4106 Advanced Investment Management Strategic

Asset Allocation session 2-3

- Andrei Simonov

Agenda

- Individuals preferences, utility function
- Measurement of risk by variance
- Diversification
- A bit of math
- Industry diversification
- International diversification
- Latest evidence
- Shortcut to math Excel!
- Risk accounting

Assumptions

- Single holding period
- Investors are risk-averse
- Investors are small
- The information about asset payoffs is common

knowledge - Assets are in unlimited supply
- Assets are perfectly divisible
- No transaction cost
- Wealth W is invested in assets

Investors preferences

- Attitude to risk
- Time horizon (do not confuse with holding period)
- Non-traded risks (liabilities, labor income,

human capital) - Constraints

Investors preferencesMean-variance framework

- Representation by utility function of wealth W
- u(W)gt0, u(W)lt0
- Taylor Expansion
- Applying Expectations operator
- Simplest utility function is quadraticuW-0.5bW2
- Problem satiation
- Arbitrary preferences Asset returns are

distributed as multivariate normal - A dominates B if E(rA)? (gt) E(rB) and sA lt(?) sB

Indifference curves

- All portfolios on a given indifference curve are

equally desirable - Any portfolio that is lying on indifference curve

that is further North-west is more desirable

than any portfolio that is lying on indifference

curve that is less Northwest - Different investors (e.g., in risk aversion)

have different indifference curves

Measuring risk by variance

- Variance
- definition probability weighted squared

deviations from the expected value - based on probability distribution
- Any drawbacks of this measure?
- People do not behave that way (read Odean)
- Overconfidence (wrong probability distribution)
- Regret (distinguish gains from losses)
- Should we use semi-variance?
- Particularly in case of delegated portfolio

management?

The measurement of risk Compare frequency

distribution of bond rates of return and rates of

returns of stocks

Source Ibbotson Assoc.

The measurement of risk by variance (example

large-c. stocksfrom frequency table)

Optimal diversification the ingredients

- Excess expected rate of return for each security

i (organized into vector) - Variance of rate of return for each security i
- Covariances of rate of return of security i with

security j (organized into matrix)

Optimal diversification (2)

- What is covariance between x and y? Estimated as
- Why does covariance come in?
- By definition of correlation, covariance is also

correlation between x and y ? standard deviation

of x ? standard deviation of y - Example of calculation from data table stocks

and bonds

Example of calculation from table stocks and

bonds

Math of mean-variance optimization

- Assume you have 1 SEK to invest into stock

(mS,sS) and long-term bond (mB,sB).

Try to do the same with 10 assets

Efficient Frontier

Using Excel to optimize

- Lord gave us Microsoft. Use it! Use Solver.

Can have many securities, add constraints. - Set up row or column of portfolio weights xi
- Variance compute xi ? cov(Ri,Rj) ? xj
- Sum both ways to get portfolio variance
- Expected return xi ? E(Ri)
- Or, if there is riskless asset, xi ? E(Ri) r
- Sum to get portfolio expected return
- Maximize
- portfolio exp. return - 1/2 ? ? portfolio

variance for given ?. ? is risk aversion. - Or maximize portfolio exp. return for given

portfolio variance (or standard deviation), - Or minimize portfolio variance for given

portfolio exp. return , - under constraint that portfolio weights sum to 1

(in the absence of riskless asset) and possibly

other constraints.

Example of spreadsheet

Random diversification Sharpediagram

Portfolio risk approaches the average covariance

between assets when the number of assets gets

large.

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Henry Lowenfeld, 1909

- It is significant to see how entirely all the

rest of the Geographically Distributed stocks

differ in their price movements from the British

stock. It is this individuality of movement on

the part of each security, included in a

well-distributed Investment List, which ensures

the first great essential of successful

investment, namely, Capital Stability. - From Investment and Exact Science, 1909.

History of Diversification

- First Mutual Fund Eendracht Maakt Magt (1774)
- Danish and Viennese banks
- Danish Tolls and Holstein
- Russia and Sweden
- Brunswick and Mecklenburg
- Postal services of Saxony
- Spanish Canals of Taouste and Imperial
- British Colonies
- Essequebo
- Berbice
- Danish American Islands

Diversification 18th Century Mutual Funds

- In the portfolio construction the fund will

observe as much as possible an equal

proportionality - Because nothing is completely certain, but

subject to fluctuations, it is dangerous to

allocate all capital to a single security - Nobody will have reason to believe that all

securities will stop paying off at the same time

thereby losing the entire invested capital

Globalization and Financial Linkages

- Common wisdom is that globalization and

integration of markets accentuates financial

linkages (correlations) - Business cycle synchronization
- Policy coordination
- Coordination of institutions
- Decrease in home bias of investors
- Globalization of firms
- Globalization and integration also allows country

specialization

Globalization and Financial Linkages

- Expansion of investment opportunities
- Lowering of transactions costs
- Trade where costs are lowest
- Competition among exchanges
- Cross-listing / depository receipts / global

shares - Cost of capital / Expected returns
- Change in covariance structure of returns

affecting portfolio risk / benefits of

diversification

What is the overall effect?

- Decrease in expected returns
- Higher correlation between asset markets
- More markets for investment
- Increase in the types of marketed securities
- Potential synchronization of business cycles
- Increased policy coordination
- Net effect?

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International Diversification 2 Time-Varying

Correlations

- Correlations between countries are highly

time-varying. - Result of Solnik can be due to segmentation

period used. - There is striking similarities between end of XIX

and XX centuries. - (Based on Goetzmann et. al. NBER W8612)

Average Correlation US UK Germany France

The Role of Emerging Markets

- Expand the investment opportunity set
- Are imperfectly correlated with existing markets
- What is the relative contribution of changing

correlations and evolution in the investment

opportunity set for diversification benefits?

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Globalization How do Correlations Change?

- Does location of a firm matter?
- Industry membership may become more important
- What happens to residual risk?

Bottom Line International Diversification Does

Not Work as it Used to...

- Trade barriers disappear (NAFTA, EU, ASEAN, etc.)
- Globalization of Business Enterprises,
- Wave of intra-industry MA (incl. cross-border

MA) - active portfolio managers will have increasing

difficulty adding - value by using a top-down strategy through

European country - allocation. (Freiman, 1998)

? New Holy Graal Industry Diversification

Industry vs. International Diversification

APT-style estimation Riai(t)SdijbijNatlMarketIn

dexj Sd(1)ijgijGlobalIndustryIndex ei where

dij (d(1))1 if firm i belongs to country

(industry) j. This can be further simlified

as Riai(t)Sdijbij(t) Sd(1)ikgik (t)

ei 2-stage estimation as in Fama-McBeth

procedure (time-series cross-section) gives us

time-series of prices of national and industry

risk. One can interpret ai(t)bij (t) is return

on geographically diversified industry portfolio.

ai(t)gij(t) is return on industry-diversified

national portfolio. Small Print (a) We miss all

other firm characteristics-size, b/m, dividend

payout ratio, leverage, etc. (b)We also assume

that securities in country i have same exposure

to domestic and foreign factors. (c) We do not

address Ericsson problem. (d) Cavaglia et. al.

(2001) consider 35 industries in 21 countries.

Industry vs. International Diversification(2)

We can use MAD (mean absolute deviation)

statistics (due to Rouwenhorst)

MAD(t)Swi(t-1) bij(t)

Random diversification international vs.

industrial

- HestonRouwenhorst, 1998 industry

diversification is better (at least for Europe) - Cavaglia et al, 2000 It is most of the time, but

sometimes it might be different (especially for

the whole world)

Random diversification international vs.

industrial (2)

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How non-diversifiable risk changes with time

(Campbell et al, 2000)?

- It increases...
- When before you were OK with 10 stocks, now you

have to use 50. - Why?
- Younger companies are on the market
- Internal capital markets are gone
- Competition
- Institutions

What about Sweden?

- Sternbrink-Tengvall thesis Swedish data

1988-2001. - Firm-level volatility remains roughly the same.
- Outside very large firms, market volatility

remains the same. With ERICSSON-likes it actually

increases a bit. - Industry-level volatility increases for all

industries.

European Equity Markets

- Increased industry importance
- Countries become less important
- Why does it still matter?
- Residual risk is increasing cost of not being

diversified is going up

Global Linkages of Other Markets

- Bond markets
- Interest rate correlations have increased in

Europe before EMU - Reduction of Bond market diversification
- Real estate markets
- Non-tradable goods
- But linked through
- business cycle correlations
- Interest rate correlations
- exchange rate correlations

International Financial Linkages- Summary -

- There is reason to believe that international

financial linkages are becoming stronger. - World is not yet a global place
- Expansion of investment opportunity set should

give some compensation for investors who seek

diversification - Number of markets
- Expansion of tradable assets new markets /

securities

Do you really have to go abroad to achieve

international diversification? (based on

Diermeier-Solnik 2001)

No, It is enough to invest into companies that do

business abroad. RiaibiLocIndSgijIndj

SdijCurrencyj ei gij is exposure to foreign

market risk, dij is exposure to foreign

currency risk.

Exposure is explained well by of foreign sales,

gij limijForSalesj

Word of caution

- Trust companieshave reckoned that by a wide

spreading of their investment risk, a stable

revenue position could be maintained, as it was

not to be expected that all the world would go

wrong at the same time. But the unexpected has

happened, and every part of the civilized world

is in trouble - Chairman of Alliance Trust Company (1929)

Optimal portfolio of riskless and risky assets

- What is riskless asset?
- No default risk
- No inflation risk
- No reinvestment risk
- What is expected value and std. dev. of returns

of the portfolio with risky and riskless asset?

Stock 100

Stock 50

ms

Exp. Return

rf

0

ss

Capital Allocation Line

- Meaning of CAL slope Revard to variability
- Combining portfolio of risky assets and rf
- Tangency portfolio (T) is the optimal risky

portfolio to mix with T-bills - Portfolios on (rf,T) positive fractions of risky

portfolio and T-bills - Portfolios on (T,?) go short in T-bills

T

How to choose the right portfolio?

Separation Property

- One can see portfolio selection problem as a

two-step routine - Finding optimal risky portfolio (meat)
- Adding enough risk-free asset to make the dish

eatable.

Non traded risks

- Human capital and death insurance
- Investment in residence
- Other consumption needs saving for retirement

and life insurance - Liabilities B/S optimization
- You must consider that these are part and parcel

of your portfolio, but with immutable weights

Human Capital

- Most of the normal individual wealth is in the

form of HUMAN CAPITAL. - Assume that human capital supply (willingness to

work) is flexible and tradeable. Value of future

cash flow decreases with time. - Share of stocks will go down with time
- The higher is the riskiness of human capital, the

less is the willingness to invest in stock - Strong effect on portfolio decisions.
- Real estate can amplify riskiness of human capital

Normative multi-period AA theory

- One risk-free asset (return r) and n risky assets

with eER and var-covar matrix V. - Investors consumption-investment problem
- Constant relative risk aversion (CRRA) utility

Optimal dynamic portfolios

- M is mean-variance portfolio
- H is hedge portfolio against changes in variable

x. - H does not matter for non-stochastic opportunity

set or log utility function.

Constraints

- Liquidity
- Regulations public or self imposed
- SEC
- Pension funds Employee Retirement Income

Security Act (ERISA) European directives - no more than 5 in any publicly traded company
- Mostly domestic assets
- Mutual funds
- No borrowing.
- Association for Investment Management and

Research (AIMR) - Taxes
- Unique needs internal restrictions

Frontier with constraints

SourceIbbotson Assoc. Portfolios with s20 No

short sales B 20 max

Time Diversification

- Can you reduce risk by holding assets longer?
- Uncertainty in annual rate of return goes down
- BUT!!! Uncertainty of total returns goes up

Source Ibbotson Assoc. R15, s20

Risk accounting(simple Value at Risk)

- Beta is just a re-scaled covariance
- here i refers to return on security i
- p refers to return of portfolio

Risk accounting

- Risk accounting
- share of standard deviation measured by means of

beta of each security with respect to portfolio

return - Interpretation of beta relative to investors

portfolio - If an investment item has a beta equal to 2 and

if 1 of the total portfolio value is invested

there, then that investment accounts for 2 of

the total risk (standard deviation) of the

portfolio. (This the basis of Value at Risk

scheme) - It is not variance or stdev of investment item

that counts - Only systematic risk matters

Optimal diversification condition of optimality

(w/o constraint)

- How can you tell whether a portfolio p is well

diversified or efficient? - For each security i, E(Ri) - r must be lined up

with cov(Ri,Rp) or, equivalently, with - ?i cov(Ri,Rp)/var(Rp)

E(Ri) - r

?i or cov(Ri,Rp)

Optimal diversification condition of optimality

- If that condition is not satisfied, the

composition of portfolio p must be changed - ?i gt 0, increase weight of security i
- ?i lt 0, decrease weight of security i

Example of risk accounting Leesons reported

positions

Example

Exercise There are only two securities

available stocks and gold. Stocks offer an

expected rate of return of 18, with a standard

deviation of 22. Gold offers an expected return

of 10 with a standard deviation of 30. The

riskless rate of interest is at 4. Calculate the

highest level of correlation between the two

risky assets that would cause an investor to hold

a nonnegative amount of gold. Answer Call x

the unknown correlation level. At the limiting

point, the weight of gold in the portfolio will

just be equal to zero. So the optimal portfolio

contains stocks at 100 or almost. In order for

the portfolio so composed to be optimal, we must

have Hence at the most.

Risk and return

- Recall if an investment item has a beta equal to

2 with respect to portfolio and if 1 of the

total portfolio value is invested there, then

that investment accounts for 2 of the total risk

(standard deviation) of the portfolio - In a portfolio that is properly constructed, all

the investment items should plot along a

(positively sloped) line, so that each bit of

risk receives its proportionate reward.

Attention Default is not in the picture!!!

Source Moodys

Practicality Estimation Risk

- Óptimization results are usually suffering from
- Huge short positions in many assets in

no-constraint case. - Corner solutions with zero positions in may

assets if constraints are imposed. - Huge positions in obscure markets with small cap
- Large shifts in positions when exp. returns or

covariances changes just a bit - All of those are coming from one common cause

difficulties in estimation of expected returns.

Practicality How to express views?

- Method is due to Black Litterman (Goldman

Sachs). The core themes equilibrium returns and

views. - Investors normally have views/preferences. They

are NOT incorporated into optimization process. - Viewsmathematically expressed preferences of

individual investors.

Equilibrium optimal portfolio

- Imagine that the investor thin that US is still

in recession. Thus, stocks will perform badly,

and bonds will perform OK. - Mathematically, it is equivalent to assuming

that bonds will go up 0.8, and stocks will drop

2.5 - Result see Table 8

Reminder from math Bayesian Updating.

- Prob(data)Prob(dataevent)Prob(event)

Prob(data no event)Prob(no event) - Prob(data,event)Prob(eventdata)Prob(data)

Prob(dataevent)Prob(event) - Posterior probability
- Prob(eventdata) Prob(dataevent)Prob(event)/

Prob(data) Prob(dataevent)Prob(event)/

(Prob(dataevent)Prob(event) Prob(data no

event)Prob(no event)) - If distributions are normal, and prior density

is g(m) ? N(m,sm2), and likelihood f(xm) ?

N(m,sx2) then posterior - G(mx) g(m) f(xm)/(?g(m) f(xm)d m)?N(M,S2)
- Posterior Variance S21/(sm-2 sx-2)
- Posterior Mean M(m/ sm2 x/ sx2) /(sm-2

sx-2)

Likelihood

Expressing Views

- Thus, one need to specify set of views and

precisions of views for each asset f(xm). - No views is equivalent to having sx?? For this

case posteriorprior. - Model will deviate further for assets where

views are stronger. - All assets are affected

Further risk control results

- Minimize tracking error
- Mkt. Exposure of the portfolio (b) (neutral

should be 1) - Look at diversification (are all eggs in one

basket?) - Results (according to GS, 95-97) (103 bp, 83

bp, -26bp)

Other uses

- Black-Litterman model is essentially Tactical

Asset Allocation model (provided that algorithm

of selecting views is specified). - But it can be used effectively in updating priors

on the distribution of the signals. - It can be used to bring in new asset classes for

which the recorded history is short or unreliable

(venture capital funds, hedge funds, emerging

markets, etc.)

Conclusion

- SAA is first-order approximation when you

determine the structure of investment portfolio. - Diversification over different asset classes,

industries, countries should be considered. - It is based on sound statistical ideas, but

practical implementation may be plagued by

instability of underlying economic processes and

difficulties in estimation expected returns. - SAA does not utilize effectively wealth of

economic information. Tactical asset allocation

make an attempt to fix that.