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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Dynamic Asset AllocationChapter 1: Introduction
Claus Munk
August 2012
AARHUS UNIVERSITY AU
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Outline
1 General considerations
2 Recommendations on asset allocation
3 About alternative approaches
4 How do individuals invest?
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Approaches to portfolio choice
• Stock (asset) picking• (Stock) market timing• Let “investment wizard” invest for you• Historical “simulation”• Stable allocation between asset classes• Mean-variance analysis• Diversification
I Within asset classesI Between asset classes
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
This course
• Consumption and portfolio strategies of individuals over thelife-cycle
• Intertemporal, utility maximizing decisions (not mean/variance)• Asset classes:
I Cash, stocks (growth/value), bonds, stock options, human capital(labor income), real estate
• Partial equilibrium models• Mathematically challenging problems, but simple solutions in
some interesting cases• Main question: How should individuals invest over their life cycle?
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Limitations
The models considered generally ignore• Finer details of return processes• Transaction costs and taxes• Portfolio constraints• Model and parameter uncertainty• Various asset classes (corporate bonds, mortgages and
mortgage-backed bonds, foreign assets, commodities)• Mandatory savings via labor market pension funds• Uncertain lifetime; insurance decisions• Multiple consumption goods
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Limitations, cont’d
• The models are used to develop intuition and obtain qualitativeand quantitative insights that are likely to carry over to the realworld
• A full, realistic, and solvable model does not exist• “All models are wrong, but some are useful”
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Recommendations on asset allocation
• Some general asset allocation recommendations freely availableon the internet
• Online “Asset Allocation Calculators”, e.g.:http://www.money-zine.com/Calculators/Investment-Calculators/Asset-Allocation-Calculator/
• Recommendations are often depending on “risk appetite” and“investment horizon”
• Most banks sell individual advice – “Interior decorator fallacy”• Many advisors seem to rely heavily on past return patterns• Common advice:
I More stocks when young and/or “aggressive”I Higher stock/bond ratio when young and/or “aggressive”
• Sophisticated advice wrt. taxes, unsophisticated advice wrt.risk/return and life-cycle aspects
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
An empirical investigation
Barber, Lehavy, McNichols, and Trueman: “Can investors profit fromthe prophets?” Journal of Finance, 2001
• More than 360,000 recommendations from 269 brokeragehouses and 4,340 analysts, US 1985-1996
• Categories 1 (best buy) – 5 (sell), see next page• Divide stocks into pf’s according to average mark:
[1,1.5], (1.5,2], (2,2.5], (2.5,3], (3,5]
• Rebalance daily and compute monthly value-weighted returns
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
Table II from Barber et al.
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
Table V from Barber et al.
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
Conclusions
• w/o transaction costs:I long pf 1, short pf 5 0.989% per monthI return decreases with infrequent or delayed rebalancing
• w/ transaction costs:I strategies require lots of tradingI risk-adjusted net returns are not reliably greater than zero
• if you want/have to buy or sell, it pays off to followrecommendations
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
Have a good weekend, comics and favorite stock websites « Topstockblog
http://topstockblog.wordpress.com/2008/10/18/have-a-good-weekend-comics-and-favorite-stock-websites/ 21-08-2009 11:49:53
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
http://www.stnicholaspam.com/sitebuilderImages/bank%20comic.jpg
http://www.stnicholaspam.com/sitebuilderImages/bank%20comic.jpg 21-08-2009 12:03:46
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
Easy to find a good portfolio manager?• αi : risk-adjusted (abnormal) return in period i• Suppose αi ∼ N (µ, σ2), iid.• Given observations α1, . . . , αN , estimates of µ and σ2 are
µ = α ≡ 1N
N∑i=1
αi , σ2 =1
N − 1
N∑i=1
(αi − α)2.
• 95% sure that µ > 0 if and only if
α
σ/√
N≥ 1.96 ⇔ N ≥
(1.96σα
)2
.
• Example:I α = 0.002 (0.2% per month ∼ 2.4% per year)I σ = 0.02 (idiosyncratic risk of 2% per month ∼
√12 · 2% ≈ 7% per
year)I then we need N ≥ 384 months, i.e., 32 years of observations!
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
Possible to predict returns?
• Mechanical prediction, “technical analysis”I ... don’t believe that!I Use scientific methods, i.e., statistical analysis and understanding
of underlying economic forces
• Stock index returns seem slightly predictable, but predictions arevery imprecise (see next page)
• Expected stock returns seem to vary somewhat over time (≈mean reversion) – included in some models
• “In the long run, stocks will outperform bonds”I seems to be the case in data, but we have very few independent
observations of “the long run”!I given mean reversion, the variance of stock returns increases more
slowly with the horizon than without mean reversionI Japan – counter example?
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
History of the Nikkei 225 stock index
All-time high Dec 29, 1989 at 38,957.44On Mar 10, 2009 down at 7,054.98 (81.9% below peak).
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
Regressing returns on price/dividend ratio
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Can investors profit from the prophets?Portfolio performance evaluationPredicting returns
Predicted v. realized returns
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
How do individuals allocate their wealth?
Main source: Campbell (2006, Journal of Finance)• Wealth distribution, participation rates, and portfolio shares for
US households 2001 (see figures)• Stock market participation depends on
I age, wealth (Wachter and Yogo, 2010, Review of Financial Studies– see tables),
I education (Christiansen, Joensen, and Rangvid, 2008, Review ofFinance – see graph),
I sex (same authors, wp),I country: Sweden 60%, US 50%, UK 40%, DK 23%,
Italy/Germany/France less than 20% (rough and old estimates)I DK 1998 (outside retirement accounts):
males 24.9%, females 21.3%, economists 42%
• Stock market participation puzzle?
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Wealth distribution1562 The Journal of Finance
Figure 1. The U.S. wealth distribution. The cross-sectional distribution of total assets, finan-
cial assets, and net worth in the 2001 Survey of Consumer Finances.
How many households participate in these markets at all? Given that theyhave decided to participate, what fraction of their assets do they allocate toeach category? How does household behavior vary with age, wealth, and otherhousehold characteristics? Because these questions can be answered withouthaving detailed information on individual asset holdings, the data problemsdescribed in Section I.A are not as serious in this context.
Following Bertaut and Starr-McCluer (2002), Haliassos and Bertaut (1995),and Tracy, Schneider, and Chan (1999), I now summarize the information in the2001 SCF that relates to these questions. Figure 1 presents the cross-sectionalwealth distribution. The horizontal axis in this figure shows the percentilesof the distribution of total assets, defined broadly to include both financialassets and nonfinancial assets (durable goods, real estate, and private businessequity, but not defined benefit pension plans or human capital). The verticalaxis reports dollars on a log scale. The three lines in the figure show the averagelevels of total assets, financial assets, and net worth (total assets less debts,including mortgages, home equity loans, credit card debt, and other debt) ateach percentile of the total assets distribution. It is clear from the figure thatmany households have negligible financial assets. Even the median householdhas financial assets of only $35,000, net worth of $86,000, and total assets of$135,000.
The figure also shows the extreme skewness of the wealth distribution.Wealthy households at the right of the figure have an overwhelming influenceon aggregate statistics. To the extent that these households behave differentlyfrom households in the middle of the wealth distribution, the aggregates tell
Source: Campbell, Journal of Finance, 2006
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General considerationsRecommendations on asset allocation
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Participation ratesHousehold Finance 1563
Figure 2. Participation rates by asset class. The cross-sectional distribution of asset class
participation rates in the 2001 Survey of Consumer Finances.
us very little about the financial decision making of a typical household: Again,while the behavior of wealthy households is disproportionately important forasset pricing models, household finance is more concerned with the behaviorand welfare of typical households.
A. Wealth Effects
Figure 2 illustrates the participation decisions of households with differentlevels of wealth. The horizontal axis is the same as in Figure 1, but the verticalaxis now shows the fraction of households that participate in particular assetclasses. In this figure I aggregate the SCF asset data into several broad cate-gories, namely, safe assets, vehicles, real estate, public equity, private businessassets, and bonds.13
Given the negligible financial assets held by households at the left of the fig-ure, it should not be surprising that these households often fail to participatein risky financial markets. Standard financial theory predicts that householdsshould take at least some amount of any gamble with a positive expected return,but this result ignores fixed costs of participation, which can easily overwhelmthe gain from participation at low levels of wealth. Figure 2 shows that most
13 Safe assets include checking, saving, money market, and call accounts, CDs, and U.S. savings
bonds. Public equity includes stocks and mutual funds held in taxable or retirement accounts
or trusts. Bonds include government bonds other than U.S. savings bonds, municipal, corporate,
foreign, and mortgage-backed bonds, cash-value life insurance, and amounts in mutual funds,
retirement accounts, trusts, and other managed assets that are not invested in stock.
Source: Campbell, Journal of Finance, 2006
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Asset sharesHousehold Finance 1565
Figure 3. Asset class shares in household portfolios. The share of each asset class in the
aggregate portfolio of households at each point in the wealth distribution, in the 2001 Survey of
Consumer Finances.
phenomenon and shows that similar patterns are obtained in several Euro-pean countries.15
B. Demographic Effects
Wealth is not the only household characteristic that may predict its will-ingness to take financial risk. Income, age, race, education, and self-reportedattitudes to risk may also be important.
Before one can understand the relative importance of these effects, one mustconfront a fundamental identification problem (Heckman and Robb (1985),Ameriks and Zeldes (2004)). At any time t a person born in year b is at yearsold, where at = t − b. Thus, it is inherently impossible to separately identify ageeffects, time effects, and cohort (birth year) effects on portfolio choice. Even ifone has complete panel data on portfolios of households over time, any patternin the data can be fit equally well by age and time effects, age and cohort effects,or time and cohort effects.
Theory suggests that there should be time effects on portfolio choice if house-holds perceive changes over time in the risks or expected excess returns of riskyassets. Theory also suggests that there should be age effects on portfolio choiceif older investors have shorter horizons than younger investors and investment
15 King and Leape (1998) capture the same phenomenon by estimating wealth elasticities of
demand for different asset classes. They find that risky assets tend to be luxury goods with high
wealth elasticities.
Source: Campbell, Journal of Finance, 2006
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Portfolios by age and wealthTable 4: Portfolio Share by Age and Wealth
We sort the sample of households in the 2001 SCF into age (columns) groups, then quartilesof financial wealth (rows) within each age group. Panel A reports the median (within eachbin) of financial wealth. Panel B reports the percentage of households that own stocks withineach bin. Panel C reports the median (within each bin) of the portfolio share in stocks.
Percentile of Financial Assets Age26–35 36–45 46–55 56–65 66–75
Panel A: Financial Assets (Thousands of 2001 Dollars)0–25 0.0 0.4 0.5 0.4 0.625–50 1.9 9.0 19.3 18.6 17.350–75 9.4 44.7 80.8 106.8 100.675–100 60.8 205.8 385.0 558.9 525.1Top 5 262.0 658.4 1343.3 2655.0 2293.0All Households 4.2 22.1 40.0 47.1 42.5
Panel B: Percentage of Households with Stocks0–25 3 9 9 5 125–50 36 52 53 46 1550–75 71 83 78 83 5375–100 86 93 96 95 88Top 5 91 97 98 99 96All Households 49 60 59 57 39
Panel C: Stocks as Percentage of Financial Assets0–25 0 0 0 0 025–50 0 6 8 0 050–75 31 47 36 38 175–100 45 59 59 54 60Top 5 60 71 63 67 59All Households 7 29 25 23 0
44
Source: Wachter and Yogo, working paper, 2007 (published 2010 in Review of Financial Studies)30 / 33
General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Portfolios by age and wealth for stockholders onlyTable 5: Portfolio Share by Age and Wealth for Stockholders
We sort the sub-sample of stockholding households in the 2001 SCF into age (columns)groups, then quartiles of financial wealth (rows) within each age group. Panel A reports themedian (within each bin) of financial assets. Panel B reports the median (within each bin)of the portfolio share in stocks.
Percentile of Financial Assets Age26–35 36–45 46–55 56–65 66–75
Panel A: Financial Assets (Thousands of 2001 Dollars)0–25 2.3 9.1 18.2 28.0 51.425–50 10.0 39.6 68.0 105.3 190.550–75 32.6 94.0 185.0 269.9 409.675–100 119.2 310.4 609.0 989.2 1079.0Top 5 459.3 850.0 2009.1 3836.3 4005.0All Households 18.0 62.8 105.3 161.4 269.5
Panel B: Stocks as Percentage of Financial Assets0–25 45 43 39 45 3925–50 54 49 47 48 4750–75 48 56 49 53 6475–100 60 68 67 54 62Top 5 67 69 69 55 58All Households 49 53 49 50 58
45
Source: Wachter and Yogo, working paper, 2007 (published 2010 in Review of Financial Studies)
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Stock market participation and educationARE ECONOMISTS MORE LIKELY TO HOLD STOCKS? 473
0
5
10
15
20
25
30
35
40
45
10 111 2 3 4 5 6 7 8 9
Figure 1. Stock market participation rates across educational groups.The figure shows the proportion (in percentage) of investors who hold stocks acrosseducational groups, 1997–2001. Subject 1: Education. Subject 2: Humanities/arts.Subject 3: Agriculture/food/forestry/ fishing. Subject 4: Business/commercial (excludingeconomists). Subject 5: Social sciences (excluding economists). Subject 6: Healthcare. Subject 7: Natural sciences/technical educations. Subject 8: Police/armedforces/transportation. Subject 9: High school. Subject 10: Basic school/preparatory school.Subject 11: Economics.
2.2 STOCK MARKET PARTICIPATION RATES
An investor is defined as participating in the stock market if the investorholds stocks with a value in excess of DKK 1,000 (around USD 141) at yearend.7 This is how we obtain the stock market participation indicators for eachindividual for each year.8
Figure 1 shows that the average rate of participation varies greatly acrossthe educational groups and in particular, at around 42% is much higher foreconomists compared to 25% or less for the other educational groups.
Figure 2 shows the time series of stock market participation rates for theentire sample and for economists. The overall participation rate is remarkablystable at around 23% with the rate for economists increasing in the sampleperiod from a low of 37% to a high of 47%.
7 Investors are defined as participating in the stock market if they have stocks in excess of asmall threshold value. This excludes individuals who, for example have been given a single stockby their employer as a Christmas present. Previous studies have applied a zero threshold value.Our conclusions are robust to the exact choice of threshold value.8 We stress that our stock market participation variable reflects an active decision by theinvestor to buy stocks or mutual funds. We do not consider a mandatory contribution to apublic pension scheme as an active stock market participation decision, as, in Denmark, theinvestor has no say over such contributions during the period under investigation.
Source: Christiansen, Joensen, and Rangvid, Review of Finance, 2008 (DK data)
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General considerationsRecommendations on asset allocation
About alternative approachesHow do individuals invest?
Diversification and rebalancingDiversification:• Many households own relatively few individual stocks
I the median number of stocks held in 2001 was threeI however many households own equity indirectly, through mutual
funds or retirement accounts• Households have a local bias wrt. domestic vs. foreign
investments and wrt. regional vs. non-regional companiesI regional bias is stronger among investors who do not own
international stocks
• Many households have large holdings in the stock of theiremployer
Rebalancing:• Males trade more frequently than females• Discount brokerage customers trade intensively• Inertia in the asset allocation of retirement savings plans.
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Dynamic Asset AllocationChapter 2: Preferences
Claus Munk
August 2012
AARHUS UNIVERSITY AU
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Outline1 Atemporal preferences
Setting and notationPreference relationsUtility indicesExpected utility and utility functions
2 Risk aversionDefinitionMeasuring risk aversion
3 Utility functionsCRRA utilityHARA utilityReasonable utility functions
4 Intertemporal preferencesGeneral resultsTime-additive expected utilityAlternatives
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Setting and notationPreference relationsUtility indicesExpected utility and utility functions
Setting and notation
• one-period, finite-state model with consumption only at time 1• consumption plan ∼ random variable c probability distributionπ on Z :
πc(z) = P(c = z) =∑
ω:cω=z
pω
• assume state-independent preferences individuals comparedifferent probability distributions
• Z : possible consumption levels• P(Z ): set of probability distributions on Z
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Setting and notationPreference relationsUtility indicesExpected utility and utility functions
ExampleState-contingent consumption plans
state ω 1 2 3state prob. pω 0.2 0.3 0.5cons. plan 1, c(1) 3 2 4cons. plan 2, c(2) 3 1 5cons. plan 3, c(3) 4 4 1cons. plan 4, c(4) 1 1 4
Corresponding probability distributionscons. level z 1 2 3 4 5cons. plan 1, πc(1) 0 0.3 0.2 0.5 0cons. plan 2, πc(2) 0.3 0 0.2 0 0.5cons. plan 3, πc(3) 0.5 0 0 0.5 0cons. plan 4, πc(4) 0.5 0 0 0.5 0
Plans 3 and 4 are equivalent (and dominated by plans 1 and 2?)6 / 38
Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Setting and notationPreference relationsUtility indicesExpected utility and utility functions
Alternative representations
Three increasingly tractable representations of preferences:• Preference relation � on P(Z )
• Utility index, U : P(Z )→ R• Utility function, u : Z → R
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Setting and notationPreference relationsUtility indicesExpected utility and utility functions
Preference relation
Preference relation � (“preferred to”) on P(Z ) satisfying• π1 � π2, π2 � π3 ⇒ π1 � π3
• for all π1, π2 either π1 � π2 or π2 � π1 (or both)
Notation:
π1 ∼ π2 means π1 � π2 and π2 � π1 [indifference]π1 � π2 means π1 � π2 and π1 6∼ π2 [strictly preferred to]
Only pairwise comparisons—hard to work with!!!
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Setting and notationPreference relationsUtility indicesExpected utility and utility functions
Utility indexUtility index U : P(Z )→ R such that
π1 � π2 ⇔ U(π1) ≥ U(π2)
Lemma 2.1 + Theorem 2.1If a preference relation � satisfies the Monotonicity Axiom and theArchimedean Axiom, then it can be represented by a utility index.
Monotonicity Axiom: Suppose π1 � π2 and a,b ∈ [0,1]. Then
a > b ⇔ aπ1 + (1− a)π2 � bπ1 + (1− b)π2
Archimedean Axiom: Suppose π1 � π2 � π3. Then a,b ∈ (0,1) existso that
aπ1 + (1− a)π3 � π2 � bπ1 + (1− b)π3
Difficult to assign values to all possible probability distributions...11 / 38
Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Setting and notationPreference relationsUtility indicesExpected utility and utility functions
Equivalent utility indices
TheoremIf U is a utility index and f : R → R is any strictly increasing function,then the composite function V = f ◦ U , defined by V(π) = f (U(π)), isalso a utility index for the same preference relation.
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Setting and notationPreference relationsUtility indicesExpected utility and utility functions
Expected utility and utility functionsUtility function u : Z → R such that
π1 � π2 ⇔∑z∈Z
π1(z)u(z) ≥∑z∈Z
π2(z)u(z),
i.e. consumption plan c1 is preferred to c2 ⇔ E[u(c1)] ≥ E[u(c2)].
Lemma 2.2 + Theorem 2.2If a preference relation � satisfies the Substitution Axiom and theArchimedean Axiom, then it can be represented by a utility function.
Substitution Axiom: For all π1, π2, π3 and all a ∈ (0,1]:
π1 � π2 ⇔ aπ1 + (1− a)π3 � aπ2 + (1− a)π3
π1 ∼ π2 ⇔ aπ1 + (1− a)π3 ∼ aπ2 + (1− a)π3
... stronger than the Monotonicity Axiom; not uncontroversial
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
Setting and notationPreference relationsUtility indicesExpected utility and utility functions
Equivalent utility functions
Theorem 2.3If u is a utility function for � and a,b are constant with a > 0, then vdefined by
v(z) = au(z) + b
is also a utility function for �.
Proof: Exercise 2.1.
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
DefinitionMeasuring risk aversion
Risk aversion
Assume Z ⊂ R and preferences can be represented by a smoothutility function u : Z → R
Risk-averse:• for all c ∈ Z and all mean-zero gambles ε, the constant
consumption c is preferred to the risky consumption c + ε
• implies a concave utility function, u′′ < 0
Assume throughout that u′ > 0 (greedy) and u′′ < 0 (risk-averse).
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
DefinitionMeasuring risk aversion
Measures of risk aversionCertainty equivalent to cons. plan c: fixed c∗ ∈ Z s.t. u(c∗) = E[u(c)]
Risk premium associated with consumption plan c: λ(c) = E[c]− c∗
Arrow-Pratt measures (invariant to positive affine transformations):
ARA(c) = −u′′(c)
u′(c)≈ −
u′(c+∆c)−u′(c)∆c
u′(c)= − [∆u′(c)]/u′(c)
∆c,
RRA(c) = −c ARA(c) = −cu′′(c)
u′(c)≈ −c
u′(c+∆c)−u′(c)∆c
u′(c)= − [∆u′(c)]/u′(c)
[∆c]/c
Risk premium for “additive fair gamble” c = c + ε:λ(c, ε) ≈ 1
2 Var[ε] ARA(c)∗ expect decreasing ARA
Relative risk premium for “multiplicative fair gamble” c = c(1 + ε):λ(c,cε)
c ≈ 12 Var[ε] RRA(c).
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
CRRA utilityHARA utilityReasonable utility functions
CRRA utility
u(c) =1
1− γc1−γ , c ≥ 0; γ > 0, γ 6= 1
Properties:• ARA(c) = γ/c (decreasing!)• RRA(c) = γ (constant!)• u′(c) = c−γ →∞ as c → 0⇒ always choose strictly positive
consumption
Equivalent utility function:u(c) = c1−γ−1
1−γ → ln c for γ → 1 (log utility)
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
CRRA utilityHARA utilityReasonable utility functions
HARA utility
ARA(c) = −u′′(c)
u′(c)=
1αc + β
Subclasses:• Negative exponential (CARA): u(c) = −e−ac , c ∈ R∗ implies ARA(c) = a (Constant! Unrealistic!)
• Satiation HARA: u(c) = −(c − c)1−γ , c < c∗ implies ARA(c) (Increasing in c! Unrealistic!)∗ includes quadratic utility (γ = −1)
• Subsistence HARA:
u(c) =
{1
1−γ (c − c)1−γ , c > c; γ > 0, γ 6= 1ln(c − c), c > c; γ = 1
∗ implies decreasing ARA and decreasing RRA (Not unrealistic!)∗ with c = 0 we are back to CRRA.
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
CRRA utilityHARA utilityReasonable utility functions
Reasonable utility and level of risk aversion?Empirical studies: CRRA (level 1-5?) or subsistence HARAThought experiment:
fixed c ∈ Z vs.
{(1− α)c with prob. 1/2(1 + α)c with prob. 1/2
Certainty equivalent c∗ for CRRA utility u(c) = c1−γ/(1− γ):
11− γ
(c∗)1−γ =12
11− γ
((1− α)c)1−γ +12
11− γ
((1 + α)c)1−γ ⇒
c∗ =
(12
)1/(1−γ) [(1− α)1−γ + (1 + α)1−γ]1/(1−γ)
c ⇒
Relative risk premium:
λ(c, α)
c=
c − c∗
c= 1−
(12
)1/(1−γ) [(1− α)1−γ + (1 + α)1−γ]1/(1−γ)
.
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
CRRA utilityHARA utilityReasonable utility functions
Reasonable utility and level of risk aversion?
Relative risk premia for a fair gamble of the fraction α of your consumption.
γ = RRA α = 1% α = 10% α = 50%0.5 0.00% 0.25% 6.70%1 0.01% 0.50% 13.40%2 0.01% 1.00% 25.00%5 0.02% 2.43% 40.72%10 0.05% 4.42% 46.00%20 0.10% 6.76% 48.14%50 0.24% 8.72% 49.29%
100 0.43% 9.37% 49.65%
Relative risk aversion outside the interval [1,10] seems extreme!
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Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
CRRA utilityHARA utilityReasonable utility functions
Two-good utility functionsEmbedded in a CRRA utility function:
u(c,q) =1
1− γf (c,q)1−γ .
CES utility:
f (c,q) =(
acψ−1ψ + bq
ψ−1ψ
) ψψ−1
.
Cobb-Douglas utility – if b = 1− a, the limit as ψ → 1:
f (c,q) = caq1−a.
Addi-log utility:
f (c,q) =
(acα + b
α
βqβ) 1α
(for β = α this is CES with α = (ψ − 1)/ψ)29 / 38
Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
General resultsTime-additive expected utilityAlternatives
Preferences for multi-date consumption plans
General results also hold for multi-date consumption plans.
Given Monotonicity and Archimedean Axioms: Utility index exists• one-period: U(c0, c1)
• discrete-time: U(c0, c1, . . . , cT )
• continuous-time: U((ct )t∈[0,T ]
)Given Archimedean and Substitution Axioms: “Multi-date utilityfunction” exists
• one-period: E[U(c0, c1)]
• discrete-time: E[U(c0, c1, . . . , cT )]
• continuous-time: E[U((ct )t∈[0,T ]
)]
32 / 38
Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
General resultsTime-additive expected utilityAlternatives
Time-additive expected utility
Often time-additivity is imposed for tractability, e.g.:
E[U(c0, c1, . . . , cT )] = E
[T∑
t=0
e−δtu(ct )
],
where u is a “single-date” utility function and δ is the subjective timepreference rate.
With multiple consumption goods (e.g. perishable and durablegoods):
E
[T∑
t=0
e−δtu(c1t , c2t )
], e.g. with u(c1, c2) =
11− γ
(cψ1 c1−ψ
2
)1−γ.
34 / 38
Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
General resultsTime-additive expected utilityAlternatives
AlternativesHabit formation utility:
E
[T∑
t=0
e−δtu(ct ,ht )
], ht = h0e−βt + α
t−1∑s=1
e−β(t−s)cs
Example:
u(c,h) =1
1− γ(c − h)1−γ ⇒ RRA = γ
cc − h
State-dependent expected utility, e.g. “keeping up with the Jones’es”:
E
[T∑
t=0
e−δtu(ct ,Xt )
], Xt ∼ consumption of other individuals
36 / 38
Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
General resultsTime-additive expected utilityAlternatives
Recursive or Epstein-Zin utilityCaptured by a recursive utility index Ut = U(ct , ct+1, ct+2, . . . ).
Fairly tractable version (Kreps-Porteus/Epstein-Zin):
Ut = f (ct ,CEt [Ut+1]) ,
where f is the CES function
f (c,q) =(
acψ−1ψ + bq
ψ−1ψ
) ψψ−1
and CEt [Ut+1] is the time t certainty equivalent of Ut+1, which wecompute from the standard CRRA utility function:
CEt [Ut+1] =(Et[(Ut+1)1−γ])1/(1−γ)
.
WLOG we can let a = 1 and think of b = e−δ as a subjective discountfactor.
37 / 38
Atemporal preferencesRisk aversion
Utility functionsIntertemporal preferences
General resultsTime-additive expected utilityAlternatives
Recursive or Epstein-Zin utility, cont’dIn sum:
Ut =
[c(1−γ)/θ
t + e−δ(
Et
[U1−γ
t+1
])1/θ]θ/(1−γ)
, θ ≡ 1− γ1− 1
ψ
.
where• γ: risk aversion parameter (greater than 1)• ψ: intertemporal elasticity of substitution (close to 1?)• δ: subjective time preference rate (small, positive)
Note:• for γ = 1/ψ: time-additive CRRA utility• for γ > 1/ψ: early resolution of uncertainty is preferred
for γ < 1/ψ: late resolution of uncertainty is preferredfor γ = 1/ψ: no preference for the timing of resolution ofuncertainty
• very fashionable, but relatively difficult to work with38 / 38
Discrete timeContinuous time
Dynamic Asset AllocationChapters 4-5: Dynamic Modeling
Claus Munk
August 2012
AARHUS UNIVERSITY AU
1 / 22
Discrete timeContinuous time
Outline
1 Discrete time
2 Continuous time
2 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Time line
t0 ≡ 0 t1 t2 tN−1 tN ≡ T
∆t ∆t ∆t
• Born at time 0, dies at time T (for sure!)• Can revise consumption and investment decisions at time points
tn = n∆t• At the terminal date T = tN = N∆t , no decisions are made• Decision time points: T = {t0, t1, . . . , tN−1}
5 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Consumption
• single good• consumption rate chosen at t : ct ct ·∆t number of goods consumed over [t , t + ∆t)
• “paid” at time t• ct must be non-negative• ct must be measurable wrt. time t information
6 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Investments
• Asset 0 (locally) risk-freeI return rt ·∆t over [t , t + ∆t)
• Assets 1, . . . ,d riskyI prices P t = (P1t , . . . ,Pdt )
>
I assume no dividendsI returns over [t , t + ∆t): R t+∆t = (R1,t+∆t , . . . ,Rd,t+∆t )
>, whereRi,t+∆t = (Pi,t+∆t − Pit )/Pit
• Portfolio...I M0t ,M1t , . . . ,Mdt : units held over [t , t + ∆t)I θ0t , θ1t , . . . , θdt : amounts invested over [t , t + ∆t)I π0t , π1t , . . . , πdt : fractions of wealth (pf. weights) invested over
[t , t + ∆t)
Time t portfolio must be measurable wrt. time t information
7 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Income
• income rate yt income of yt ·∆t number of goods over [t , t + ∆t)
• “received” at time t• assumed exogenously given
8 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Financial wealth and budget constraint• Financial wealth arriving at time t ∈ T : Wt =
∑di=0 Mi,t−∆tPit
(before income and consumption at time t)• Budget constraint: Wt + (yt − ct )∆t =
∑di=0 MitPit
Wt+∆t −Wt = θ0t rt ∆t + θ>t Rt+∆t + (yt − ct ) ∆t
= [θ0t rt + θ>t µt + yt − ct ] ∆t + θ>
t σ tεt+∆t√
∆t ,
where returns are decomposed as
Rt+∆t = µt ∆t + σ t εt+∆t√
∆t
with εt+∆t having mean 0 and variance I so that
Et [Rt+∆t ] = µt ∆t , Vart [Rt+∆t ] = σ t σ>t ∆t .
Assume σ t non-singular no redundant assets• Can replace θ’s with π’s:
πit =θit
Wt + (yt − ct )∆t.
9 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Objective
Assume time-additive expected utility.
At time t = tk :
Jt = sup(ctn ,πtn )N−1
n=k
Et
[N−1∑n=i
e−δ(tn−t)u(ctn )∆t + e−δ(T−t)u(WT )
].
Note: JT = u(WT ) representing bequest or “terminal consumption”
J: indirect utility or value function
10 / 22
Discrete timeContinuous time
Model set-upDynamic programming
The Bellman equationIndirect utility satisfies the Bellman equation
Jt = supct ,πt
{u(ct )∆t + e−δ∆t Et [Jt+∆t ]
}.
• Backward recursive procedure, starting with JT = u(WT )
• Maximization at each t for each possible “state”• Only tractable if “state” is captured by low-dimensional Markov
process, say, x = (x t ), and wealth Wt , so that
J(Wt ,x t , t) = supct ,πt
{u(ct )∆t + e−δ∆t Et [J(Wt+∆t ,x t+∆t , t + ∆t)]
}I can show “envelope condition” u′(ct ) = JW (Wt , x t , t)
• Some discrete-time problems do have explicit solutions, butcontinuous-time models are more elegant and powerful
• Bellman-equation useful for “numerical dynamic programming”
12 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Assets• Asset 0 locally risk-free
I price P0t = e∫ t
0 rs ds
I return rt (annualized; continuously compounded)• Assets 1, . . . ,d risky
I prices P t = (P1t , . . . ,Pdt )>
I assume no dividendsI returns in discrete-time model
Pi,t+∆t − Pit
Pit= µit ∆t + σ>
it εt+∆t√
∆t
I returns in continuous-time model (without jumps)
dPit
Pit= µit dt + σ>
it dz t , i.e.
dP t = diag(P t )[µt dt + σ t dz t
],
where z = (z t ) is a standard Brownian motion of dimension d(WLOG).
15 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Wealth dynamics• From discrete time:
Wt+∆t −Wt = [θ0t rt + θ>t µt + yt − ct ] ∆t + θ>
t σ tεt+∆t√
∆t .
• Continuous-time equivalent:
dWt = [θ0t rt + θ>t µt + yt − ct ] dt + θ>
t σ t dz t .
Since now Wt =∑d
i=0 θit = θ0t + θ>t 1:
dWt =[rtWt + θ>
t (µt − rt1)︸ ︷︷ ︸σ t λt
+yt − ct]
dt + θ>t σ t dz t ,
where λt is a market price of risk.Since now πt = θt/Wt :
dWt = Wt[rt + π>
t σ tλt]
dt + [yt − ct ] dt + Wtπ>t σ t dz t .
16 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Objective
Indirect utility:
Jt = sup(c,π)∈At
Et
[∫ T
te−δ(s−t)u(cs) ds + e−δ(T−t)u(WT )
].
Maximizing over processes (c,π) satisfying• ct ≥ 0 and WT ≥ 0 (cannot die in debt)• integrability constraints• Wt ≥ −K (rules out doubling strategies)
17 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Further assumptions
Suppose relevant info is captured by• Wt : (financial) wealth• xt : (one-dimensional) diffusion process,
so that yt = y(xt , t), rt = r(xt ), µt = µ(xt , t), σ t = σ (xt , t) λt = λ(xt ).
And suppose
dxt = m(xt ) dt + v(xt )> dz t + v(xt ) dzt ,
where z = (zt ) is a (one-dimensional) standard Brownian motion,independent of z.
19 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Deriving the HJB equation
The Bellman-eq. in a discrete-time approximation:
J(W , x , t) = supct≥0,πt∈Rd
{u(ct )∆t + e−δ∆t EW ,x,t [J(Wt+∆t , xt+∆t , t + ∆t)]
}Now
1 multiply by eδ∆t ,2 subtract J(W , x , t),3 divide by ∆t , and4 let ∆t → 0 the HJB equation
δJ(W , x , t) = supct≥0,πt∈Rd
{u(ct ) + drift of J(Wt , xt , t) } .
20 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Deriving the HJB equation, cont’d
The drift of J(Wt , xt , t) is determined by Ito’s Lemma
δJ = supc≥0,π∈Rd
{u(c) +
∂J∂t
+ JW
(W[r(x) + π>σ (x , t)λ(x)
]+ y(x)− c
)+
12
JWW W 2π>σ (x , t)σ (x , t)>π + Jx m(x)
+12
Jxx (v(x)>v(x) + v(x)2) + JWx Wπ>σ (x , t)v(x)}.
Split up the maximization:
δJ = LcJ + LπJ +∂J∂t
+ (y(x) + r(x)W ) JW + Jx m(x) +12
Jxx (v(x)>v(x) + v(x)2),
where
LcJ = supc≥0{u(c)− cJW } ,
LπJ = supπ∈Rd
{WJWπ>σ (x , t)λ(x) +
12
JWW W 2π>σ (x , t)σ (x , t)>π + JWx Wπ>σ (x , t)v(x)}.
21 / 22
Discrete timeContinuous time
Model set-upDynamic programming
Verification theorem
You can solve the dynamic optimization problem as follows1 solve max-problem embedded in HJB-eq. c∗,π∗ in terms of J and its derivatives
2 substitute c∗,π∗ into HJB-eq., ignore the sup, and solve forJ(W , x , t); a non-linear PDE c∗,π∗ in terms of W , x , t
3 check conditions are satisfied (ignored in this course!)
Note:• can be extended to multi-dim. x , but more complicated equations• looks very complicated, but nice and fairly simple solutions in
some interesting cases• hard to handle portfolio constraints
22 / 22
General analysisCRRA utility
Other utility functions
Dynamic Asset AllocationChapter 6: Constant investment opportunities
Claus Munk
August 2012
AARHUS UNIVERSITY AU
1 / 32
General analysisCRRA utility
Other utility functions
Outline
1 General analysis
2 CRRA utility
3 Other utility functions
2 / 32
General analysisCRRA utility
Other utility functions
AssumptionsSolution
In this chapter assume
• constant investment opportunities, i.e.,I r constantI σ constantI µ constant
λ = σ−1(µ− r1) constant• no income, yt = 0
I require Wt ≥ 0 at all times t ∈ [0,T ]I then portfolio weights are well-defined
• As in Merton (1969)
5 / 32
General analysisCRRA utility
Other utility functions
AssumptionsSolution
The HJB equation to solveGiven the assumptions, the wealth dynamics is
dWt =(Wt[r + π>
t σλ]− ct
)dt + Wtπ
>t σ dz t ,
and the indirect utility function is
J(W , t) = sup(cs,πs)s∈[t,T ]
EW ,t
[∫ T
te−δ(s−t)u(cs) ds + e−δ(T−t)u(WT )
].
The associated HJB equation:
δJ(W , t) = LcJ(W , t) + LπJ(W , t) +∂J∂t
(W , t) + rWJW (W , t)
with
LcJ(W , t) = supc≥0{u(c)− cJW (W , t)} ,
LπJ(W , t) = supπ∈Rd
{WJW (W , t)π>σλ +
12
JWW (W , t)W 2π>σσ>π
}.
The terminal condition is J(W ,T ) = u(W ).7 / 32
General analysisCRRA utility
Other utility functions
AssumptionsSolution
Optimizing consumption...
LcJ(W , t) = supc≥0{u(c)− cJW (W , t)}
Assuming non-negativity constraint is not binding, FOC gives
u′(c) = JW (W , t) ⇔ c = Iu (JW (W , t))
LcJ(W , t) = u (Iu(JW (W , t)))− Iu(JW (W , t))JW (W , t).
8 / 32
General analysisCRRA utility
Other utility functions
AssumptionsSolution
Optimizing portfolio...
LπJ(W , t) = supπ∈Rd
{WJW (W , t)π>σλ +
12
JWW (W , t)W 2π>σσ>π
}
FOC gives
JW (W , t)Wσλ + JWW (W , t)W 2σσ>π = 0 ⇔
π = − JW (W , t)WJWW (W , t)
(σ>)−1λ
LπJ(W , t) = −12‖λ‖2 JW (W , t)2
JWW (W , t),
where ‖λ‖2 = λ>λ.
9 / 32
General analysisCRRA utility
Other utility functions
AssumptionsSolution
Returning to HJB equation...
If the PDE
δJ(W , t) = u(Iu(JW (W , t))
)− JW (W , t)Iu(JW (W , t)) +
∂J∂t
(W , t)
+ rWJW (W , t)− 12‖λ‖2 JW (W , t)2
JWW (W , t)
with terminal condition J(W ,T ) = u(W ) has a solution such that thestrategy (c,π) defined above is feasible (satisfies the technicalconditions), then we have found the optimal consumption andinvestment strategy and J(W , t) is indeed the indirect utility function.
10 / 32
General analysisCRRA utility
Other utility functions
AssumptionsSolution
Properties of optimal investment
• Two-fund separation (Theorem 6.1):I fraction − JW (W ,t)
WJWW (W ,t) 1>(σ>)−1λ of wealth in tangency pfI remaining wealth in risk-free asset
• similar to mean-variance analysis, but based on more soundassumptions
• for any two risky assets i and j , the ratio πi/πj should be thesame for all investors
I if bonds are considered risky assets, this conflicts with typicaladvice concerning the stock/bond-ratio.
11 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
CRRA preferencesThe indirect utility function is now defined as
J(W , t) = sup(cs,πs)s∈[t,T ]
EW ,t
[ ∫ T
te−δ(s−t) ε1
c1−γs
1− γ︸ ︷︷ ︸u(cs)
ds + e−δ(T−t) ε2W 1−γ
T
1− γ︸ ︷︷ ︸u(WT )
].
Interesting special cases:• ε1 > 0, ε2 = 0: WLOG ε1 = 1• ε1 = 0, ε2 > 0: WLOG ε2 = 1 and δ = 0• ε1 > 0, ε2 > 0: only ε2/ε1 matters (WLOG ε1 = 1)
In all cases
ct = ε1/γ1 JW (W , t)−1/γ , LcJ = ε
1/γ1
γ
1− γ J1−1/γW .
The HJB equation is thus
δJ(W , t) = ε1/γ1
γ
1− γ JW (W , t)1− 1γ +
∂J∂t
(W , t)+rWJW (W , t)−12‖λ‖2 JW (W , t)2
JWW (W , t).
with terminal condition J(W ,T ) = ε2W 1−γ/(1− γ).14 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Solving HJBQualified conjecture:
J(W , t) =g(t)γW 1−γ
1− γ
Based on idea:(c∗,π∗) optimal with wealth W ⇒ (kc∗,π∗) optimal with wealth kW
Conjecture is correct if g solves
g′(t) = Ag(t)− ε1/γ1 , g(T ) = ε
1/γ2 ,
where A = δ+r(γ−1)γ + 1
2γ−1γ2 ‖λ‖2.
Assume A 6= 0 (A > 0 is natural). Then solution is
g(t) =1A
(ε
1/γ1 +
[ε
1/γ2 A− ε1/γ
1
]e−A(T−t)
)> 0.
15 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
The solution
Theorem 6.2 (Merton 1969)
Assume A 6= 0, constant r , µ, and σ, and no income. Then the indirectutility function for CRRA utility is
J(W , t) =g(t)γW 1−γ
1− γ,
g(t) =1A
(ε
1/γ1 +
[ε
1/γ2 A− ε1/γ
1
]e−A(T−t)
).
The optimal investment and consumption strategy is given by
Π(W , t) =1γ
(σ>)−1λ =1γ
(σσ>)−1(µ− r1),
C(W , t) = ε1/γ1
Wg(t)
= A(
1 +[(ε2/ε1)1/γA− 1
]e−A(T−t)
)−1W .
16 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Wealth dynamics with optimal strategies
dW ∗t = W ∗
t
[(r +
1γ‖λ‖2 − ε1/γ
1 g(t)−1)
dt +1γλ> dz t
].
• geometric Brownian motion (although with a time-dependentdrift)
• future values of wealth are lognormally distributed• wealth stays positive• Expected wealth
E[W ∗t ] = W ∗
0 exp{
1γ
(r − δ +
γ + 12γ‖λ‖2
)t} 1 +
[(ε2/ε1)1/γA− 1
]e−A[T−t]
1 + [(ε2/ε1)1/γA− 1] e−AT
I if ε2 = 0, then E[W ∗t ] decreases in t for t → T
I generally, life-cycle pattern in wealth depends on all preferenceparameters and the market parameters r and ‖λ‖
18 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Optimal consumption
c∗t = ε1/γ1
W ∗t
g(t).
leads to
dc∗t = c∗t
[1γ
(r − δ +
γ + 12γ‖λ‖2
)dt +
1γλ> dz t
].
Expected future consumption is
E[c∗t ] = W0A
1 +[(ε2/ε1)1/γA− 1
]e−AT
exp{
1γ
(r − δ +
γ + 12γ‖λ‖2
)t}.
Since r − δ + γ+12γ ‖λ‖
2 is positive for realistic parameters consumption expected to increase throughout life
19 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Optimal consumption vs. observed consumptionModel prediction contrasts with empirical studies: hump-shapedconsumption pattern over the life-cycle
Source: Gourinchas and Parker, Econometrica 2002.
Possible explanations: mortality risk, labor income, constraints,...20 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Properties of optimal portfolio
π∗t =1γ
(σ>)−1λ =1γ
(σσ>)−1(µ− r1)
• intuitive dependence on parameters• constant weights continuous trading• ‘sell winners, buy losers’
(everybody cannot do that simultaneously – must be others withdifferent preferences)
• an asset’s return history is unimportant• independent of investment horizon
I contrasts with popular recommendationsI because model does not exhibit the “long-run dominance of stocks”
that is seen in the data (though not recently in Japan)?
22 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Outperformance probabilitiesFor a single stock (index):
Prob(
PT
P0> erT
)= N
((µ− r − σ2/2)
√T
σ
)
40%
50%
60%
70%
80%
90%
100%
0 5 10 15 20 25 30 35 40
investment horizon, years
outp
erf
orm
ance p
robabili
ty
6%
9%
12%
15%
Note: r = 4%, σ = 20%, different µ’s23 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Underperformance probabilitiesFor a single stock (index):
Prob(
PT
P0< erT − K
)= N
(ln(erT − K
)−(µ− 1
2σ2)
T
σ√
T
).
Excess return on bond 1 year 10 years 40 years0% 44.0% 31.8% 17.1%
25% 6.4% 22.2% 16.1%50% 0.0% 13.1% 15.1%75% 0.0% 5.7% 14.0%100% 0.0% 1.3% 13.0%
The table shows the probability that a stock investment over a period of 1, 10, and 40 years providesa percentage return which is at least 0, 25, 50, 75, or 100 percentage points lower than the risk-freereturn. The numbers are computed using the parameter values µ = 9%, r = 4%, and σ = 20%.
24 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Welfare loss (in more general setting)• Utility for given strategy (c,π) from time t on is
V c,π(Wt , xt , t) = Et
[∫ T
te−δ(s−t)u(cs) ds + e−δ(T−t)u(W c,π
T )
].
• Obviously V c,π(Wt , xt , t) ≤ J(Wt , xt , t) ≡ V c∗,π∗(Wt , xt , t).• Wealth-equivalent percentage loss `t defined implicitly by
V c,π(Wt , xt , t) = J(Wt [1− `t ], xt , t).
`t : percentage of time t wealth that the individual will sacrifice tobe able to apply the optimal strategy (c∗,π∗) instead of thestrategy (c,π) from time t on.
• An equivalent measure is ˜t defined by
V c,π(Wt [1 + ˜t ], xt , t) = J(Wt , xt , t).
26 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Loss from missing the right π• Assume Merton model with δ = 0, ε1 = 0, and ε2 = 1. Assume a
single risky asset.• For any fixed portfolio weight π in the risky asset, the wealth
dynamics is
dWπt = Wπ
t [(r + πσλ) dt + πσ dzt ] ,
• It can be shown (see Exercise 6.3) that
Vπ(W , t) ≡ Et
[1
1− γ(Wπ
T )1−γ]
=1
1− γ(gπ(t))γ W 1−γ ,
where
gπ(t) = exp{−γ − 1
γ
(r + πσλ− γ
2π2σ2
)(T − t)
}.
• Percentage wealth loss `πt is
`πt = 1− e−1
2γ (λ−γπσ)2(T−t) ≈ 12γ
(λ− γπσ)2(T − t),
where the approximation ex ≈ 1 + x for x near 0 is used.27 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Loss from missing the right πWelfare losses for different levels of risk aversion:
40%
50%
60%
70%
80%
90%
100%
RRA=1
RRA=2
RRA=3
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
-25% 0% 25% 50% 75% 100% 125% 150% 175% 200%
RRA=1
RRA=2
RRA=3
RRA=6
The percentage wealth-equivalent utility loss `πt from applying a suboptimal constant portfolioweight instead of the optimal portfolio weight. The investment horizon is T − t = 10 years, theSharpe ratio of the stock is λ = 0.3, and the volatility of the stock is σ = 0.2.
28 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Loss from missing the right πWelfare losses for different investment horizons:
30%
40%
50%
60%
70%
80%
T=1
T=10
T=20
0%
10%
20%
30%
40%
50%
60%
70%
80%
-25% 0% 25% 50% 75% 100% 125% 150% 175% 200%
T=1
T=10
T=20
The percentage wealth-equivalent utility loss `πt from applying a suboptimal constant portfolioweight instead of the optimal portfolio weight. The relative risk aversion is γ = 2, the Sharpe ratio ofthe stock is λ = 0.3, and the volatility of the stock is σ = 0.2.
29 / 32
General analysisCRRA utility
Other utility functions
SolutionWealth and consumptionOptimal investmentsSuboptimal investments
Infrequent rebalancing• Continuous rebalancing is infeasible
• Maybe rebalance to π∗ with frequency ∆t• Monte Carlo simulations
St+∆t
St= exp
{(r + σλ−
12σ2
)∆t + σ(zt+∆t − zt )
},
Wt+∆t = πWt exp{(
r + σλ−12σ2
)∆t + σ(zt+∆t − zt )
}+ (1− π)Wt exp{r∆t}
= Wt er∆t{
1 + π
[exp
{(σλ−
12σ2
)∆t + σ(zt+∆t − zt )
}− 1
]}.
• Approximation of expected utility: E[u(WT )] ≈ 1M
∑Mm=1 u (W m
T ) .
• Example with r = 0.02, σ = 0.2, λ = 0.3, γ = 2, T − t = 10 years π∗ = 0.75 = 75%
I With W0 = 1, indirect utility will be −0.65377.I MC simulation with 2000 “antithetic” pairs of paths and ∆t = 0.25
years average utility was −0.65547, loss only 0.26% of wealthI More in Exercise 6.5
30 / 32
General analysisCRRA utility
Other utility functions
Solution with other utility functions?
• Log utility: similar; put γ = 1 (see Theorem 6.3)• Subsistence HARA: Exercise 6.4• Habit formation: (maybe) later• Epstein-Zin preferences: (maybe) later
32 / 32
General utility functionCRRA utility function
Losses from suboptimal strategies
Dynamic Asset AllocationChapter 7: Stochastic investment opportunities
Claus Munk
August 2012
AARHUS UNIVERSITY AU
1 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
Outline
1 General utility function
2 CRRA utility function
3 Losses from suboptimal strategies
2 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
Setting and notationAnalysisMulti-dimensional state variable
Assumptions
• No labor income, yt ≡ 0• A locally risk-free and d risky assets; no portfolio constraints• A one-dimensional state variable x = (xt ) exists so that
rt = r(xt ), µt = µ(xt , t), σ t = σ (xt , t),
λt = λ(xt ) = σ (xt , t)−1 (µ(xt , t)− r(xt )1)
• The state variable follows the process
dxt = m(xt ) dt + v(xt )>dz t + v(xt ) dzt
If v(xt ) 6= 0 incomplete market
5 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
Setting and notationAnalysisMulti-dimensional state variable
The problem to be solvedWealth dynamics for given consumption strategy c and portfolioweight process π:
dWt =(Wt[r(xt ) + π>
t σ (xt , t)λ(xt )]− ct
)dt + Wtπ
>t σ (xt , t)dz t
Indirect utility:
J(W , x , t) = supc≥0,π∈Rd
EW ,x,t
[∫ T
te−δ(s−t)u(cs) ds + e−δ(T−t)u(WT )
]The HJB-equation:
δJ = LcJ + LπJ +∂J∂t
+ rWJW +Jxm +12
Jxx(‖vv‖2 + v2),
LcJ = supc≥0{u(c)− cJW} ,
LπJ = supπ∈Rd
{JW Wπ>σλ +
12
JWW W 2π>σσ>π+JWxWπ>σv}
6 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
Setting and notationAnalysisMulti-dimensional state variable
Optimal consumption
The first order condition wrt. c is
u′(c) = JW (W , x , t) ⇒ c = Iu (JW (W , x , t))
andLcJ = u (Iu (JW (·)))− Iu (JW (·)) JW (·).
8 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
Setting and notationAnalysisMulti-dimensional state variable
Optimal portfolioThe first order condition wrt. π is
JW Wσλ + JWW W 2σσ>π + JWxWσv = 0
⇒ π = − JW
WJWW
(σ>)−1
λ︸ ︷︷ ︸speculative/myopic
− JWx
WJWW
(σ>)−1 v︸ ︷︷ ︸
hedge term
• Three-fund separation (THM. 7.1): combine (1) locally riskfreeasset, (2) tangency portfolio, and (3) hedge portfolio, where
πtan =1
1>(σ>)−1λ
(σ>)−1
λ, πhdg =1
1>(σ>)−1v(σ>)−1 v
• Investors no longer hold different assets in same proportion!• THM. 7.2: Hedge pf has maximal absolute correlation with x• THM. 7.3: Of all pfs with same expected return as π∗, π∗ has
minimal variability of consumption over time(tangency pf minimizes the variability of wealth)
9 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
Setting and notationAnalysisMulti-dimensional state variable
We get
LπJ = · · · = −12
J2W
JWW‖λ‖2 − 1
2J2
WxJWW
‖v‖2 − JW JWx
JWWv>λ.
The HJB-equation becomes
δJ =u (Iu (JW (·)))− Iu (JW (·)) JW (·)− 12
J2W
JWW‖λ‖2 − 1
2J2
WxJWW
‖v‖2
− JW JWx
JWWv>λ +
∂J∂t
+ rWJW + Jxm +12
Jxx(‖v‖2 + v2)
with terminal condition J(W , x ,T ) = u(W ).
10 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
Setting and notationAnalysisMulti-dimensional state variable
More about hedging• Intuition: To keep consumption stable over states and time, a
risk-averse investor will choose a portfolio with high positivereturns [ high wealth] in states with bad future investmentopportunities and conversely
• No hedge ifI JWx = 0: true for t → T and for log utility (don’t want to hedge)I v(x) ≡ 0: investor is not able to hedge
• What risks are to be hedged? If neither r nor ‖λ‖2 arestate-dependent, then the solution to
δJ = u (Iu (JW (·)))− Iu (JW (·)) JW (·)− 12
J2W
JWW‖λ‖2 +
∂J∂t
+ rWJW
is independent of x and will also solve the full HJB-equation.Then JWx = 0 so no hedge! (THM. 7.4)Intuitive, given the instantaneous CML is captured by r and ‖λ‖.
11 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
Setting and notationAnalysisMulti-dimensional state variable
Setting and problem to solve
Assume k -dimensional state variable x = (x t ) with
dx t = m(x t ) dt + v(x t )>dz t + v(x t ) d z t
where z is a k -dim. Brownian motion independent of z.
The HJB-equation for general time-additive utility:
δJ = LcJ + LπJ +∂J∂t
+ rWJW + m>Jx +12
tr(Jxx
[v>v + v v>])
,
LcJ = supc≥0{u(c)− cJW} ,
LπJ = supπ∈Rd
{JW Wπ>σλ +
12
JWW W 2π>σσ>π + Wπ>σvJWx
}with terminal condition J(W ,x ,T ) = u(W ).
13 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
Setting and notationAnalysisMulti-dimensional state variable
Optimal consumption and investment strategyConsumption: as before...Investment:
π∗t = − JW
WJWW
(σ>)−1
λ−(σ>)−1 v
JWx
WJWW
= − JW
WJWW
(σ>)−1
λ−k∑
j=1
(σ>)−1
v1j...
vdj
JWxj
WJWW
(k + 2)-fund separation.Since
LπJ = · · · = −12
J2W
JWW‖λ‖2 − 1
2JWWJ>
Wxv>vJWx − λ>vJW JWx
JWW,
the HJB-equation becomes
δJ =u (Iu(JW ))− JW Iu(JW )−12
J2W
JWW‖λ‖2 −
12JWW
J>Wx v>vJWx − λ>v
JW JWx
JWW
+∂J∂t
+ rWJW + m>Jx +12
tr(
Jxx
[v>v + v v>
])14 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
CRRA utilityAssume utility functions are
u(c) = ε1c1−γ
1− γ, u(W ) = ε2
W 1−γ
1− γ, γ > 0 and γ 6= 1
Conjecture
J(W , x , t) =1
1− γg(x , t)γW 1−γ
leads to
0 = ε1γ
1 −(δ
γ+γ − 1γ
r(x) +γ − 12γ2
‖λ(x)‖2)
g(x , t) +
(m(x)−
γ − 1γ
λ(x)>v(x)
)gx (x , t)
+∂g∂t
(x , t) +12
gxx (x , t)(‖v(x)‖2 + v(x)2
)+γ − 1
2v(x)2 gx (x , t)2
g(x , t)
with the terminal condition g(x ,T ) = ε1/γ2 .
Explicit solution in affine/quadratic models when ε1 = 0 or v = 0;otherwise numerical solution
17 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
Optimal strategiesOptimal consumption strategy is
c∗t = ε1γ
1Wt
g(xt , t)
• Time- and state-dependent fraction of wealth;good inv.opp.’s high consumption
• We need g(x , t) ≥ 0 for solution to make senseOptimal investment strategy is
π∗t =1γ
(σ (xt , t)>
)−1λ(xt )︸ ︷︷ ︸
speculative
+gx (xt , t)g(xt , t)
(σ (xt , t)>
)−1 v(xt )︸ ︷︷ ︸hedge
• Time- and state-dependent strategy
18 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
Optimal strategies, cont’dHedge portfolio is matching the sensitivity of the optimal W/c wrt.hedgeable shocks:
dg(xt , t) = g(xt , t)[. . . dt +
gx (x , t)g(x , t)
v(xt )> dz t +
gx (x , t)g(x , t)
v(xt ) dzt
].
The dynamics of the value of a given portfolio π is
dVπt = Vπ
t[(
r(xt ) + π>t σ (xt , t)λ(xt )
)dt + π>
t σ (xt , t) dz t].
Wealth dynamics with optimal strategies:
dW ∗t = W ∗
t
[(r(xt ) +
1γ‖λ(xt )‖2+
gx
gv(xt )
>λ(xt )− ε1/γ1 g(xt , t)−1
)dt
+
(1γλ(xt )+
gx
gv(xt )
)>
dz t
]19 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
CRRA, terminal wealth only (ε1 = 0, ε2 = 1, δ = 0)Write
g(x , t) ≡ g(x , t ; T ) = exp{−γ − 1
γH(x ,T − t)
},
then H(x , τ) has to solve the simpler PDE
0 =r(x) +1
2γ‖λ(x)‖2 − ∂H
∂τ(x , τ) +
(m(x)− γ − 1
γλ(x)>v(x)
)Hx (x , τ)
+12
Σx (x)Hxx (x , τ)− γ − 12γ
(Σx (x) + (γ − 1)v(x)2
)Hx (x , τ)2 (*)
with the condition H(x , 0) = 0.
THM. 7.5: The indirect utility function is
J(W , x , t) =1
1− γ e−(γ−1)H(x,T−t)W 1−γ =1
1− γ
(WeH(x,T−t)
)1−γ,
and the optimal investment strategy is
Π(W , x , t) =1γ
(σ (x , t)>
)−1λ(x)− γ − 1
γHx (x ,T − t)
(σ (x , t)>
)−1v(x),
where H(x , τ) solves (*) with initial condition H(x , 0) = 0.20 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
CRRA, ε1, ε2 ≥ 0, complete market (v(x) = 0)
THM. 7.6: Let H(x , τ) solve (*) when v(x) = 0, with H(x , 0) = 0. Define
g(x , t ; s) = exp{− δγ
(s − t)− γ − 1γ
H(x , s − t)}.
Then
g(x , t) = ε1γ
1
∫ T
tg(x , t ; s) ds + ε
1γ
2 g(x , t ; T ).
So the optimal strategies are
C(W , x , t) = ε1/γ1
Wg(x , t)
=
(∫ T
tg(x , t ; s) ds +
(ε2
ε1
) 1γ
g(x , t ; T )
)−1
W ,
Π(W , x , t) =1γ
(σ (x , t)>
)−1λ(x)− γ − 1
γD(x , t ,T )
(σ (x , t)>
)−1v(x),
where
D(x , t ,T ) =
∫ Tt Hx (x , s − t)g(x , t ; s) ds + (ε2/ε1)
1γ Hx (x ,T − t)g(x , t ; T )∫ T
t g(x , t ; s) ds + (ε2/ε1)1γ g(x , t ; T )
.
21 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
Special case: Logarithmic utility (γ = 1)The indirect utility function is given by
J(W , x , t) = g(t) ln W + h(x , t), g(t) =1δ
(ε1 + [ε2δ − ε1] e−δ(T−t)
)The optimal investment strategy is given by
π∗t =(σ (xt , t)>
)−1λ(xt )
a log-investor do not hedge stochastic variations in the investmentopportunity set.The growth-optimal portfolio: the portfolio strategy of all assets held by the log utilityinvestor; maximizes the expected average compound growth rate of portfolio value, i.e.,the expectation of 1
T−t ln (WT /Wt ) .
The optimal consumption strategy is
c∗t = ε1Wt
g(t)= δ
(1 + [ε2δ − 1] e−δ(T−t)
)−1W
22 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
Affine model• One-dimensional state variable
dxt = m(xt ) dt + v(xt )>dz t + v(xt ) dzt
• Affine model: r(x), m(x), ‖λ(x)‖2, ‖v(x)‖2, v(x)>λ(x), andv(x)2 are all affine functions of x , in particular
r(x) = r0 + r1xm(x) = m0 + m1x
v(x)2 = v0 + v1x
‖λ(x)‖2 = Λ0 + Λ1x
‖v(x)‖2 = V0 + V1xv(x)>λ(x) = K0 + K1x
• CRRA utility24 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
Then (*) has a solution of the form H(x , τ) = A0(τ) + A1(τ)x where
A′1(τ) = r1 +Λ1
2γ+
(m1 −
γ − 1γ
K1
)A1(τ)−
γ − 12γ
(V1 + γv1)A1(τ)2, A1(0) = 0
A0(τ) =
(r0 +
Λ0
2γ
)τ +
(m0 −
γ − 1γ
K0
)∫ τ
0A1(s) ds −
γ − 12γ
(V0 + γv0)
∫ τ
0A1(s)2 ds.
Suppose (m1 −
γ − 1γ
K1
)2+ 2
γ − 1γ
(r1 +
Λ1
2γ
)(V1 + γv1) > 0,
and define
ν =
√(m1 −
γ − 1γ
K1
)2+ 2
γ − 1γ
(r1 +
Λ1
2γ
)(V1 + γv1).
Then
A1(τ) =2(
r1 + Λ12γ
)(eντ − 1)(
ν + γ−1γ
K1 −m1
)(eντ − 1) + 2ν
,
A0(τ) =
(r0 +
Λ0
2γ
)τ +
(m0 −
γ − 1γ
K0
)∫ τ
0A1(s) ds −
γ − 12γ
(V0 + γv0)
∫ τ
0A1(s)2 ds.
25 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
With ε1 = 0, δ = 0, ε2 = 1:Indirect utility and optimal portfolio become
J(W , x , t) =1
1− γ
(WeA0(T−t)+A1(T−t)x
)1−γ
π∗t =1γ
(σ (xt , t)>
)−1λ(xt )−
γ − 1γ
(σ (xt , t)>
)−1 v(xt )A1(T − t).
26 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
With ε1, ε2 ≥ 0 and complete market (v(x) = 0):Let g(x , t ; s) = exp
{− δγ (s − t)− γ−1
γ (A0(s − t) + A1(s − t)x)}
.Indirect utility function and optimal strategies are
J(W , x , t) =1
1− γ
(ε
1γ
1
∫ T
tg(x , t ; s) ds + ε
1γ
2 g(x , t ; T )
)γW 1−γ ,
C(W , x , t) =
(∫ T
tg(x , t ; s) ds +
(ε2
ε1
) 1γ
g(x , t ; T )
)−1
W ,
Π(W , x , t) =1γ
(σ (x , t)>
)−1λ(x)− γ − 1
γD(x , t ,T )
(σ (x , t)>
)−1 v(x),
with D(x , t ,T ) =∫ T
t A1(s−t)g(x,t ;s) ds+(ε2/ε1)1γ A1(T−t)g(x,t ;T )∫ T
t g(x,t ;s) ds+(ε2/ε1)1γ g(x,t ;T )
.
With consumption and incompleteness? Numerical solution for geven in affine model!?!
27 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
Assumptions
• One-dimensional state variable
dxt = m(xt ) dt + v(xt )>dz t + v(xt ) dzt
• Quadratic model:
r(x) = r0 + r1x + r2x2
‖λ(x)‖2 = Λ0 + Λ1x + Λ2x2
m(x) = m0 + m1xv(x)>λ(x) = K0 + K1x
and ‖v(x)‖2 and v(x)2 constant• Consider a CRRA investor
29 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
Then (*) has solution H(x , τ) = A0(τ) + A1(τ)x + 12 A2(τ)x2, where
A′0(τ) = r0 +Λ0
2γ+
(m0 −
γ − 1γ
K0
)A1(τ)
+12
(‖v‖2 + v2
)A2(τ)− γ − 1
2γ
(‖v‖2 + γv2
)A1(τ)2,
A′1(τ) = r1 +Λ1
2γ+
(m0 −
γ − 1γ
K0
)A2(τ)
+
[m1 −
γ − 1γ
K1 −γ − 1γ
(‖v‖2 + γv2
)A2(τ)
]A1(τ),
A′2(τ) = 2r2 +Λ2
γ+ 2
(m1 −
γ − 1γ
K1
)A2(τ)− γ − 1
γ
(‖v‖2 + γv2
)A2(τ)2
with initial conditions A0(0) = A1(0) = A2(0) = 0.
30 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
Suppose that(m1 −
γ − 1γ
K1
)2
+γ − 1γ
(2r2 +
Λ2
γ
)(‖v‖2 + v2
)> 0
and define
ν = 2
√(m1 −
γ − 1γ
K1
)2
+γ − 1γ
(2r2 +
Λ2
γ
)(‖v‖2 + v2),
q =
(m0 −
γ − 1γ
K0
)(2r2 +
Λ2
γ
)−(
m1 −γ − 1γ
K1
)(r1 +
Λ1
2γ
).
Then
A2(τ) =2(
2r2 + Λ2γ
)(eντ − 1)(
ν + 2 γ−1γ
K1 − 2m1
)(eντ − 1) + 2ν
,
A1(τ) =r1 + Λ1
2γ
2r2 + Λ2γ
A2(τ) +4qν
(eντ/2 − 1
)2
(ν + 2 γ−1γ
K1 − 2m1)(eντ − 1) + 2ν,
and we can compute A0(τ) by integrating.31 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
With ε1 = 0, δ = 0, ε2 = 1:
Indirect utility and optimal portfolio become
J(W , x , t) =1
1− γ
(WeA0(T−t)+A1(T−t)x+ 1
2 A2(T−t)x2)1−γ
,
Π(W , x , t) =1γ
(σ (x , t)>
)−1λ(x)
− γ − 1γ
(σ (x , t)>
)−1v (A1(T − t) + A2(T − t)x) .
32 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
With ε1, ε2 ≥ 0 and complete market (v(x) = 0):
Let g(x , t ; s) = exp{− δγ
(s − t)− γ−1γ
(A0(s − t) + A1(s − t)x + 1
2 A2(s − t)x2)}
.
Indirect utility function and optimal strategies are
J(W , x , t) =1
1− γ
(ε
1γ
1
∫ T
tg(x , t ; s) ds + ε
1γ
2 g(x , t ; T )
)γW 1−γ ,
C(W , x , t) =
(∫ T
tg(x , t ; s) ds +
(ε2
ε1
) 1γ
g(x , t ; T )
)−1
W ,
Π(W , x , t) =1γ
(σ (x , t)>
)−1λ(x)−
γ − 1γ
D(x , t ,T )(σ (x , t)>
)−1v ,
where
D(x, t, T ) =
∫ Tt (A1(s − t) + A2(s − t)x)g(x, t ; s) ds +
(ε2ε1
) 1γ (A1(T − t) + A2(T − t)x) g(x, t ; T )∫ T
t g(x, t ; s) ds +(ε2ε1
) 1γ g(x, t ; T )
.
33 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
One-dimensional state variableAffine model with one state variableQuadratic model with one state variableMultidimensional state variable
• Similar results• Explicit solutions in affine/quadratic models when ε1 = 0 or
v = 0; otherwise numerical solution• For example, in a multi-dimensional affine setting with utility of
terminal wealth, the optimal investment strategy is of the form
π(x , t) =1γ
(σ (x , t)>
)−1λ(x)−γ − 1
γ
(σ (x , t)>
)−1√
V (x)D>A1(T−t).
• Hard to solve high-dimensional ODE’s analytically (for A1), butnumerical solution is not too difficult...
35 / 37
General utility functionCRRA utility function
Losses from suboptimal strategies
How costly are deviations from the optimal investment strategy? Larsen & Munk (2012)• Nice results in affine/quadratic models for some relevant
sub-optimal investment strategies, e.g., strategies ignoringintertemporal hedge term and/or ignoring a certain asset (class).
37 / 37