Bus 172a                                                                   Professor Ali Reza
end up with $67,000. But suppose that you switch between the CP and the NYSE index
        once a month based on your fore...
the holes”) – the completeness fund uses as benchmark a customized portfolio that has
            the characteristics of s...
The advantage of ETFs relative to mutual funds is that ETFs are traded throughout the day,
      like any other stock. The...
To overcome the objection raised with peer group evaluation, composite measures were developed
to take into account risk a...
Information ratio, IR (also called Appraisal ratio): It is the ratio of Jensen’s αp and the standard
    error of the regr...
The reason for the difference in evaluation is that Janus portfolio is so far from being well-
diversified. As a result S ...
Note that
(13)    (wP1 R P1 - wB1 R B1 ) = {(wP1 - wB1) R B1}+ [(RP1 - RB1) w P1]
(14)    (wP2 R P2 - wB2 R B2 ) = (wP2 - ...
3.    So we are left with the situation that the stock is volatile but trendless. Whether one
          makes or loses mon...
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Equity Portfolio Mgt


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  1. 1. Bus 172a Professor Ali Reza Lecture Notes: Equity Portfolio Management Portfolio management involves 4 steps: 1. Specify objectives 2. Specify constraints 3. Formulate policy 4. Monitor and update the portfolio Objectives Constraints Policies Return Requirement Liquidity Asset Allocation Risk Tolerance Horizon Diversification Regulations Income generation Taxes Unique needs Under policies, by far the most important is Asset Allocation – how much to invest in each major category (money markets [usually called “cash”]; fixed-income [bonds]; stocks; real estate; precious metals; others). Note: you may want to check www.quicken.com/retirement/planner to come up with along-term allocation plan for your self. Policy statement: institutions (e.g., pension plans) are governed by boards that have official statements of investment policy. These often provide info about objectives and constraints. A portfolio manager should stick with the policy and requirements as stated by the investor. Departing from them, even if it leads to higher returns, is just not acceptable. To demonstrate why, the investor may have other investments that are acquired with this policy in mind. Note on reasons for investing in stocks: Often people offer the following reasons for investing in stocks: a. Stocks are not risky in the long-run. Wrong! While the variance of stock rates of return declines over the long run, the number of dollars at risk rises. Consider the case where you toss a coin 10 times. You may get 6H and 4T (Heads and Tails), for 60% Heads – a significant departure from 50% expected. We have only “2 Heads in excess” of Tails. Now toss 1000 times. You may get 505 H and 495 T, 50.5% Heads – very close to the expected 50%. But now we have a larger “10 Hs in excess” of Tails. As the number of independent events rises the variance diminishes but the total number away from the expected value rises – the number of dollars away from the mean rises. b. Stocks are a good hedge against inflation. Wrong! The argument favoring stocks as hedge goes as follows: stocks represent ownership of real physical capital so profits are either unaffected or even enhanced with inflation. It turns out that the empirical evidence is against that: stock returns are mildly negatively correlated with inflation. Market timing (example by R. Merton): Consider an investment of $1,000 on 1/1/1927 until 12/31/1978. If invested in 30-day commercial paper (CP) and all proceeds reinvested, you’d end up with $3,600. If invested in the NYSE index and reinvest all dividends you’d 1
  2. 2. end up with $67,000. But suppose that you switch between the CP and the NYSE index once a month based on your forecast of which of the two would outperform the other. If your forecast was right 100% of the time, how much do you think you’d end up with? (Answer at the end of these notes). It is a huge number. That is why so many people attempt market timing. By the way this higher return is NOT compensation for risk; it is compensation for superior analysis/forecasting – that is one reason so many people work hard to become good analysts/forecasters. On Forecasting: a weather forecaster in Palm Springs who always predicts no rain may be right 90% of the time, but this is not evidence of forecasting ability. Similarly, the appropriate measure of market forecasting is not the overall proportion of correct forecasts – if a forecaster always predicts a market advance, he’ll be right 2/3 of the time simply because the market has in the past risen 2/3 of the time; this forecast is not worth much. Passive and Active Management: Portfolio management is either Passive or Active. Passive – long-term buy-and-hold strategy. Usually the attempt is made for the portfolio return to track that of an index (this approach is often referred to as “indexing”). The purpose of such a strategy is not to beat the index, but to match it – to minimize the deviation of the portfolio return from the return of the index Note that passive management is really not that passive – as the stock market rises and falls, even an index fund has to make changes in its portfolio. Rebalancing is needed as the portfolio receives dividends and these must be reinvested; firms merge or are dropped from the index; etc. Active – attempt to outperform (on a risk-adjusted basis) a benchmark portfolio, e.g., the S&P500 index. The benchmark portfolio is a portfolio whose characteristics (beta, firm size, industry weights, P/E ratio, dividend yield …) match the risk-return objective of the client. Active management costs about 1.5% of the assets invested. If markets are efficient, it is difficult to overcome this cost and still match (let alone beat) the market (on a risk-adjusted basis). It turns out that active management investments dominate the passive ones (in dollar terms). But the index sector is growing faster than the active sector, at least in part because of lower cost of the passive approach. Also when it comes to fixed-income indexing, active management is much more popular than for equity-indexing – it seems that people prefer active strategy when it comes to bond investing (PIMCO expanded because of its active management in fixed-income investing). Index benchmarks: a. Buy the index itself, that is, buy all the securities in the index, in proportion to their weights. But if there are lots of stocks included, transaction costs may become too burdensome, especially since one has to reinvest the dividends b. Buy a sample of the index, i.e., buy a representative sample of the securities in the index. This overcomes the problems associated with buying a large number of securities in the index, but since it is a sample, the portfolio may not track the benchmark closely; this may require rebalancing and thus add to transaction costs. A skillful manager can balance these costs with tracking c. Optimization: use an optimization technique to construct the portfolio. Using the price changes and correlations between securities, select those securities for inclusion in the portfolio that minimize the tracking error with the benchmark (see below for tracking error) d. Completeness funds. These are portfolios to complement active portfolios that are overweighed some stock types. The sponsor then wants the remaining funds to be invested passively in market segments that are not invested in the active portfolio (“fill 2
  3. 3. the holes”) – the completeness fund uses as benchmark a customized portfolio that has the characteristics of stocks excluded from the active portfolio. For example, an institutional investor uses an active portfolio manager to invest part of its funds in hi- tech stocks. To ensure diversification, the investor will want the remaining funds to be invested in a completeness fund which has as a customized benchmark stocks that exclude hi-tech stocks but cover the rest of the market. Note: when one uses a benchmark to track one’s portfolio, then the appropriate systematic risk of the managed portfolio is its beta relative to the benchmark, not relative to the overall market. In such a situation, the benchmark acts as the “market”. It should be noted that in general the beta of a security or portfolio changes depending on the “market” one uses. So, e.g., if GM beta = 0.75 when the market is measured by the S&P500, GM beta may be 0.56 (or some other number) if the market is measured by the global stock market index. Tracking error: This is the nonsystematic risk of the portfolio. It is defined as the extent to which fluctuations in the managed portfolio are not correlated with returns in the benchmark. Consider the rate of return to the managed portfolio P: (1) RPt = Σ wj Rjt , where there are j=1,…, N securities; t measures time; and t = 1,…, T In Eq 1, RPt measures the managed portfolio return at time t; Rjt is the rate of return on security j and fraction wj is invested in security j. The differentials of the return of the portfolio P from the benchmark portfolio B is (2) Δ t = RPt – RBt ; t = 1,…,T For a sample of T observations of the return differentials, the variance of Δ t is (3) var = Σ (Δ t - Δ )2/ (T-1) ; the summation is from 1 through T, and Δ is the mean of Δ t . Then we can calculate the tracking error as σ = [ var ]1/2 In general, to reduce tacking error one must spend more time and expense. So again, a skillful manager must balance the tradeoff. Index Funds and ETFs: There are about 200 ETFs and many times that index funds. An ETF (exchange traded fund) is similar to ADRs (American Depository Receipts which are used to invest in foreign stocks). The ETF is more like an open-end fund: the ETF share price will not deviate (unlike closed-end funds which often do) from the NAV. ETF shares are bought and sold in two different ways. Large investors trade the ETF directly, while small investors trade shares on the market through a broker. If the ETF share price is less than the NAV, large investors buy the ETF shares, redeem them to obtain the underlying securities, and sell these underlying securities in the market. This closes the gap between the ETF share price and the NAV. There are ETFs tracking the S&P500 (“spider” which is based on a basket of all securities in the index), iShares (replicates several international markets), sector ETFs (focus on specific industries). 3
  4. 4. The advantage of ETFs relative to mutual funds is that ETFs are traded throughout the day, like any other stock. The cost of running an ETF is minimal. The main disadvantage of ETF is brokerage commission. The main disadvantage of mutual funds is that redemption takes place once a day at the closing value of the NAV. Strategic, Tactical and Insured Asset Allocation: Strategic allocation is used to decide asset class weights for the long run. Historical long- run returns, variances and covariances are used to construct efficient frontiers (portfolios). The investor then selects the asset mix that meets her/his needs. The mix is periodically rebalanced to keep the weights as specified initially. Tactical allocation causes frequent changes in the asset class allocation in response to changing market conditions. The attempt is to take advantage of changing market conditions, given that the investor’s risk tolerance and investment constraints have stayed the same. Insured allocation assumes that returns and risks have remained unchanged but the investor’s goals and constraints have changed (due to age, family profile, wealth, …). Treynor-Black (TB) Model of Portfolio Construction: (For details see, J. Treynor and F. Black, “How to Use Security analysis to Improve Portfolio Selection”, J. of Business, Jan. 1973). The TB idea is that with active management one can do security analysis and find underpriced securities. Construct a portfolio of these securities. But since it is likely that this portfolio may have a large unsystematic risk, diversify it by allocating some of the funds to the market portfolio (the Passive portfolio, if you will) – so the final portfolio consists of the underpriced securities discovered through security analysis and the market portfolio. This new portfolio will of course lie above the market portfolio since the market portfolio is no longer viewed as efficient. As a result the Capital Market Line of the CAPM is no longer a constraint. It should be noted that as one diversifies the underpriced securities portfolio by mixing it with the market portfolio, the expected return of the new portfolio declines (as compared to the underpriced securities portfolio); but the benefit lies in the lower risk of the new mix portfolio. [This approach is quite a bit more involved than sketched here. For example, one can determine the optimal composition of the underpriced securities portfolio and the market index. So you should consult the original article if you decide to pursue this approach] Answer to the Merton problem: $5.35 –Billion Portfolio Evaluation How do we decide whether our portfolio has done well? We need to compare it with something. What? Among the most popular measures are: Peer group; Treynor’s measure; Sharpe’s measure; Jenson’s measure; Information ratio (also called Appraisal ratio); and M2 measure. Treynor, Sharpe, Jenson and M2 measures are all related to the CAPM model. Peer group: select a representative universe of portfolios, investment managers or investors and compare their performance with that of the portfolio being evaluated. We can divide the performance of the investors into percentiles and see where our portfolio fits. The problem is that despite careful attention and classification, each of the portfolios is likely to have a different risk level. 4
  5. 5. To overcome the objection raised with peer group evaluation, composite measures were developed to take into account risk as well. The following measures are among these. Treynor (T): Essentially this measure compares the managed portfolio P with the SML (security market line). If it lies above (below) the SML, it has outperformed (underperformed) the market index. The Treynor measure itself is a ratio: (4) T = (Rp – Rf) / βp ; Rp = portfolio return ; Rf = riskfree rate ; βp = portfolio beta The higher is T the better has the portfolio performed. Note that the Rp-Rf is the risk premium, so T measures risk premium per unit of systematic risk. All risk averse investors want to minimize this ratio so we don’t need to know anything else about the utility function of the investors. Treynor’s measure assumes that the appropriate measure of risk for P is beta – in other words, the assumption is that P is well-diversified. It is worth mentioning that if the portfolio is the market index M itself, then βM =1 and the T does indeed measure the slope of SML. Thus, for any portfolio, T is a measure of the SML. Sharpe (S): this measure does away with the implicit assumption that P is well-diversified – it may or may not be. So instead of using beta, Sharpe’s measure uses P’s standard deviation: (5) S = (Rp- Rf) / σP ; σP = portfolio stand dev The S ratio is simply the slope of the Capital Market Line (CML). If P outperforms the market, then it will lie above the CML – i.e., the slope of the line from Rf on the vertical axis (in the return-standard deviation space) will be greater than the slope of the CML. The higher is S the better has the portfolio performed. Jensen (α): Consider a benchmark portfolio B and the managed portfolio P. Since the idea is to evaluate the performance of P, then the return on the benchmark B is (6) Rb = Rf + (Rm- Rf) βp ; note that the beta used is P’s, not the benchmark’s. Now consider the difference between the returns on the benchmark B and the managed portfolio P (7) αp = Rp – Rb ; now substitute Eq (6) for Rb of Eq(7) to obtain: (8) αp = Rp – [Rf + (Rm- Rf) βp ] This is Jensen’s alpha (or Jensen’s measure). Again, note that this relationship is based on the SML. If P has outperformed the benchmark, αp > 0. There is an alternative explanation: Consider the regression of Rp on Rm (the characteristic line), Rp = αp + βp Rm + ep . Deduct Rf from both sides to obtain (9) Rp – Rf = αp + βp (Rm – Rf) + ep . This again gives you the Jensen alpha. 5
  6. 6. Information ratio, IR (also called Appraisal ratio): It is the ratio of Jensen’s αp and the standard error of the regression in Eq(9), σp IR = αp / σp The Treynor measure finds the position of the managed portfolio versus the SML – it the portfolio lies above the SML, it has outperformed the market; if below the SML the portolfio has underperformed the market. The Sharpe measure does the same thing, except that the postion of the the portfolio is relative to the CML. With the Jensen measure we find the position of the portfolio; alpha is simply the distance between the portfolio and the SML. M2 (for Modigliani and Modigliani) measure: this has become popular in recent years. It was first proposed by Graham and Harvey (W.P. #4890, Oct-1994, National Bureau of Economic Research), but M-M popularized it (J of Portfolio Management, Winter 1997). A problem with the Sharpe measure is that it can be used to rank portfolio performance but its value is not easy to interpret. For example, suppose that a managed portfolio P has a Sharpe ratio = 0.72 whereas the market portfolio M has a Sharpe ratio = 0.77. P has underperformed the market, but is the difference 0.05 economically meaningful? Can’t tell. But if we knew that the two portfolios had identical risks, then all we’d have to do would be to compare their returns. M2 does that. Suppose that P is 1.33 times as volatile as the market. Construct a new portfolio P* by mixing P with the riskfree security: invest 75% in P and 25% in the riskfree security – P* would now be exactly as volatile as M. They both would have the same variance. So we can compare their returns: M2 = RP* _ Rp It should be noted that this measure too is based on the CML (See Figure on p.7). Comparison of T, S , α and M2 : Since T and α are based on the SML, they always provide identical answers – if using T it is determined that P has outperformed the market, using α one gives the same answer. The same applies to S and M2 since both are based on the CML. And if P is perfectly diversified, then all four will give the same answer. But if P is not well diversified, one could (often does) get different rankings using T versus S. In particular, if P is not well-diversified, one could have T telling us that P outperformed the market while S saying it underperformed the market. Ex: the Janus Investment Fund (Venture) had an average rate of return of Rj=29.47% during the period July-95 through June-00 (monthly data); its standard deviation was SDj =31.92%. For the S&P500 the return was Rm=22.96% with a std dev of SDm=14.95%. The betas were for this period βj = 1.01 and βm = 1.00. The correlation of Rm and Rj was 0.224 – suggesting that Janus was very poorly diversified. Calculate T and S for Janus and for Market (Rf= 5.28% during this period): Tj = 29.47-5.28/1.01 = 23.95 Tm = 22.96-5.28/1 = 17.68 ; thus Janus outperforms the market according to T Sj = 29.47-5.28/31.92 = 0.758 Sm = 22.96-5.28/14.95 = 1.182 ; thus Janus underperforms the market according to S. 6
  7. 7. The reason for the difference in evaluation is that Janus portfolio is so far from being well- diversified. As a result S penalizes this portfolio for bearing so much unsystematic risk. ____________________________________________________ Geometry of M2 E(R) CML M P P* Rf 0 SDM SDP Std Dev The above diagram shows how M2 evaluates portfolios. Portfolio P has returned E(Rp) >E(Rm). But P has a std dev which is higher than that of M (the market). But mixing the appropriate amounts of the riskfree security and P we can obtain the new portfolio P* which has exactly the same std dev as that of M. Since the riskfree asset has a lower rate of return than P. then E(RP*) < E(RP). This mixing is shown as a straight line from Rf to P. As the dashed arrow shows, M2 transports point P to point P*. Now we have tow portfolios with the same std deviations and portfolio M dominates P* and, therefore, P as well. In other words, P has underperformed M. ____________________________________ Attribution: Instead of focusing on risk-adjusted return, one may want to identify which decisions resulted in superior performance – i.e., what can one attribute to the performance. We can measure “ability” broadly (asset allocation) or very narrowly (individual security selection). Attribution starts from the broadest asset allocation and progressively focuses on ever-finer details. Difference between the benchmark and the managed portfolio may be expressed as the sum of contributions to performance of a series of decisions at the various levels of the portfolio construction process. For example, attribution decomposing performance into 3 components: (1) broad asset allocation; (2) industry choice; and (security choice within each industry. Ex: Two sectors are considered each for the benchmark (B) and the managed portfolio (P). (10) RB = wB1 R B1+ w B2 R B2 (11) RP = wP1 R P1+ w P2 R P2 For example: B1 is the autos sector; P1 is the auto tires sector B2 is the muni bonds; P2 is the corporate bonds Thus the rates of return can be different for B and P. Next measure the difference in returns: (12) RP - RB = (wP1 R P1 - wB1 R B1 ) +( w P2 R P2 - w B2 R B2 ) 7
  8. 8. Note that (13) (wP1 R P1 - wB1 R B1 ) = {(wP1 - wB1) R B1}+ [(RP1 - RB1) w P1] (14) (wP2 R P2 - wB2 R B2 ) = (wP2 - wB2) R B2 + (RP2 - RB2) w P2 Consider Eq(13). The term in { } shows the impact of allocation between autos and tires because is shows the deviation of the portfolio P’s weight (wP1) from benchmark’s weight (wB1 ) multiplied by the return on the benchmark. It is the weight difference that has added to (or deducted from) P’s performance. The term in [ ] shows the impact of sector selection because it shows excess return of one sector compared to benchmark’s; the more invested in the auto tires (i.e., the greater is w P1) the higher will be the impact. Suppose that P earned 5.663% and B earned 4.066%, for a difference of 1.597% A. Contribution of Allocation Portfolio weight Benchmark weight Excess weight Benchmar Contribution to k return performance (1) (2) (3) =(1)–(2) (4) (5) = (3)x(4) Equity 0.7 0.6 0.1 5.81 0.581 Bonds 0.3 0.4 -0.1 1.45 -0.145 Difference – Contribution of allocation 0.436 B. Contribution of Selection Portfolio Benchmark Excess Portfolio return Contribution to performance performance performance performance (1) (2) (3) =(1)–(2) (4) (5) = (3)x(4) Equity 7.28 5.81 1.47 0.7 1.029 Bonds 1.89 1.45 0.44 0.3 0.132 Difference – Contribution of Selection 1.161 Of the total difference1.597% in the return of P in excess of the return of B, 0.436% is due to allocation and 1.161% due to selection of the bonds. Dollar-cost averaging (DCA): DCA involves allotting a fixed dollar figure, say $100, to buying a stock (or mutual fund shares) every month. This in effect is some kind of portfolio policy/management. It should be noted that many investors follow this rule (or some modified version of it) without calling it dollar-cost averaging – e.g., savings account in which we deposit a certain amount every month; contribution to 401k (pension) plans; etc. Suppose the investment is in a stock that is volatile. Since a fixed sum is used to buy shares, when price is low we buy a larger number of shares and when price is high we buy a smaller number of shares. The result is therefore that we end up with the portfolio that is composed of a large number of shares bought cheap and a small number of shares bought expensive. Is this strategy profitable? Consider the following possibilities: 1. The stock has an upward trend. Regardless of volatility, since on average the stock price rises, we will make money following the DCA. But if our forecast is that the price will rise on average, we are better buying all the shares we want immediately without waiting. Of course, if we wait until a trough is reached and invest all our money then we can make even more money – but the forecasting requirement in this case is formidable. 2. The stock has a downward trend. If we pursue the DCA policy we will lose. If we can forecast this outcome, we should short the stock right away. 8
  9. 9. 3. So we are left with the situation that the stock is volatile but trendless. Whether one makes or loses money over the planning horizon depends on what path the price takes between now and the end of the horizon. a. If the stock price rises for the early periods and then falls, one ends up losing money. This is because one buys few shares in the beginning when price is high, but starts buying more and more shares as price declines. So one ends up with lots of stocks that have lost value, both bought cheap and bought expensive. The same applies if price was stable early on, then rose (even a little), then dropped. In either case, the critical issue is the price at selling time b. If in the beginning price falls (or is stable), but then rises toward the end, one stands to earn a positive return. Again, to a large degree it is the price that prevails when one wants to liquidate one’s position that determines the outcome. Whether one does or does not make money under the DCA policy depends on the final price. In other words, it is a matter of market timing. Suppose that your plan had been to cash out your portfolio on June 18, 2005. You observe that the price of your shares has hit a new low; if you sell you lose; if not you have violated your plan. But suppose that you decide the heck with the plan; you’d rather not lose (at least not much). So you postpone the time you sell; but that means that you are forecasting a turnaround in the price – this is market timing. If you have that ability then you should have foreseen the price drop long before and sold your portfolio earlier. You may wonder whether there is a better strategy than selling all your stock at once. One could decide to do reverse DCA – sell $100 worth of stocks each month; but this means that when price is high you sell few shares, and you dump a whole bunch when price is low. As it happens too often, many focus on return without looking at risk – so it is with DCA. What DCA can do is to have an impact on risk. If returns are not correlated over time, spreading your investment over time – instead of investing all your funds at once – may lower the risk of your portfolio. 9