04/21/23Strategic Asset Allocation1
Strategic Asset Allocation Strategic Asset Allocation sessionsession 1 1
Andrei Simonov
04/21/23Strategic Asset Allocation2
AgendaAgenda
Introduction, Course Outline, Requirements, Resources
Reminder: SAADefinitionsHistorical Records of Returns on different
securitiesCrisis in Investment Industry
04/21/23Strategic Asset Allocation3
Introduction Introduction The field of Finance and Investments
– Individual agents making decisions to supply capital to the markets
– Firms getting capital from the financial markets (when, where, how?)
– Capital Markets acting as market clearing device. Goal of the course:
– To familiarize you with ”real world” of investments.
– To give broad overview of modern investment issues. By June one should know what does that mean to be investment professional.
04/21/23Strategic Asset Allocation4
Overview Overview of the courseof the course Strategic Asset Allocation Asset Pricing Models Tactical Asset Allocation Volatility & Skewness Information Processing by markets Market Neutral Investments Behavioral Finance
04/21/23Strategic Asset Allocation5
Resources and requirements:Resources and requirements: Courseweb page. I put some stuff on my page, but
links are on courseweb Articles (package+web site)
Provide deeper insight, latest developments No econometrics, just general idea
Wall Street Journal or Financial Times Access to Internet, some Excel experience, basic
knowledge of econometrics It is assumed that basic courses are still remembered
by you. Groups of 2-3 (pls let TA know by the end of the
week)
04/21/23Strategic Asset Allocation6
CasesCases What case report is NOT:
– Not copy of textbook or article.
– Not exercise in history of economics or finance. I do not care (at least, in that class) who got Nobel Prize for what...
Ideal case report is similar to consulting report:– Analysis of data that is in the case (preferrably statistical analysis)
– Covering all relevant issues (pros and cons)
– Take the position and defend it!
– Case report is not War and Peace. Be brief!
– Please understand what you are writing about.
– Cases are due before the discussion session. Do not spend more than 2 days on ANY case! Class discussion is part of the case work.
04/21/23Strategic Asset Allocation7
My assumptions about you:My assumptions about you:You know and understand basic regression
analysis (what is R2, statistical significance, etc.)
You remember conditions of optimality from Econ 101
You remember basics from Finance I You are willing to learn...
04/21/23Strategic Asset Allocation8
AgendaAgenda
Individual’s 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
04/21/23Strategic Asset Allocation9
First Approximation Model of First Approximation Model of Investors’ Behavior: Assumptions:Investors’ Behavior: Assumptions:
Single holding periodInvestors are risk-averseInvestors are ”small”The information about asset payoffs is
common knowledgeAssets are in unlimited supplyAssets are perfectly divisibleNo transaction costWealth W is invested in assets
04/21/23Strategic Asset Allocation10
Investors´ preferencesInvestors´ preferences
Attitude to risk Time horizon (do
not confuse with holding period)
Non-traded risks (liabilities, labor income, human capital)
Constraints
PortfolioAdvisor & Investor Type: Cash Bond Stock Bonds/StockFidelityConservative 50 30 20 1.500Moderate 20 40 40 1.000Aggressive 5 30 65 0.462Merrill LynchConservative 20 35 45 0.778Moderate 5 40 55 0.727Aggressive 5 20 75 0.267New York TimesConservative 20 40 40 1.000Moderate 10 30 60 0.500Aggressive 0 20 80 0.250
04/21/23Strategic Asset Allocation11
Investor’s preferences:Mean-Investor’s preferences:Mean-variance frameworkvariance framework
Representation by utility function of wealth W– u’(W)>0, u’’(W)<0
Taylor Expansion:
Applying Expectations operator:
Simplest utility function is quadratic:u=W-0.5bW2
Problem: satiation Arbitrary preferences: Asset returns are distributed as
multivariate normal A dominates B if E(rA) (>) E(rB) and A <() B
2))~
(~
))(~
((''2
1))
~(
~))(
~(('))
~(()
~( WEWWEuWEWWEuWEuWu
2))~
((''2
1))
~(()
~( WEuWEuWuE
2222
2))
~(()
~(
2)
~()
~(
2)
~()
~( b
WEfWEb
WEWEb
WEWuE
04/21/23Strategic Asset Allocation12
Indifference curvesIndifference 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
Solid line, b=1, dashed line, b=2
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0 1 2 3 4 5 6
Std. Dev.
Exp.
return
04/21/23Strategic Asset Allocation13
Measuring risk by varianceMeasuring 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?
04/21/23Strategic Asset Allocation14
How to live with risk?How to live with risk? Know and classify risks into
asset classes. On what basis? Price risk (Country (incl.
Political risk), Industry,statistical categories)
Credit risk, counterparty risk Tail risk or risk of ruin
Most important classification concept: statistical correlation pitfalls of correlations quasi-arbitrage opportunities
(“convergence trades”): LTCM and limits of arbitrage (Shleifer &Visny)
Large vs. Small Cap: Total Return (01.1954=$1)
$1
$3
$5
$7
$9
$11
$13
$15
$17
1954 1959 1964 1969
Large CapSmall CapLarge CapSmall Cap
HIGH CORRELATIONr=0.99
LOW CORRELATION
r=0.5
04/21/23Strategic Asset Allocation15
The same story:The same story: Nasdaq vs. S&P 500Nasdaq vs. S&P 500
04/21/23Strategic Asset Allocation16
04/21/23Strategic Asset Allocation17
Henry Lowenfeld, 1909Henry 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.
04/21/23Strategic Asset Allocation18
Globalization and Financial Globalization and Financial LinkagesLinkages 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
04/21/23Strategic Asset Allocation19
What is the overall effect?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?
04/21/23Strategic Asset Allocation21
International Diversification 2: International Diversification 2: Time-Varying CorrelationsTime-Varying Correlations
1872-2000 US France GermanyUK 0.265 0.351 0.143US 0.163 0.083France 0.189
Average correlation =0.199Integration, 1872-1914, 1972-2000 US France GermanyUK 0.345 0.467 0.369US 0.301 0.284France 0.520
Average correlation =0.381Segmentation, 1914-1971 US France GermanyUK 0.193 0.311 0.097US 0.101 0.041France 0.135
Average correlation =0.146
1. Correlations between countries are highly time-varying.
2. Result of Solnik can be due to segmentation period used.
3. There is striking similarities between end of XIX and XX centuries.
(Based on Goetzmann et. al. NBER W8612)
04/21/23Strategic Asset Allocation22
The Role of Emerging MarketsThe 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?
04/21/23Strategic Asset Allocation24
Decomposing the Benefits of International Diversificationequally-weighted portfolio variance / average market variance
0.0
0.2
0.4
0.6
0.8
1.0
1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Rat
io p
ortf
olio
vol
atili
ty /
aver
age
mar
ket
vola
tilit
y
Core Countries (limited diversification)
Average four countries
All Countries (unlimited diversification)
04/21/23Strategic Asset Allocation26
Bottom Line: International Diversification Bottom Line: International Diversification Does Not Work as it Used to...Does Not Work as it Used to...
•Trade barriers disappear (NAFTA, EU, ASEAN, etc.)•Globalization of Business Enterprises,•Wave of intra-industry M&A (incl. cross-border M&A)
“…active portfolio managers will have increasing difficulty addingvalue by using a top-down strategy through European countryallocation.” (Freiman, 1998)
New Holy Graal: Industry Diversification
04/21/23Strategic Asset Allocation27
Industry vs. International DiversificationIndustry vs. International DiversificationAPT-style estimation:
Ri=i(t)+ijijNatlMarketIndexj+ ijijGlobalIndustryIndex+ i
where ij ()=1 if firm i belongs to country (industry) j. This can be further simlified as
Ri=i(t)+ijij(t)+ ikik (t) + i
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 i(t)+ij (t) is return on geographically diversified industry portfolio. i(t)+ij(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.
04/21/23Strategic Asset Allocation28
Random diversification: Random diversification: international vs. industrialinternational vs. industrial
Eiling, Gerard, Hillion & de Roon (2009)Eiling, Gerard, Hillion & de Roon (2009)
Adding currency deposits to industry portfolios is a winning recepie.
Our conditional results show that international equity returns are primarily driven by industry and currency risk factors…The dominance of global industry factors over country factors is in line with the seven developed countries in our sample being among the most integrated equity markets in the world. Finally, we find that currency returns significantly improve the mean-variance efficiency of country, industry and world market portfolio returns.
04/21/23Strategic Asset Allocation29
04/21/23Strategic Asset Allocation30
How non-diversifiable risk changes How non-diversifiable risk changes with time (with time (CampbellCampbell et al et al, , 2000)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
04/21/23Strategic Asset Allocation31
Do you really have to go abroad to achieve Do you really have to go abroad to achieve international diversification? (based on international diversification? (based on Diermeier-Solnik 2001)Diermeier-Solnik 2001)
No, It is enough to invest into companies that do business abroad.
Ri=i+iLocInd+ijIndj+ ijCurrencyj+ i
ij is “exposure” to foreign market risk, ij is “exposure” to foreign currency risk.
Adjusted R2
Country
International Market
Exposure
Foreign Currency Exposure
France 0.13 0.06Germany 0.31 0.09Italy 0.40 0.00Japan 0.22 0.24Netherlands 0.49 0.19Switzerland 0.32 0UK 0.22 0.17US 0.02 0.02
Exposure is explained well by % of foreign sales,
ij =i+ijForSalesj
04/21/23Strategic Asset Allocation32
Word of caution:Word of caution:
“Trust companies…have 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)
04/21/23Strategic Asset Allocation33
04/21/23Strategic Asset Allocation34
Non traded riskNon traded riskss 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
04/21/23Strategic Asset Allocation35
Human CapitalHuman 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
04/21/23Strategic Asset Allocation36
Normative multi-period AA: theoryNormative multi-period AA: theory
One risk-free asset (return r) and n risky assets with e=E[R] and var-covar matrix V.
Investor’s consumption-investment problem:
Constant relative risk aversion (CRRA) utility:
1'
1
,1
,
~1
~ ..
,...,, maxmax
tttttt
T
t
t
wCTtt
wC
rCWWts
CUECCCUEtttt
Rw1w
1 ,ln
1,0 ,1)(
1
C
CCU
04/21/23Strategic Asset Allocation37
Optimal dynamic portfolios: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.
,...)(', , 111
x
CC
C
rE
WCW
xC
WCWU
UBA
ννVH1RVM
HMHMw
04/21/23Strategic Asset Allocation38
ConstraintsConstraints 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
04/21/23Strategic Asset Allocation39
Standard Deviation (Risk)
Expected Return
0.0 42.03.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0 36.0 39.02.0
19.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
S&P500
IA Small Stocks
Russell 2000
MSCI Europe
MSCI Pacific TR
IA Corporate
IA 20Yr GvtIA 5YR Gvt
IA 1 Year GvtNon-US LT Gvt
30 Day TBILL
Gold
CRSP MidCap
Real Estate
S&P500 (20.0%)
IA Small Stocks (7.5%)MSCI Europe (19.7%)
MSCI Pacific TR (7.5%)
IA 5YR Gvt (4.5%)
IA 1 Year Gvt (20.0%)
Real Estate (20.0%)
S&P500 (51.8%)
IA Small Stocks (2.6%)
MSCI Europe (11.9%)
MSCI Pacific TR (10.2%)Real Estate (23.5%)
Frontier with constraintsFrontier with constraints
Source:Ibbotson Assoc.Portfolios with =20%No short salesB: 20% max
04/21/23Strategic Asset Allocation40
Time ”Diversification”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
Position 56
S&P500 (37.6%)
IA Small Stocks (23.2%)
MSCI Europe (18.7%)MSCI Pacific TR (12.5%)
Real Estate (8.1%)
Time
Compound Annual ReturnPosition 56
Return Percentiles
Nov2001
Nov2020
Dec2005
Dec2010
Dec2015
-20.0%
60.0%
-10.0%
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
95th Percentile Expected Value 5th Percentile
Time
Wealth (USD)Position 56
Wealth Percentiles
Nov2000
Nov2020
Dec2005
Dec2010
Dec2015
1
50
1
10
95th Percentile Expected Value 5th Percentile
Source: Ibbotson Assoc.R=15%, =20%
04/21/23Strategic Asset Allocation41
Practicality: Estimation RiskPracticality: 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 returnsexpected returns.
04/21/23Asset Pricing Models42
Another example (GS 2003):Another example (GS 2003):
Here are forecasts given by “Wall Street Protagonist” (Columns 1 and 2) and set of portfolio weights that are generated by it.
RETURN STD DEVUnconstrained weights
No Short Sales
Japanese Gov Bonds 4.7% 4.2% -2.02 0.00EU Gov Bonds 5.1% 3.6% -3.21 0.00US Gov Bonds 5.2% 4.6% -4.84 0.00US Equities 5.4% 15.5% -0.11 0.00Global Fixed income 6.0% 3.6% 14.93 0.00EU Equities 6.1% 16.6% -2.58 0.00US Inv Grade Corp Bonds 6.3% 5.4% -3.86 0.00EAFE 8.0% 15.3% 3.14 0.00Hedge Fund Portfolio 8.0% 5.2% 0.58 0.55US High Yield 8.9% 7.3% -0.10 0.36Private Equity 9.0% 28.9% 0.01 0.00Emerging Debt 9.0% 17.6% -0.29 0.00REITs 9.0% 13.0% 0.04 0.08Japanese Equity 9.5% 19.6% -0.72 0.01Emerging Equities 11.8% 23.4% 0.03 0.00
Portfolio ER 4.90% 5.10%Portfolio Volatility 18.20% 8.40%
04/21/23Asset Pricing Models43
Equilibrium and individual Equilibrium and individual asset’ expected returnasset’ expected returnFrom previous section one can expect that
ERP is between 4 and 6% and is fairly stable with time
One can make forecast for individual assets that are different from long term premium.
But by forecasting one asset class, we are implicitly making forecast for other asset classes as well.
04/21/23Asset Pricing Models44
Practicality: How to express Practicality: How to express views?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.
Views=mathematically expressed preferences of individual investors.
Step Action Purpose1 Calculate equilibrium returns Set neutral reference point2 Define weight for news Dampen impact of aggressive news3 Set target tracking error Control risk wrt benchmark4 Set target market exposure control directional effect5 Get portfolio weight Find allocation that maximize performance6 Examine risk distribution Is risk diversifies?
04/21/23Asset Pricing Models45
Equilibrium optimal portfolioEquilibrium 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:
04/21/23Asset Pricing Models46
Updating of discrete Updating of discrete probabilitiesprobabilities
1. We have a probability estimate for event H:prior probability P(H)
2. New information D is gained
3. Update the estimate using Bayes’ theorem:posterior probability P(H|D)
04/21/23Asset Pricing Models47
The Bayes’ theoremThe Bayes’ theorem
The updating is done using the Bayes’ theorem:
( | ) ( )( | )
( )
P D H P HP H D
P D
04/21/23Asset Pricing Models48
Example: Using Bayes’ Example: Using Bayes’ theoremtheorem1,5 % of the population suffer from schizophrenia
P(S) = 0.015 (prior probability)Brain atrophy is found in
– 30 % of the schizophrenic P(A|S) = 0.3
– 2 % of normal people P(A|S) = 0.02If a person has brain atrophy, the probability that he is schizophrenic
(posterior probability) is:
( | ) ( )( | )
( | ) ( ) ( | ) ( )
0.3 0.0150.186
0.3 0.015 0.02 0.985
P A S P SP S A
P A S P S P A S P S
Picture: Clemen s. 250
Figure: Posterior probability with different prior probabilities.
04/21/23Asset Pricing Models49
Updating of continuous Updating of continuous distributionsdistributions
Choose a theoretical distribution, P(X=x|), for the physical process of interest.
Assess uncertainty about parameter : prior distribution, f()
Observe data x1
Update using Bayes’ theorem: posterior distribution of , f(|x1)
Note:
Uncertainty about X has two parts:
1. Due to the process itself, P(X=x|).
2. Uncertainty about , f(), later updated to f(|x1).
04/21/23Asset Pricing Models50
Updating of continuous Updating of continuous distributionsdistributionsBayes’ theorem for continuous :
f(x1|) is called the likelihood function of with a given observed data x1.
In most cases the posterior distribution can not be calculated analytically, but must be solved numerically.
11
1
( | ) ( )( | )
( | ) ( )
f x ff x
f x f d
04/21/23Asset Pricing Models51
Normal distributionNormal distribution1. The physical process of interest is normal distributed:
X ~ N(, 2) ( is assumed to be known)2. Prior distribution for :
~ N(m0, 20)(notation: 2
0 = 2 / n0)3. Observe a sample of the physical process:
– sample size: n1
– sample mean: x1
4. The posterior distribution, calculated using the Bayes’ theorem, gets reduced to: ~ N(m*, 2*), where
2 20 1 1 0 0 0 1 1
2 21 0 0 1
2 2 22 0 1
2 21 0 0 1
*
*
m n x n m n xm
n n n
n
n n n
04/21/23Asset Pricing Models52
Expressing ViewsExpressing Views Thus, one need to specify set of “views” and “precisions” of
views for each asset f(x|). “No views” is equivalent to having x For this case
posterior=prior. Model will deviate further for assets where views are stronger. All assets are affected:
04/21/23Asset Pricing Models53
Further risk control & results:Further risk control & results: Minimize tracking error Mkt. Exposure of the portfolio () (neutral should be 1) Look at diversification (are all eggs in one basket?) Results (according to GS, 95-97): (+103 bp, +83 bp, -26bp)
04/21/23Asset Pricing Models54
Other uses: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.)
04/21/23Asset Pricing Models55
What is expected risk premium?What is expected risk premium?
Expected risk premium= Var(rM)/E[W/]Plays central role in any discussion about
the marketWhat is that? How to measure it? What
will it tell us about mankind and economy (or Asset Pricing Model)?– Historical perspective– Equilibrium perspective
04/21/23Asset Pricing Models56
Asset Classes Returns: US HistoryAsset Classes Returns: US History
0.1
1
10
100
1000
10000
Jan1926
Jan1931
Jan1936
Jan1941
Jan1946
Jan1951
Jan1956
Jan1961
Jan1966
Jan1971
Jan1976
Jan1981
Jan1986
Jan1991
Jan1996
Jan2001
U.S. LT Gvt TR
U.S. 30 Day TBill TR
S&P 500 TR
U.S. Inflation
U.S. Small Stk TR
04/21/23Asset Pricing Models57
Asset Classes Returns: Swedish Asset Classes Returns: Swedish HistoryHistory
0.1
1
10
100
1000
10000
100000
1000000
1900 1920 1940 1960 1980 2000
1 S
EK
inve
sted
in 1
900
DMS Global Sweden Bill TR
DMS Global Sweden Inflation
DMS Global Sweden Equity TR
DMS Global Sweden Bond TR
04/21/23Asset Pricing Models58
Goetzmann&Jorion: International EvidenceGoetzmann&Jorion: International Evidence
04/21/23Asset Pricing Models59
Goetzmann&Jorion: International EvidenceGoetzmann&Jorion: International Evidence
Median Market RR 0.75%GDP-weighted RR 4%
04/21/23Asset Pricing Models60
Estimates from historical dataEstimates from historical data
Ibbotson & Sinquefield (76): Real ERP=5%
Ibbotson & Chen (2000) 4%
Fama & French (2002) longer period 4.4%Jagannathan, McGrattan,Scherbina (2000)
– 1926-70 ERP=7%– 1971-99 ERP=0.7%
04/21/23Asset Pricing Models61
Equilibrium Approach: C-CAPMEquilibrium Approach: C-CAPMIndividuals have preferences over consumption
C described by CRRA u=-C1-
Certainty case: marginal utility of consumption today =discounted marginal utility of consumption tomorrow times teturn of asset ri:
C-t[(1+ri)/(1+r)] C-t+1In case of uncertainty
C-t[(1+ri)/(1+r)] C-t+1ntroducing consumption growth
g=C(t+1)/C(t)-1
04/21/23Asset Pricing Models62
C-CAPM(2)C-CAPM(2)
grrrgVargEr
gVargrgEr
gVargEgrgErEgEr
ggrgrgr
grE
ifif
ii
iii
iii
i
,covE2
)1(
:assets riskless andrisky both for truebe should eq. This2
)1(,covE
:ddisregarde becan termsquadratic then small, ist If2
)1(,covE
:operator nsExpectatio Applying2
)1(111 :expansionTaylor
111
2
2
r
r
r
r
04/21/23Asset Pricing Models63
C-CAPM: Equity premium puzzleC-CAPM: Equity premium puzzle
Mehra&Prescott(85); Mankiv &Zeldes(91):– 1890-1979[1948-88]: Risk premium=0.06 [.08]– Std of consumption growth =0.036 [0.014]– Std of market returns=0.167 [0.14]– Correlation between consumption growth and
market returns = 0.40 [0.45]– 0.06=*0.40*0.167*.036 => =25 [90]
04/21/23Asset Pricing Models64
Equity premium puzzleEquity premium puzzle
Gamble: take 20% paycut if state of the world is ”bad” (prob=1/2) and stay at your current salary in good state, or agree on permanent cut of X%:
0.5*(0.8+1)=x
04/21/23Asset Pricing Models65
Equity premium puzzleEquity premium puzzle
Gamble: take 20% paycut if state of the world is ”bad” (prob=1/2) and stay at your current salary in good state, or agree on permanent cut of X%:
0.5*(0.8+1)=x
If =25 then x=17.7%Realistic estimate for gamma=3
04/21/23Asset Pricing Models66
How can we solve it?How can we solve it?
Habit formation u=-(Ct-Ct-1)1-
– Increases demand for bonds, lower Rf– “Keeping up with the Joneses”: instead C(t-1)
there is AVERAGE consumption in the reference group.
Idiosynchratic labor riskDisaster states and survivorship bias.Liquidity premiumLimited Participation