Advanced Portfolio Theory - Assignment Example

Portfolio Theory Student’s Name Instructor’s Name Course Name Date of Submission Question 2 An index model can be described as a statistical model for calculating security returns. SIM usually identifies two major sources of uncertainty for the stock return. These risk are the systematic risk which is uncertainty which is assumed to be well presented by the single stock returns (Bodie 2013). The second risk is unique risk which is in the security of specific random components of the stock. Usually the stock here tends to move in the same direction being driven by similar economic factors. The single index model have stable relationship between stock and we can conclude that stability of stock weightings of an optimum portfolio generated using the single index model are very stable due to their relationship (Huang, et al., 2015). Testing of the optimum portfolio weightings of time varying can be done using Markowitz optimal portfolio using the matrix algebra. The Markowitz mean variance approach is one of the best approaches which can be used to construct optimal portfolio from chosen assets through diversifying risk and choosing weights for every assets in the portfolio variance for the maximization of the returns with a given risk level (Fan et al., 2013). Therefore, the Markowitz mean variance model is very ideal in measuring and creating the required weighted portfolio. From the expected return of the portfolio, the weighted average of the individual securities this can be calculated using the formula:- E (rp) = WA E (rA) + WBE (rB) + Wfrf In this case we have WA, WA, and wf are the portfolio weights of the stock both A and B and T-bills respectively. Plugging numbers in the formula from the first question above, will have E(rp) = .3X13 +0. 45X18 + 0.25*8 = 8 For the beta of the portfolio, we will use similar weighted average of the betas of the each security and the formula for the portfolio will be:- = βp = WAβA +WBβB + Wfβf Since the T- bills for beta is zero Will have = βP = .30 × .8 + .45 × 1.2 + 0 = .78 The variance of this portfolio is:- .302 × 302 + .452 × 402 + .252 × 0 = 405 Question three Regression OLS model The regression model results is shown in the table below:- Table 1.0: OLS Regression results Variable Coefficient Std. Error t-Statistic Prob.   FP 0.027849 0.000659 42.26201 0.0000 C 0.022863 0.034381 0.665009 0.5115 R-squared 0.984565     Mean dependent var 1.217933 Adjusted R-squared 0.984014     S.D. dependent var 0.847121 S.E. of regression 0.107107     Akaike info criterion -1.565643 Sum squared resid 0.321211     Schwarz criterion -1.472229 Log likelihood 25.48464     Hannan-Quinn criter. -1.535759 F-statistic 1786.077     Durbin-Watson stat 1.379008 Prob(F-statistic) 0.000000 From the table above, α = 0.022863, β = 0.027849 R2 = 0.984565. Therefore, using the first model the edge ratio will be β = 0.027849. The second model is the estimation of the hedge ratio using the VAR model and the results from the Eviews is shown in Table 2 Below. Table 2: The Bivariate VAR Model Estimates Equation (2) SP Equation (3) FP SP(-1)  1.552737  37.76572  (0.58313)  (24.1148) [ 2.66276] [ 1.56608] SP(-2) -1.010655 -26.86954  (0.57143)  (23.6308) [-1.76865] [-1.13706] FP(-1) -0.011652 -0.070008  (0.01491)  (0.61678) [-0.78123] [-0.11351] FP(-2)  0.021983  0.677736  (0.01413)  (0.58441) [ 1.55556] [ 1.15968] C  0.050706  1.262276  (0.09062)  (3.74739) [ 0.55957] [ 0.33684]  R-squared  0.914831  0.886019  Adj. R-squared  0.900019  0.866196  Sum sq. resids  1.584460  2709.664  S.E. equation  0.262468  10.85410  F-statistic  61.76287  44.69680  Log likelihood  0.477176 -103.7435  Akaike AIC  0.323059  7.767395  Schwarz SC  0.560953  8.005289  Mean dependent  1.154143  40.91679  S.D. dependent  0.830076  29.67281 From this model, we recall that in order to get the optimal hedge ratio the formula is given by:- Where:-δsf covariance ƐsƐf δf = Ɛf In order to calculate the hedge ratio here Will have:- Table 3: Optimal Hedge ratio using Bivariate VAR Model Values Covariance (ƐsƐf) 0.000195 Variance (Ɛf) 0.000209 h* 0.933 The third step is to calculate the hedge ratio using the vector error model from the data of spot and future prices. The results are shown in the table below:- Table 5: The Vector Error Correction Estimates Model Error Correction: D(SP) D(FP) Coint Eq1  0.668483  56.74549  (0.65994)  (25.5853) [ 1.01294] [ 2.21790] D(SP(-1))  0.886632  28.68181  (0.52491)  (20.3501) [ 1.68912] [ 1.40942] D(SP(-2)) -1.860029 -74.80484  (0.52636)  (20.4066) [-3.53374] [-3.66573] D(FP(-1)) -0.016662 -0.652466  (0.01430)  (0.55435) [-1.16529] [-1.17699] D(FP(-2))  0.037930  1.416303  (0.01228)  (0.47623) [ 3.08778] [ 2.97399] C -0.096724 -4.010205  (0.04251)  (1.64807) [-2.27533] [-2.43328]  R-squared  0.502149  0.568357  Adj. R-squared  0.383613  0.465585  Sum sq. resids  0.938071  1409.951  S.E. equation  0.211353  8.193933  F-statistic  4.236252  5.530273  Log likelihood  7.045513 -91.71023  Akaike AIC -0.077445  7.237795  Schwarz SC  0.210518  7.525759  Mean dependent -0.089000 -3.074074  S.D. dependent  0.269204  11.20865  Determinant resid covariance (dof adj.)  0.324400  Determinant resid covariance  0.196242  Log likelihood -54.63920  Akaike information criterion  5.084385  Schwarz criterion  5.756301 From the equation of calculating the optimal hedge ratio, we recall that in order to get the optimal hedge ratio the formula is given by:- Where:-δsf covariance ƐsƐf δf = Ɛf For the calculation of the hedge ratio using the above formula will have:- Table 6. Optimal Hedge Ratio from the VEC Model Value Covariance (εs εf ) 0.000194 Variance (Ɛf) 0.000207 h* 0.937 In order for us to establish the efficiency of the model two and the model three, it is important for us to investigate or use the fourth model to examine the characteristics of the residuals. We will be able to plot the residuals of these two equations to establish the effect. The graphs are shown below:- Equation 2 Equation 3 Equation 3 From the above table, we conclude that there is no presence of the ARCH effect in the data and we can conclude with the interpretation of the results. The calculation of the hedge ratio in this case will be:- Table 7: GARCH Model of Estimation GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1) Variable Coefficient Std. Error z-Statistic Prob.   FP 0.026674 0.001025 26.01631 0.0000 C 0.045452 0.020403 2.227714 0.0259 Variance Equation C 0.000108 0.000145 0.749360 0.4536 RESID(-1)^2 0.615843 0.413277 1.490145 0.1362 GARCH(-1) 0.388111 0.198727 1.952983 0.0508 R-squared 0.981697     Mean dependent var 1.217933 Adjusted R-squared 0.981043     S.D. dependent var 0.847121 S.E. of regression 0.116636     Akaike info criterion -2.767670 Sum squared resid 0.380909     Schwarz criterion -2.534138 Log likelihood 46.51506     Hannan-Quinn criter. -2.692961 Durbin-Watson stat 1.008923 From the table above, we can find β=0.026674. In addition, α =0.045452, p-value in both cases is < 0.05 indicating that the null hypothesis is rejected and acceptance of alternative hypothesis. Question four The Markowitz procedure and the Separation Property is what we used to first compute w1* to determine P* (given the risk-free rates and the risk and return characteristics of the risky assets) and then compute y* to determine C* (using the risk-free rate, the risk and return of the optimal risky portfolio and risk aversion). The Separation Property refers to separating the choice of the weights of the risky assets from the choice of the weight of the risk-free asset (Fan et al., 2013). Although we talked about it in lecture, we did not do a complete Index Model Optimization. The procedure is described in the text starting on page 261. The advantage of an index model optimization, compared to the Markowitz procedure, is the reduced number of estimates required (Rabhi, Belkhir and Soltani 2016). For N assets, the Markowitz procedure requires: N Variance (or standard deviation) calculations: σi2 (N2 – N)/2 Covariance calculations: σij The total is N + (N2 – N)/2 In N = 25, then the number of estimates is 25 + (252 -25)/2 = 325 For the index model, we can compare each asset to the index. So solve for: σi2 = βi2σM2 + σ2(εi) σij = βiβjσM2 For N assets, the index procedure requires: N Beta calculations: βi N Variance of the errors calculations: σ2(εi) 1 Expected return of the market estimate: E(rM) 1 Variance of the market estimate: σM2 The total is 2 x N + 2 In N = 25, then the number of estimates is 50 + 2 = 52 Another advantage of using an index model is that the relationship between stocks and an index (measured by βi) may be more stable than the relationships between stocks (measured by σij) (Rabhi, Belkhir and Soltani 2016). The disadvantage of the index model arises from the model’s assumption that return residuals are uncorrelated and therefore can be ignored. This assumption will be incorrect if the index employed omits a significant risk factor. For example, we know that the single-index market model (essentially the CAPM) seems to omit risk-factors associated with market-cap and relative price (growth or value stock) (Rabhi, Belkhir and Soltani 2016). In this case, the errors are correlated. Therefore the math that allows for the calculate of σi2 = βi2σM2 + σ2(εi) and σij = βiβjσM2 does not work Following the development by Markowitz, William Sharpe was able to come up with the single index model which can help in calculating the security returns in most of common index. For this purpose, scholars or investment analyst uses a wide range of common stock index. The following equation is able to express single index model and it is written as follows:- Ri = α +βi +Ɛi Where we have:- Ri – Return on security Rm – Market index security which is independent on performance αi – this is the part of return on security βi - is the constant Ɛi is the random error term Single index model help in dividing the security return into two components which is the market related component and the unique part (Wang et al., 2015). In SIM, it is important to note that the unique part of the model represent factors that affect individual companies while the second part represent macro events that influence the entire market. Model assumes that the market securities only relates with their common response to the return on the market. That is, the error residuals for the stock security I are usually uncorrelated with those of security Y and can be expressed as Cov (ei, ey>) = 0. The objective of the SIM is normally similar to that of the Markowitz model as both help in tracing the efficient frontier (set) of a given portfolios from which the investor would choose an optimal portfolio from (Wang et al., 2015). This model is a simplification of variance-Covariance matrix which is needed for the Markowitz model. Multi index model on the other hand is given by the formula:- E (Ri) = αi + bi (RM) + CiNF + ei In this equation, NF is the nonmarket factor. The multi index model uses more information compared to single index model hence it is able to perform much better. In general, the multi index model falls between the full variance-covariance method of the Markowitz and the single index model proposed by Sharpe. MIM pursue both the beta adjustment in a different approach compared to SIM. It tackles the problem in relation to the stock security not only to the market but through including a market index and other indexes that can quantify other movements in the market. The covariance of security return with other market impacts can also be added to the model directly by quantifying the effect through the use of more indexes. In SIM, security are only related in their common response to the market. Shows that security normally covaries together only because of their common relationship to the market index. The market risk is the only factor that influences the security covariance and this can be written as follows:- The model split security risk into two as mentioned above, that is Total risk = market risk + Unique risk The multiple index model majorly focuses on the equilibrium relationship of expected risk and risk on different risky assets. It is based on the capital asset pricing model with similar assumptions like:- No transaction cost There is no personal taxes No single investor who can influence the prices of stock The capital market are in equilibrium Information are readily available in the market Investors can easily borrow and lend money at risk free rate Reference Bodie, Z., 2013. Investments. McGraw-Hill. Fan, Y., Härdle, W.K., Wang, W. and Zhu, L., 2013. Composite quantile regression for the single-index model. Huang, Z., Shao, Q., Pang, Z. and Lin, B., 2015. Adaptive testing for the partially linear single-index model with error-prone linear covariates. Statistical Methodology, 25, pp. 51-58. Rabhi, A., Belkhir, N. and Soltani, S., 2016. On Strong Uniform Consistency of Conditional Hazard Function in the Functional Single-Index Model. International Journal of Mathematics and Statistics™, 17(2), pp.85-109. Wang, T., Zhang, J., Liang, H. and Zhu, L., 2015. Estimation of a groupwise additive multiple-index model and its applications. Statistica Sinica, pp.551-566. Read More