Portfolio Theory and Asset Allocation: Some Practical Considerations Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 1 Investment Horizon Mean-Variance analysis is a static problem ,! It can be applied to any horizon ,! But, the choice of horizon affects the inputs ,! With a monthly investment horizon, the expected returns and variances should be monthly It is easy to adjust the expected returns to the horizon ,! Expected returns scale with the horizon H The horizon play a more subtle role when estimating risk ,! Finance textbooks often advocate the square-root of time rule: p s(H-period return) = H × s(1-period return) ,! What is the implicit assumption here? Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 2 Investment Horizon Should an investor with a 10-year investment horizon allocate more to stocks than an investor with a 1-year horizon? ,! Intuitively, “stocks are safer in the long run” so she can afford to “wait out” any market declines Our portfolio allocation formula says that the optimal allocation to the risky portfolio depends on the ratio of its mean to its variance: 1 E(r) − r w = f A s2 ,! So, what is the answer to the previous question? ,! As an aside, what does this imply about Sharpe Ratios? ,! What does it actually look like empirically? Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 3 Investment Horizon H-period returns on the aggregate market portfolio 1927-2012: var(rH ) Horizon var(rH ) var(r1) 1-month 0.003 1-year 0.040 13.54 2-year 0.086 28.82 3-year 0.117 39.34 4-year 0.147 49.17 5-year 0.165 55.47 10-year 0.240 80.63 What is going on here? ,! “Stocks are safer in the very long-run” ,! Do the finance textbooks give the right advice? ,! To further complicate things, the variance isn’t constant through time ... Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 4 Investment Horizon Daily US equity volatility constructed from 5-minute intraday returns: CAPITAL AQR MANAGEMENT ,! Volatility clearly time-varying ,! Higher in bad economic times and periods of market distress ,! What about longer horizons? Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 5 Investment Horizon S&P 500 realized volatilities, 1990-2008: Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 6 Investment Horizon S&P 500 options implied volatilities, 2008-2010: Schwert (2011, European Financial Management ) Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 7 Dimensionality In the US there are literally tens of thousands of traded securities ,! Clearly, we cannot solve the MV optimization problem with all of these different securities Even with a "reasonable" number of securities, say N=100, the number of parameters to be estimated is prohibitively large: si’s N 100 E(ri)’s N 100 1 Cov(ri;r j)’s 2 N(N − 1) 4950 1 Total 2 N(N + 3) 5150 ,! The number of covariances to be estimated grows like N2 Index Models provide a way to reduce the dimensionality ,! A small number of common factors drive the correlations of all the assets/returns Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 8 Single Index Model The single index, or market, model assumes that returns can be decomposed into a single “systematic” (market related) part and “non-systematic” (non-market related) part: ri − r f = ai + bi(rm − r f ) + ei ,! The non-systematic ei part is truly asset-specific ,! For two different securities i and j: cov(ei;e j) = 0: ,! Is that a reasonable simplification? ,! We will later generalize this to allow for K > 1 common factors Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 9 Single Index Model The covariances implied by the single index model are: 2 cov(ri;r j) = bib jsm 8i 6= j For the variances: 2 2 2 2 si = bi · sm + sei ,! The variances are unaffected by the model The correlation between the two assets i and j is therefore: 2 bib jsm ri; j = sis j Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 10 Single Index Model Instead of N × (N − 1)=2 individual covariances, for the single index model we only need: ,! N betas, one for each of the assets ,! The variance of the market (the common factor) For a K-factor model we would need N × K betas and K factor variances ,! The covariances are deduced solely from the K common (macro) factors that affect all securities ,! For a small number of factors, the reduction in the number of parameters to be estimated can be substantial ,! This also facilitate the implementation of traditional security analysis. Why? Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 11 Single Index Model The beta of asset i with portfolio m in the single index model is formally defined by: cov(ri;rm) bi = var(rm) Where do these betas come from? ,! They may be estimated by time series regressions ,! Assume the same single index model holds for T consecutive time periods: ri;t − r f ;t = ai + bi(rm;t − r f ;t ) + ei;t ; t = 1;:::;T ,! What are the assumptions here? ,! We will revisit this issue in more detail later ... Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 12 BKM Beta Estimation Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 13 BKM Beta Estimation bHP = 2:03 bDELL = 1:23 Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 14 BKM Beta Estimation bBP = 0:47 bSHELL = 0:67 bS&P500=? Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 15 BKM Beta Estimation Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 16 Six Country Example Consider the problem of investing in the following six countries: France, Germany, Japan, Switzerland, UK and the USA ,! 504 monthly aggregate equity index returns from 1970 to 2011: FRA GER JAP 0.3 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.1 0.1 0.1 0.05 0 0.05 0 −0.1 0 −0.05 −0.05 −0.2 −0.1 −0.15 −0.1 −0.3 −0.2 −0.15 −0.4 −0.25 −0.2 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 SWZ UK USA 0.25 0.6 0.2 0.2 0.5 0.15 0.15 0.4 0.1 0.1 0.3 0.05 0.05 0.2 0 0 0.1 −0.05 −0.05 0 −0.1 −0.1 −0.1 −0.15 −0.15 −0.2 −0.2 −0.2 −0.3 −0.25 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 17 Six Country Example Six countries: France, Germany, Japan, Switzerland, UK and the USA Sample means, variances and covariances: 0 1:02 1 0 44:0 30:9 17:9 23:8 26:5 17:0 1 B 1:00 C B 30:9 41:8 16:3 24:6 22:1 16:0 C B C B C B 0:94 C B 17:9 16:3 39:0 15:2 16:4 10:0 C µˆ = B C Sˆ = B C B 1:06 C B 23:8 24:6 15:2 28:8 21:3 13:7 C B C B C @ 1:03 A @ 26:5 22:1 16:4 21:3 41:8 17:0 A 0:88 17:0 16:0 10:0 13:7 17:0 20:5 ,! What are the annual means and variances? ,! Correlations? Now, let’s try to use a single index model ,! We need to specify the common factor Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 18 Six Country Example, Single Index Model Take the “global” market index as the common factor ,! An equally weighted average of the six country portfolios ,! Var(rG) = 22:1 The resulting b’s and covariances implied by this single index model are: 0 1:211 1 0 44:0 30:6 23:2 25:7 29:3 19:0 1 B 1:147 C B : 41:8 22:0 24:3 27:7 18:0 C B C B C B 0:869 C B :: 39:0 18:4 21:0 13:6 C b = B C S = B C B 0:963 C B ::: 28:8 23:3 15:1 C B C B C @ 1:098 A @ :::: 41:8 17:2 A 0:712 ::::: 20:5 Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 19 Six Country Example, Single Index Model Historical and implied single index model correlations: 0 1 0:72 0:43 0:67 0:62 0:57 1 B : 1 0:40 0:71 0:53 0:55 C B C B :: 1 0:45 0:41 0:35 C Cˆ =B C versus B ::: 1 0:61 0:56 C B C @ :::: 1 0:58 A ::::: 1 0 1 0:71 0:56 0:72 0:68 0:63 1 B : 1 0:54 0:70 0:66 0:62 C B C B :: 1 0:55 0:52 0:48 C CIndex Model = B C B ::: 1 0:67 0:62 C B C @ :::: 1 0:59 A ::::: 1 Not much difference here ,! Why? Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 20 BKM Recap Six-stock MVE portfolios, BKM Figure 8.5: ,! Estimated efficient frontiers very close I Why? Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 21 Single Index Model and Time-Varying Correlations One reason that MV theory fails in practice is that the world is non-stationary ,! The past is not always a good predictor of the future A common belief among market practitioners is that in times of high uncertainty (market volatility), asset correlations tend to increase ,! As correlations increase, the benefits to diversification decrease ,! Case in point, 2008-09 financial crises Recall that within the single index model: b b s2 r = i j m i; j 2 2 2 1=2 2 2 2 1=2 (bi · sm + se;i) (b j · sm + se; j) 2 ,! All else equal, an increase in the volatility of the common factor sm, implies an increase in the pairwise correlations (provided bi > 0 and b j > 0) ,! Index models can help make sense of time-varying correlations Econ 471/571, F19 - Bollerslev Portfolio Theory and Asset Allocation in Practice 22 Single Index Model and Diversification Consider an equally weighted portfolio of N assets The variance of the portfolio equals: N N N 2 1 2 1 sp = 2 ∑ si + 2 ∑ ∑ sis jri j N i=1 N i=1 j=1 i6= j 1 ' s¯ 2 + s¯ 2 × r¯ N ,! What happens as more and more assets are included in the portfolio? ,! The average variance