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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
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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
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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).
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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.
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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.
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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.
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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.
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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.
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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 )
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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.
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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.
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