 0 DARA R( ) = * A( ) = = R'( ) = 0 CRRA Quadratic utility function: U( ) = 4 0 U' ( ) = 4 5 U' '( ) = 5 U' '( ) 5 A( ) = = = 0.05 U' ( ) 4 5 A' ( ) = > 0 IARA R() = *A() = *(0.05- ) = 0.05 R' ( ) = 0.05 > 0 IRRA

12 Exponential utility function: U ( ) = e U '( ) = * e U ''( ) = * e U' '( ) * e A( ) = = U' ( ) * e 3 A'( ) = DARA = 3 R() = * A() = R' () 9 = 4 < DRRA Exhibit 5: Portfolio risk Risk of chosen portfolios - standard deviation.0%.8%.6%.4%.%.0% 0.8% 0.6% 0.4% 0.% 0.0%.93%.59%.40%.8%.35% portfolio size

13 Exhibit 6: Annual returns on portfolios (actual data of -6.% return on market portfolio) Annual returns on portfolios -% -7% -% -5.3% -6.% -7% -% -7% -3.6% -3.4% -4.5% -3% portfolio size Exhibit 7: Annual returns on portfolios (scenario of +5% return on market portfolio) Annual returns on portfolios (scenario) 6% 4% % 0% 4.3% 5.% 4.0%.5% 5.0% 8% 6% 4% % 0% portfolio size 3

14 Exhibit 8: Descriptive portfolio statistics Descriptive Statistics: port5delta Variable N N* Mean Median TrMean StDev port5del 59-0,008-0,0078-0,0086 0,099 Variable SE Mean Minimum Maximum Q Q3 port5del 0,000-0,0586 0, ,0434 0,00797 Descriptive Statistics: port0delta Variable N N* Mean Median TrMean StDev port0de 59-0,009-0, ,006 0,059 Variable SE Mean Minimum Maximum Q Q3 port0de 0, , ,0464-0,038 0,0096 Descriptive Statistics: port0delta Variable N N* Mean Median TrMean StDev port0de 59-0, , , ,0396 Variable SE Mean Minimum Maximum Q Q3 port0de 0, , , , ,00834 Descriptive Statistics: port50delta Variable N N* Mean Median TrMean StDev port50de 59-0, , ,0004 0,084 Variable SE Mean Minimum Maximum Q Q3 port50de 0, , , , ,00605 Descriptive Statistics: ftsedelta Variable N N* Mean Median TrMean StDev ftsedelt 59-0, , , ,0353 Variable SE Mean Minimum Maximum Q Q3 ftsedelt 0, ,0576 0, , ,0070 4

15 Exhibit 9: Portfolio regression analysis Regression Analysis: port5delta versus ftsedelta The regression equation is port5delta = -0, ,956 ftsedelta Predictor Coef SE Coef T P Constant -0, , ,83 0,407 ftsedelt 0, , ,50 0,000 S = 0,0433 R-Sq = 45,0% R-Sq(adj) = 44,8% Analysis of Variance Source DF SS MS F P Regression 0, , , 0,000 Residual Error 57 0, ,00005 Total 58 0,09598 Regression Analysis: port0delta versus ftsedelta The regression equation is port0delta = -0, ,0 ftsedelta Predictor Coef SE Coef T P Constant -0, , ,44 0,5 ftsedelt,0087 0, ,9 0,000 S = 0,0086 R-Sq = 73,8% R-Sq(adj) = 73,7% Analysis of Variance Source DF SS MS F P Regression 0,0489 0, ,07 0,000 Residual Error 57 0,078 0, Total 58 0, Regression Analysis: port0delta versus ftsedelta The regression equation is port0delta = -0, ,939 ftsedelta Predictor Coef SE Coef T P Constant -0, , ,4 0,6 ftsedelt 0,9390 0, ,0 0,000 S = 0, R-Sq = 8,8% R-Sq(adj) = 8,8% Analysis of Variance Source DF SS MS F P Regression 0,0463 0, ,0 0,000 Residual Error 57 0, , Total 58 0,

16 Regression Analysis: port50delta versus ftsedelta The regression equation is port50delta = -0, ,835 ftsedelta Predictor Coef SE Coef T P Constant -0, ,0000-0,48 0,63 ftsedelt 0, ,069 5,8 0,000 S = 0, R-Sq = 9,% R-Sq(adj) = 9,% Analysis of Variance Source DF SS MS F P Regression 0, , , 0,000 Residual Error 57 0,0039 0,00003 Total 58 0,