dynamic asset allocation - aarhus universitet...general considerations recommendations on asset...

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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Outline

1 General considerations

2 Recommendations on asset allocation

3 About alternative approaches

4 How do individuals invest?

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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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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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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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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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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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http://www.money-zine.com/Calculators/Investment-Calculators/Asset-Allocation-Calculator/

http://www.money-zine.com/Calculators/Investment-Calculators/Asset-Allocation-Calculator/



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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Table II from Barber et al.

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Table V from Barber et al.

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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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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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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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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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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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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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Regressing returns on price/dividend ratio

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Predicted v. realized returns

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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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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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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.


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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.


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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



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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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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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


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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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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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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




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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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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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




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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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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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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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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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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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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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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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)

.

27 / 38




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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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



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




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




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




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




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


1 / 22


Outline

1 Discrete time

2 Continuous time

2 / 22


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



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



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



Income

• income rate yt income of yt ·∆t number of goods over [t , t + ∆t)

• “received” at time t• assumed exogenously given

8 / 22



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



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



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



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



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



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



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



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



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



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


1 / 32



Outline

1 General analysis

2 CRRA utility

3 Other utility functions

2 / 32



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



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


].

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



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



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



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



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



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




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




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




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




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




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




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




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




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




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




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




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




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




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



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


1 / 37



Outline

1 General utility function

2 CRRA utility function

3 Losses from suboptimal strategies

2 / 37



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




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


]The HJB-equation:


+ 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




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




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




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




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




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:


+ 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




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



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




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




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




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




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




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




Affine model• One-dimensional state variable


• 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




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




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




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


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




Assumptions

• One-dimensional state variable


• 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




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




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




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




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


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




• 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



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

dynamic asset allocation - aarhus universitet...general considerations recommendations on asset...

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