4106 Advanced Investment Management Strategic Asset Allocation session 2-3

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4106 Advanced Investment Management Strategic
Asset Allocation session 2-3

• Andrei Simonov

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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

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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

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Investors preferences

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

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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

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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

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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?

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The measurement of risk Compare frequency
distribution of bond rates of return and rates of
returns of stocks
Source Ibbotson Assoc.
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The measurement of risk by variance (example
large-c. stocksfrom frequency table)
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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)

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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

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Example of calculation from table stocks and
bonds
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Math of mean-variance optimization

• Assume you have 1 SEK to invest into stock
  (mS,sS) and long-term bond (mB,sB).

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Try to do the same with 10 assets
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Efficient Frontier
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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.

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Example of spreadsheet
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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.

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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

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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

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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

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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

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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)

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Average Correlation US UK Germany France
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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?

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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
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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.
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Industry vs. International Diversification(2)
We can use MAD (mean absolute deviation)
statistics (due to Rouwenhorst)
MAD(t)Swi(t-1) bij(t)

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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)

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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

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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.

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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

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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

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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

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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
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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)

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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
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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
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How to choose the right portfolio?
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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.

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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

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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

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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

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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.

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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

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Frontier with constraints
SourceIbbotson Assoc. Portfolios with s20 No
short sales B 20 max
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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
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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

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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

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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)
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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

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Example of risk accounting Leesons reported
positions
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Example
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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.
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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.

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Attention Default is not in the picture!!!
Source Moodys
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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.

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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.

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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

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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
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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

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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)

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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.)

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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.