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4106 Advanced Investment Management Strategic Asset Allocation session 2-3

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Title: 4106 Advanced Investment Management Strategic Asset Allocation session 2-3


1
4106 Advanced Investment Management Strategic
Asset Allocation session 2-3
  • Andrei Simonov

2
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

3
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

4
Investors preferences
  • Attitude to risk
  • Time horizon (do not confuse with holding period)
  • Non-traded risks (liabilities, labor income,
    human capital)
  • Constraints

5
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

6
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

7
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?

8
The measurement of risk Compare frequency
distribution of bond rates of return and rates of
returns of stocks
Source Ibbotson Assoc.
9
The measurement of risk by variance (example
large-c. stocks from frequency table)
10
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)

11
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

12
Example of calculation from table stocks and
bonds
13
Math of mean-variance optimization
  • Assume you have 1 SEK to invest into stock
    (mS,sS) and long-term bond (mB,sB).

14
Try to do the same with 10 assets
15
Efficient Frontier
16
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.

17
Example of spreadsheet
18
Random diversification Sharpe diagram
Portfolio risk approaches the average covariance
between assets when the number of assets gets
large.
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21
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.

22
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

23
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

24
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

25
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

26
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?

27
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28
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)

29
Average Correlation US UK Germany France
30
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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35
Globalization How do Correlations Change?
  • Does location of a firm matter?
  • Industry membership may become more important
  • What happens to residual risk?

36
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
37
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.
38
Industry vs. International Diversification(2)
We can use MAD (mean absolute deviation)
statistics (due to Rouwenhorst)
MAD(t)Swi(t-1) bij(t)

39
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)

40
Random diversification international vs.
industrial (2)
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45
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

46
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.

47
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

48
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

49
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

50
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
51
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)

52
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
53
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
54
How to choose the right portfolio?
55
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.

56
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

57
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

58
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

59
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.

60
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

61
Frontier with constraints
SourceIbbotson Assoc. Portfolios with s20 No
short sales B 20 max
62
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
63
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

64
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

65
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)
66
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

67
Example of risk accounting Leesons reported
positions
68
Example
69
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.
70
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.

71
Attention Default is not in the picture!!!
Source Moodys
72
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.

73
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.

74
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

75
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
76
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

77
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)

78
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.)

79
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.
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