Overview of Portfolio

8/14/2019 Overview of Portfolio

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RISK PREFERENCES

The trade off between Risk and Return

Most, if not all, investors are risk averse

To get them to take more risk, you have to offer higher expected returns

Conversely, if investors want higher expected returns, they have to be willingto take more risk.

Ways of evaluating risk

Most investors do not know have a quantitative measure of how much risk thatthey want to take

Traditional risk and return models tend to measure risk in terms of volatility orstandard deviation

The Mean Variance View of Risk

In the mean-variance world, variance is the only measure of risk. Investorsgiven a choice between tow investments with the same expected returns butdifferent variances, will always pick the one with the lower variance.


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Estimating Mean and Variance

In theoretical models, the expected returns and variances are in terms offuture returns.

In practice, the expected returns and variances are calculated usinghistorical data and are used as proxies for future returns.

Illustration 1: Calculation of expected returns/standard deviation using

historical returns

GE The Home Depot

Year Price at Dividends Returns Price at Dividends Returns

end of year during year end of year during year

1989 $ 32.25 $ 8.13

1990 $ 28.66 $ 0.95 -8.19% $ 12.88 $ 0.04 58.82%

1991 $ 38.25 $ 1.00 36.95% $ 33.66 $ 0.05 161.79%

1992 $ 42.75 $ 1.00 14.38% $ 50.63 $ 0.07 50.60%

1993 $ 52.42 $ 1.00 24.96% $ 39.50 $ 0.11 -21.75%

1994 $ 51.00 $ 1.00 -0.80% $ 46.00 $ 0.15 16.84%

Average 13.46% 53.26%

Standard Deviation 18.42% 68.50%

Concept Check:

While The Home Depot exhibited higher variance in returns, much ofthe variance seems to come from the stock price going up dramatically

between 1989 and 1992? Why is this upside considered risk? Should risk not be defined purely in terms of "downside" potential

(negative returns)?

Variance of a Two-asset Portfolio


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portfolio = wA A + (1 - wA) B

2portfolio = wA2 2A + (1 - wA)

2 2B + 2 wA wB A B

where

wA = Proportion of the portfolio in asset A

The last term in the variance formulation is sometimes written in terms of the covariancein returns between the two assets, which is

AB = A B

The savings that accrue from diversification are a function of the correlationcoefficient.

Illustration 2: Extending the two-asset case - GE and The Home Depot

Step 1: Use historical data to estimate average returns and standard deviations in

returns for the two investments.

Stock Average Return (1990-94)Standard Deviation (1990-

94)

General Electric 13.46% 18.42%

The Home Depot 53.26% 68.50%

Step 2: Estimate the correlation and covariance in returns between the two investments

using historical data.

YearReturns on

GE(RGe)

Returns on HD

(RH)

(RGE-

Avge(RGE))2

(RH-

Avge(RH))2

(RGE- Avge(RGE)

(RH-Avge(RH)

1990 -8.19% 58.82% 0.04686 0.00309 (0.01203)

1991 36.95% 161.79% 0.05518 1.17786 0.25494

1992 14.38% 50.60% 0.00008 0.00071 (0.00024)

1993 24.96% -21.75% 0.01322 0.56265 (0.08625)

1994 -0.80% 16.84% 0.02034 0.13265 0.05194

Total 0.13568 1.87696 0.20835

Variance in GE Returns = 0.13568/4 = 0.0339 Standard Deviation in GE Returns =0.03390.5 = 0.1842

Variance in HD Returns = 1.87696/4 = 0.4692 Standard Deviation in HD Returns =0.46920.5 = 0.6850


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Covariance between GE and The Home Depot Returns = 0.20835/4 = 0.0521

Correlation between GE and The Home Depot Returns = GH = GH / G H =0.0521/(0.1842*0.6850) = 0.4129

Step 3: Compute the expected returns and variances of portfolios of the two securities

using the statistical parameters estimates above

Consider, for instance, a portfolio composed of 50% in GE and 50% in The Home Depot

Average Return of Portfolio = 0.5 (13.46%) + 0.5 (53.26%) =

Variance of Portfolio = (0.5)2 (18.42%)2 + (0.5)2 (68.50%)2 + 2 (0.5) (0.5) (0.4129)

(18.42%)(68.50%) = 1518%

Standard Deviation of Portfolio = 38.96%


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From Two Assets To Three Assets to n Assets

The variance of a portfolio of three assets can be written as a function of thevariances of each of the three assets, the portfolio weights on each and thecorrelationsbetween pairs of the assets. The variance can be written as follows

-

p2= wA

2 2A + wB2 2B + wC

2 2C+ 2 wA wB AB A B+ 2 wA wC AC A C+ 2 wB wC BC B C

where

wA,wB,wC = Portfolio weights on assets

2A ,2B ,

2C = Variances of assets A, B, and C

AB , AC , BC = Correlation in returns between pairs of assets (A&B, A&C,B&C)

The Data Requirements

Number of covariance terms = n (n-1) /2

where n is the number of assets in the portfolio

Number of Covariance Terms as a function of the number of assets in

portfolio

Number ofAssets

Number ofcovariance terms

2 1

5 10

10 45

20 190

100 4950

1000 499500

Some Closing Thoughts on Risk

Most Investors do not measure their risk preferences in terms of standarddeviation


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For other investors, risk has to be assessed by using

Scoring Systems (where investors are asked for information or questionsto answers which can be used to analyze how much risk an investor iswilling to take)

Risk categories (High; Average; Low) Life cycle theories of investing

A Life Cycle View of Risk

General Propositions:

As investors age, there will be a general increase in risk aversion, leading togreater allocation to safer asset classes.

PORTFOLIO VALUE

The value of a portfolio constrains yourchoices at later stages. This is because trading individual securities creates costs - brokerage

costs, bid-ask spreads and price impact

There is a critical mass value, below which it does not pay to activelymanage a portfolio - it is far better to invest in funds.

The larger a portfolio, the more choices become available in terms ofassets - this is largely because some components of trading costs - the

brokerate costs and the spread - may get smaller for larger portfolios. If a portfolio becomes too large, it might start creating a price impact

which might cause trading costs to start increasing again.


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Taxes do matter: Individuals should care about after-tax returns

Stock and Bond Returns: 1926-1989 - Before and After Taxes

Stocks Bonds

Market Returns $ 534.46 $ 17.30

After Transactions Cost $ 354.98 $ 11.47

After Income Taxes $ 161.55 $ 4.91

After Capital Gains Taxes $ 113.40 $ 4.87

After Inflation $ 16.10 $ 0.69

Transactions Costs: 0.5% a year; Income taxes: at 28%; Capital Gains at 28%every 20 years;

The Effect of Turnover on After-tax Returns

CASH NEEDS & TIME FRAME

- What is a long time horizon?

- Determinants of time horizon

* Age

* Level of Income


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* Stability of Income

* Cash Requirements

- Time Horizon and Asset Choice

Proposition: The cost of keeping funds in near-cash investments increases with

the time horizon of the investor.

THE IMPORTANCE OF ASSET ALLOCATION

The first step in all portfolio management is the asset allocation decision. The asset allocation decision determines what proportions of the

portfolio will be invested in different asset classes. Asset allocation can be passive,

o It can be based upon the mean-variance framework


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o It can be based upon simpler rules of diversification or marketvalue based

When asset allocation is determined by market views, it is active assetallocation.

Passive Asset Allocation: The Mean Variance View of Asset AllocationEfficient Portfolios

Return Maximization Risk Minimization

Objective Function

Maximize Expected Return Minimize return variance

Constraint

where,

2

= Investor's desired level of variance

E(R) = Investor's desired expected returns

Markowitz Portfolios

The portfolios that emerge from this process are called Markowitz portfolios.These portfolios are considered efficient, because they maximize expectedreturns given the standard deviation, and the entire set of portfolios is referredto as the Efficient Frontier. Graphically, these portfolios are shown on theexpected return/standard deviation dimensions in figure 5.7 -

Figure 5.7: Markowitz Portfolios

Application to Asset Allocation

If we have information on the expected returns and variances of differentasset classes and the covariances between asset classes, we can devise

efficient portfolios given any given level of risk.


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For example, if the following is the information of 4 asset classes:

Asset Class MeanStandard

deviation

U.S. stocks 12.50% 16.50%

U.S. bonds 6.50% 5.00%

Foreign Stocks 15.00% 26.00%

Real Estate 11.00% 12.50%

Correlation Matrix for Asset Classes

U.S. Stock U.S. BondsForeign

StocksReal Estate

U.S. Stocks 1.00 0.65 0.35 0.25

U.S. Bonds 1.00 0.15 -0.10

Foreign Stocks 1.00 0.05

Real Estate 1.00

The More General Lesson: Diversification Pays

Passive Asset Allocation: Market Value Based Allocation


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Active Asset Allocation: The Market Timers

The objective is to create a portfolio to take advantage of 'forecasted' marketmovements, up or down. Strategies could include:

* Shifting from (to) overvalued asset classes to (from) undervalued assetclasses if you expect the market to go up (down).

* Buying calls (puts) or buying (selling) futures on a market if you expect the

market to go up (down).

Assumption: You can forecast market movements

Advantage: If you can forecast market movements, the rewards are immense.

Disadvantage: If you err, the costs can be significant.

Does Active Asset Allocation Work?

Tactical asset allocation funds do not do well ..


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Fairly unsophisticated strategies beat these funds..


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SECURITY SELECTION

Once the asset allocation decision has been made, the portfolio managerhas pick the securities that go into the portfolio.

Again, the decision can be made on a passive basis or on active basis. Active security selection can take several forms:

o it can be based upon fundamentalso it can be based upon technical indicatorso it can be based upon information

Passive Security Selection: The Index Fund


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Index funds are created by holding stocks in a wider index in proportion to theirmarket value. No attempt is made to trade on a frequent basis to catch marketupswings or downswings or select 'good' stocks.

Assumptions: Markets are efficient. Attempts to time the market and pick good

stocks are expensive and do not provide reasonable returns. Holding a welldiversified portfolio eliminates unsystematic risk.

Advantages: Transactions costs are minimal as is the cost of searching forinformation.

B. Markowitz Portfolio: A Markowitz efficient portfolio is created bysearching through all possible combinations of the universe of securities to findthat combination that maximizes expected return for any given level of risk.

Assumptions: The portfolio manager can identify the inputs (mean, variance,covariance) to the model correctly and has enough computer capacity to runthrough the optimization exercise.

Advantages: If historical data is used, the process is inexpensive and easilymechanised.

Disadvantages: The model is only as good as its inputs.

II. ACTIVE STRATEGIES

The objective is to use the skills of your security analysts to select stocks thatwill outperform the market, and create a portfolio of these stocks. The securityselection skills can take on several forms.

(1) Technical Analysis, where charts reveal the direction of future pricemovements

(2)Fundamental Analysis , where public information is used to pickundervalued stocks

(3)Access to private information, which enables the analyst to pinpoint mis-valued securities.

Assumption: Your stock selection skills help you make choices which, onaverage, beat the market.


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Inputs: The model will vary with the security selection model used.

Advantage: If there are systematic errors in market valuation andyour modelcan spot these errors, the portfolio will outperform others in the market.

Disadvantage: If your security selection does not pay off, you have expendedtime and resources to earn what another investor could have made with randomselection.Security Selection strategies vary widely and can lead tocontradictory recommendations..

Technical investors can beo momentum investors, who buy on strength and sell on weaknesso reversal investors, who do the exact opposite

Fundamental investors can beo value investors, who buy low PE orlow PBV stocks which trade

at less than the value of assets in placeo growth investors, who buy high PE and high PBV stocks which

trade at less than the value of future growth Information traders can believe

o that markets learn slowly and buy on good news and sell on badnews

o that markets overreact and do the exact opposite

They cannot all be right in the same period and no one approach can be right inall periods.

A Caveat.. There are not very many great stock pickers either...


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III. Trading and Execution

The cost of trading includes the brokerage cost, the bid-ask spread andthe price impact


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The Trade offs on Trading

There are two components to trading and execution - the cost ofexecution (trading) and the speed of execution.

Generally speaking, the tradeoff is between faster execution and lowercosts.

For some active strategies (especially those based on information) speedis of the essence.

Maximize: Speed of Execution

Subject to: Cost of execution < Excess returns from strategy

For other active strategies (such as long term value investing) the costmight be of the essence.

Minimize: Cost of Execution

Subject to: Speed of execution < Specified time period.

The larger the fund, the more significant this trading cost/speed tradeoffbecomes.

MEASURING PERFORMANCE

* Who should measure performance?


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> 0: Outperformed

< 0: Underperformed

where,

Actual Return = Returns on the portfolio (including dividends)

Expected Return = Riskfree rate at the start of the period + Beta of portfolio *(Actual return on market during the period - Riskfree Rate)

This abnormal return is calledJensen's Alpha. It can also be computed byregressing the returns on the portfolio against a market index, and thencomparing the intercept to Rf (1- Beta).

Variants: Define Rp to be the return on the portfolio and Rm to be the return onthe market.

Treynor Index = (Rp - Rf)/ Beta of the portfolio

> (Rm - Rf) : Outperformed

< (Rm - Rf) : Underperformed

Sharpe Index = (Rp - Rf)/ Variance of the portfolio

> (Rm - Rf)/ m : Outperformed

< (Rm - Rf)/ m : Underperformed

Information Ratio = Jensens alpha / Unsystematic Risk

> 0: Outperformed the market

< 0 : Underperformed

Tracking Error as a Measure of Risk

Tracking error measures the difference between a portfolios return andits benchmark index. Thus portfolios that deliver higher returns than the

benchmark but have higher tracking error are considered riskier. Tracking error is a way of ensuring that a portfolio stays within the same

risk level as the benchmark index.


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It is also a way in which the active in active money management can beconstrained.

Performance Attribtion

This analysis can be carried one step forward, and the overall performance of amoney manager can be decomposed into market timing and securityselection components.

If money managers are good market timers,o they should hold high beta stocks, when the the return on the

market > risk free rateo they should hold low beta stocks, when the return on the market

Overview of Portfolio

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