Dynamic Asset Allocation...We follow Davis & Norman (1990, Math. Operations Research) and consider following model I a single risky asset (a stock) and a riskfree asset, I proportional

The model Solution w/o transaction costs Solution w/ transaction costs Some extensions

Dynamic Asset AllocationChapter 18: Transaction costs

Claus Munk

Aarhus University

August 2012


Transaction costs

We follow Davis & Norman (1990, Math. Operations Research)and consider following model

I a single risky asset (a stock) and a riskfree asset,I proportional transaction costs when trading the stock,I constant investment opportunities are constant,I investor has an infinite time horizon,I CRRA utility of consumption

Many relevant extensionsI Fixed costs? Fixed and proportional costs?I Multiple assets?I Finite time horizon?I Trading costs for durable goods?I Transaction costs + stochastic investment opportunities?


Outline

1 The model

2 Solution w/o transaction costs

3 Solution w/ transaction costs

4 Some extensions


AssumptionsRisk-free bank account constant r ; no costs.A single risky asset (the stock) with price dynamics

dPt = Pt [µdt + σ dzt ] .

Buying one unit costs (1 + a)Pt , selling one unit provides(1− b)Pt , where a,b ≥ 0.Investment strategy in the stock is represented by the pair ofprocesses (L,U)

I Lt : cumulative amounts of stock purchased on [0, t ]I Ut : cumulative amounts of stock sold on [0, t ]

(amounts measured by the listed price, costs subtracted from thebank account)S0t : bank account; S1t : value of the stocks owned at time t(measured at the listed unit price at time t).The dynamics is

dS0t = (rS0t − ct) dt − (1 + a)dLt + (1− b)dUt , S00 = x , (1)dS1t = µS1t dt + σS1t dzt + dLt − dUt , S10 = y . (2)

Here ct is the consumption rate at time t .


Assumptions, cont’dSolvency condition: after eliminating his position in the stock, youmust have non-negative wealth.

I If S1t > 0, the requirement is S0t + (1− b)S1t ≥ 0, i.e.,S1t ≥ − 1

1−b S0t .I If S1t < 0, the requirement is S0t + (1 + a)S1t ≥ 0, i.e.,

S1t ≥ − 11+a S0t .

The solvency region is therefore

S ={(x , y) ∈ R2 : x + (1− b)y ≥ 0, x + (1 + a)y ≥ 0

}.

The set of admissible consumption and trading strategies is

U(x , y) = {(c,L,U) : (S0t ,S1t) ∈ S for all t ≥ 0 (a.s.), ct ≥ 0}

For preferences, assume infinite horizon and power utility withγ > 1 denoting the relative risk aversion. Let

J(x , y) = sup(c,L,U)∈U(x,y)

Ex,y

[∫ ∞0

e−δt 11− γ

c1−γt dt

].


Take Merton to the limit...For the case without transaction costs (a = b = 0), we solved thesimilar problem for a finite time horizon in Chapter 6.Let λ = (µ− r)/σ. If the constant

A =δ + r(γ − 1)

γ+

12γ − 1γ2 λ2

is positive, the limit as T →∞ of the solution is

J(x , y) =1

1− γA−γ(x + y)1−γ ,

c∗ = A[x + y ], π∗ =λ

γσ,

where x + y is the total wealth.We have π∗t = S1t

S0t+S1tand hence

S1t

S0t=

π∗

1− π∗=

λ

γσ − λ,

corresponding to a straight line through the origin in the(S0,S1)-space, the Merton line.


Theorem 18.1The value function J(x , y) has the following properties:

1 J is concave, i.e., for θ ∈ [0, 1]

J (θx1 + [1− θ]x2, θy1 + [1− θ]y2) ≥ θJ(x1, y1) + [1− θ]J(x2, y2).

2 J is homogeneous of degree 1− γ, i.e., for k > 0

J(kx , ky) = k1−γJ(x , y).

It follows that J(x , y) = kγ−1J(kx , ky) for any k > 0. Consequently,

Jx(x , y) ≡∂J∂x

(x , y) =∂

∂x

(kγ−1J(kx , ky)

)= kγJx(kx , ky)

and, similarly,

Jy (x , y) ≡∂J∂y

(x , y) = kγJy (kx , ky).

So, for all k > 0,Jy (kx , ky)Jx(kx , ky)

=Jy (x , y)Jx(x , y)

.

Hence, Jy/Jx is constant along any straight line through the origin.


Heuristic solutionAssume trading strategies are of the form

Lt =

∫ t

0`s ds, Ut =

∫ t

0us ds; ls, us ∈ [0,K ]

for some constant K . In particular, dLt = `t dt and dUt = ut dt . HJB-eq.:

δJ(x , y) = supc≥0,`∈[0,K ],u∈[0,K ]

{ 11− γ c1−γ + Jx [rx − c − (1 + a)`+ (1− b)u]

+ Jy [µy + `− u] +12

Jyyσ2y2}

= supc≥0

{1

1− γ c1−γ − cJx

}+ sup`∈[0,K ]

{(Jy − (1 + a)Jx) `}

+ supu∈[0,K ]

{((1− b)Jx − Jy ) u}+ rx Jx + µyJy +12σ2y2Jyy .

The first-order conditions imply

` =

{K , if Jy ≥ (1 + a)Jx ,

0, otherwise,u =

{0, if Jy > (1− b)Jx ,

K , otherwise.


Heuristic solution, cont’d

Trading strategy:

Jy ≥ (1 + a)Jx : buy stocks(1 + a)Jx > Jy > (1− b)Jx : do not trade stocks

Jy ≤ (1− b)Jx : sell stocks.

This divides the solvency region into three regions: a buying region, ano trade region, and a selling region.

Boundaries:∂B: points with Jy (x , y) = (1 + a)Jx(x , y) ... a straight line! (slopedenoted 1/ωB)∂S: points with Jy (x , y) = (1− b)Jx(x , y) ... a straight line! (slopedenoted 1/ωS; ωB ≥ ωS)

Draw graph...



Do not trade when

1ωB≤ S1t

S0t≤ 1ωS.

The fraction of wealth invested in the stock is πt = S1t/(S0t + S1t),which will then satisfy

11 + ωB

≤ πt ≤1

1 + ωS.

Except for extreme cases, the Merton weight π∗ = λγσ falls between

the boundaries.



Inside the no trade region, HJB-eq. simplifies to

δJ = supc≥0

{1

1− γ c1−γ − cJx

}+ rx Jx + µyJy +

12σ2y2Jyy

=γ

1− γ J1− 1

γx + rx Jx + µyJy +

12σ2y2Jyy .

Exploit homogeneity of the value function:

J(

xy, 1)

=

(1y

)1−γ

J(x , y) ⇒ J(x , y) = y1−γJ(

xy, 1)≡ y1−γψ

(xy

).

HJB-eq. becomes

12σ2ω2ψ′′(ω) + (r − µ+ γσ2)ωψ′(ω)

−(δ + (γ − 1)µ− 1

2σ2γ(γ − 1)

)ψ(ω)+

γ

1− γ ψ′(ω)1− 1

γ = 0, ω ∈ [ωS , ωB].



In the selling region, we must have J(x , y) constant along any line of slope−1/(1− b), so that J(x , y) = F (x + [1− b]y) for some function F . ThenJx = F ′ and Jy = (1− b)F ′ so that Jy = (1− b)Jx .

Inserting Jx and Jy , we see that

ψ′(ω)(ω + 1− b) = (1− γ)ψ(ω),

⇒ ψ(ω) = A1

1− γ (ω + 1− b)1−γ

for a constant A.Hence, J(x , y) = y1−γψ(x/y) = A 1

1−γ (x + [1− b]y)1−γ .

Similarly,

ψ(ω) = B1

1− γ (ω + 1 + a)1−γ

for some constant B in the buying region, i.e.,J(x , y) = B 1

1−γ (x + [1 + a]y)1−γ .


The final solutionTo sum up, we have to find constants ωB, ωS ,A,B and a function ψ so that

12σ2ω2ψ′′(ω) + (r − µ+ γσ2)ωψ′(ω) +

γ

1− γ ψ′(ω)1− 1

γ

−(δ + (γ − 1)µ− 1

2σ2γ(γ − 1)

)ψ(ω) = 0, ω ∈ [ωS , ωB],

ψ(ω) = A1

1− γ (ω + 1− b)1−γ , ω ≤ ωS ,

ψ(ω) = B1

1− γ (ω + 1 + a)1−γ , ω ≥ ωB.

Davis & Norman show that (under a technical condition) a solution to thisproblem will lead to the optimal strategies as described above.

The optimal consumption rate will be

c∗t = S1t(ψ′(S0t/S1t)

)−1/γ.

Davis & Norman confirm that a solution to the problem exists.


The final solution, cont’d

At the boundaries, we have the so-called value-matching conditions

ψ(ωS) = A1

1− γ (ωS + 1− b)1−γ , ψ(ωB) = B1

1− γ (ωB + 1 + a)1−γ .

The so-called smooth-pasting conditions ensure that the derivative of ψ atωS is the same from the left and from the right, and equivalently at ωB .Therefore

ψ′(ωS) = A(ωS + 1− b)−γ , ψ′(ωB) = B(ωB + 1 + a)−γ .

Numerical solution techniques are required!


Finite time horizon

Note: vertical axis shows stock-bond ratioSource: Gennotte and Jung, Management Science, 1994


Proportional and fixed costs

Source: Øksendal and Sulem, SIAM Journal of Control and Optimization, 2002


Multiple assets

Source: Muthuraman and Kumar, Mathematical Finance, 2006

Introduction Model and problem Optimal policies Gains from tax-optimization Comparative statics The end

Asset allocation over the life cycle:

How much do taxes matter?

Holger Kraft1 Marcel Marekwica2 Claus Munk3

1Goethe University Frankfurt, Germany

2Copenhagen Business School, Denmark

3Aarhus University, Denmark


Outline

1 Introduction

2 Model and problem

3 Optimal policies

4 Gains from tax-optimization

5 Comparative statics

6 The end


Contribution

Derive and study optimal consumption and portfolio decisions ina life cycle model with labor income and taxation of realizedcapital gains

Assess the importance of tax-timing, i.e., exploiting therealization-based feature of capital gains taxationvs. “mark-to-market taxation”

Assess the importance of taking into account taxes on financialreturns at all


Asset allocation with income

Hakansson (Econ-70), Merton (JET-71): risk-free income

Bodie-Merton-Samuelson (JEDC-92): spanned income; laborsupply decisions

Viceira (JF-01): unspanned income

Cocco-Gomes-Maenhout (RFS-05): life cycle income profiles

Munk-Sørensen (JFE-10): interest rates and income

Yao-Zhang (RFS-05), Van Hemert (REE-10), Kraft-Munk(MS-11): housing decisions and income

Note: no taxes!


Asset allocation with taxes

Constantinides (Econ-83): wash sales; shorting-the-box

Dammon-Spatt-Zhang (RFS-01): our model w/o income (well...)

Gallmeyer-Kaniel-Tompaidis (JFE-06): multiple stocks

Ehling et al. (wp-10): asymmetric taxation of gains and losses

DeMiguel-Uppal (ManSci-05): “exact share identification” vs.“average purchase price”

Dammon-Spatt-Zhang (JF-04), Zhou (JEDC-09),Gomes-Michaelidis-Polkovnichenko (RED-09): taxable vs.tax-deferred accounts; assume either inappropriate income or“mark-to-market taxation” of capital gains in taxable account

Note: realization-based taxation not studied in life cycle setting withreasonable model of labor income!Note: no welfare analysis!


Our main conclusions

For investors assuming “mark-to-market taxation” the welfaregains from switching to the fully optimized portfolio policy areless than 0.5% of wealth

Tax-timing considerations have modest impact on optimal portfoliosExpected utility little sensitive to small variations in portfolios[Brennan-Torous (EN-99), Rogers (FiSt-01)]

An investor completely ignoring taxation of investment profits willgain less than 2% by switching to the fully optimal portfolio policy

Exception: very old investors with strong bequest motives ifcapital gains are forgiven at death (as in the U.S.)


Assumptions

Discrete-time model with one-year time intervals

Single consumption good, constant inflation rate i = 3%

Individual lives for at most T = 100 years with deterministicunconditional survival probabilities Ft

Ct : nominal consumption in period t

Time-additive expected CRRA utility (bequest considered later):

E

[T∑

t=t0

βt · Ft · U

(Ct

(1 + i)t

)], U(c) =

c1−γ

1− γ

Benchmark parameters γ = 4, β = 0.96


SecuritiesAlways a one-period risk-free asset, pre-tax return r = 4%, taxrate τi = 35%

One risky asset (stock index) with price Pt

capital gain gt+1 =Pt+1

Pt− 1; mean µ = 6%, std-dev σ = 20%

1 + gt+1 lognormally distributedrealized capital gains taxed at rate τg = 20%constant dividend yield d = 2%, taxed at rate τi

qt : number of stocks hold from t to t + 1tax basis P∗t is an average historical purchases price:

P∗t =

Pt if P∗t−1 ≥ Pt (realize loss; buy new)qt−1P∗

t−1+max(qt−qt−1,0)Pt

qt−1+max(qt−qt−1,0)if P∗t−1 < Pt (unrealized gains)

Capital gains subject to taxation at time t is

Gt =[χ{P∗

t−1≥Pt}qt−1 + χ{P∗t−1<Pt}max (qt−1 − qt , 0)

]· (Pt − P∗t−1)


Labor income

Pre-tax labor income It , taxed at rate τi

Stochastic income in working life, i.e., up to age J = 65

Income growth rate ft+1 =It+1

It− 1 has std-dev σL = 15% and mean

µ(t + 1) calibrated to age-profile of high-school graduates in theU.S.Capital gains 1 + gt+1 and income growth 1 + ft+1 jointlylognormally distributed with correlation ρ = 0

Retirement income is a fraction λ = 68.2% of pre-retirementincome (inflation-adjusted):

IJ+m = λ(1 + i)mIJ


The optimization problemObjective: max{Ct ,qt ,Bt}T

t=t0E[∑T

t=t0 βt · Ft · U

(Ct

(1+i)t

)]Disposable wealth:

Wt = It (1− τi) + qt−1Pt (1 + d (1− τi)) + Bt−1 (1 + (1− τi) r)

Budget constraint: Ct + τgGt + qtPt + Bt ≤Wt

Conditions: qt ≥ 0,Bt ≥ 0,Ct > 0

State variables: Wt , Pt , P∗t−1, qt−1, and ItReduce complexity by normalization. New state variables:

after-tax income-to-wealth ratio it = (1 − τi)It

Wt,

entering relative equity exposure st =qt−1Pt

Wt,

basis-price ratio p∗t =P∗

t−1Pt

Cons/wealth ratio ct =CtWt

, exiting equity exposure αt =qt PtWt

, andrelative bond investment bt =

BtWt

only depend on t , it , st , and p∗t


Numerical approach

Solve problem numerically using backward induction on a grid forthe state variables

31 grid points each for s, p∗, and i ; 80 timepoints 80 · 313 ' 2.4 million optimization problems

With a Matlab implementation on a single pc, it takes around oneweek to solve the optimization problem


Inputs

Assume benchmark parameter values introduced earlier

Assume individual is currently of age t0 = 20

No initial equity holdings

Initial wealth consists entirely of labor income over the first year,i0 = 100%


Optimal consumption: state-dependence

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Income−to−wealth ratio

Con

sum

ptio

n−w

ealth

rat

ioConsumption policies, age=25, s=50%

p*=0.85p*=0.55p*=1.2


Optimal consumption: state-dependence

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Con

sum

ptio

n−w

ealth

rat

ioConsumption policies, s=50%, p*=0.85

Age=25Age=40Age=60Age=8045 degree line


Optimal investment: state-dependence

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.4

0.5

0.6

0.7

0.8

0.9

1


Equ

ity e

xpos

ure

Investment policies, age=25, s=50%

p*=0.85p*=0.55p*=1.2


Optimal investment: state-dependence

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.4

0.5

0.6

0.7

0.8

0.9

1


Equ

ity e

xpos

ure

Investment policies, s=50%, p*=0.85

Age=25Age=40Age=60Age=80


Life cycle profile

20 30 40 50 60 70 80 90 1000

1

2

3

Age

Con

sum

ptio

n, In

com

e

Life cycle profile

20 30 40 50 60 70 80 90 1000

10

20

30

Wea

lth le

vel

ConsumptionIncomeWealth

Note: averages based on 10,000 simulations


Life cycle profile: income/wealth

20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Age

Inco

me−

to−

wea

lth

Evolution of income−to−wealth ratio over the life cycle

Mean5th percentile95th percentile

Note: based on 10,000 simulations


Life cycle profile: basis-price ratio

20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

Age

Bas

is−

pric

e ra

tio

Evolution of initial basis−price ratio over the life cycle




Life cycle profile: consumption-wealth ratio

20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

Age

Con

sum

ptio

n−w

ealth

rat

io

Evolution of consumption−wealth ratio over the life cycle




Life cycle profile: equity exposure

20 30 40 50 60 70 80 90

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Age

Equ

ity e

xpos

ure

Evolution of equity exposure over the life cycle




Suboptimal policies considered

No tax-timing:

Best consumption and portfolio policies assuming“mark-to-market taxation of capital gains”, i.e., Gt = qt(Pt −Pt−1)

Optimal consumption policy + best portfolio policy assuming“mark-to-market taxation”

No taxes on returns at all:

Best consumption and portfolio policies assuming zero taxes onreturns (capital gains, interest and dividend payments)

Optimal consumption policy + best portfolio policy assuming zerotaxes on returns


Measure of suboptimality

Welfare gain:extra total wealth (current wealth + future income) the investorfollowing a suboptimal policy needs to obtain the same utility if shecontinues with the suboptimal policy instead of shifting to the optimalpolicy


State-dependent welfare gains: no tax-timing

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.2

0.25

0.3

0.35

0.4

0.45


Wel

fare

gai

nWelfare gain for investor ignoring tax−timing, age=25, s=67%

p*=0.85p*=0.55p*=1.25


State-dependent welfare gains: no taxes

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51.3

1.4

1.5

1.6

1.7

1.8

1.9

2


Wel

fare

gai

nWelfare gain for investor ignoring taxes, age=25, s=67%

p*=0.85p*=0.55p*=1.25


Welfare effects over the life cycle: no tax-timing

20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Age

Wel

fare

gai

n

Welfare gain investor ignoring tax−timing

Suboptimal portfolio strategySuboptimal consumption−portfolio strategy



Welfare effects over the life cycle: no tax-timing

cons/wealth equity exposure welfare gains

Age optimal heuristic optimal heuristic cons+port port only

20 78.9% 78.9% 100% 100% 0.23% 0.17%

30 20.8% 20.7% 100% 100% 0.32% 0.24%

40 12.2% 12.2% 97% 98% 0.41% 0.30%

50 10.0% 10.2% 90% 91% 0.42% 0.29%

60 10.5% 10.7% 86% 86% 0.35% 0.21%

70 14.6% 14.9% 87% 87% 0.24% 0.10%

80 28.1% 28.6% 91% 90% 0.17% 0.04%

90 62.3% 62.7% 93% 94% 0.08% 0.00%



Welfare effects over the life cycle: no taxes

20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

Age

Wel

fare

gai

n

Welfare gain for investor ignoring taxes

Suboptimal portfolio strategySuboptimal consumption−portfolio strategy



Welfare effects over the life cycle: no taxes

cons/wealth equity exposure welfare gains

Age optimal heuristic optimal heuristic cons+port port only

20 78.9% 79.8% 100% 100% 1.50% 0.96%

30 21.6% 22.3% 100% 100% 1.91% 1.28%

40 12.8% 13.4% 93% 88% 1.97% 1.41%

50 10.7% 11.2% 82% 74% 1.57% 1.18%

60 11.3% 11.7% 77% 70% 1.00% 0.78%

70 15.8% 16.2% 81% 75% 0.53% 0.40%

80 30.1% 30.5% 89% 85% 0.24% 0.16%

90 68.3% 68.4% 95% 95% 0.06% 0.03%



Robustness

In the paper we demonstrate that the results are robust to relevantvariations in key inputs:

initial income/wealth ratio

volatility of labor income

tax rates

risk aversion, Epstein-Zin preferences


Robustness wrt. bequest?

So far, no utility of bequest– in line with empirical study of Hurd (Econ-89)

With bequest motive of “strength” k , maximize

E

[T∑

t=0

βt · Ft · U(

Ct

(1 + i)t

)+ β · (Ft − Ft+1) · k · U

(W B

t+1

(1 + i)t+1

)]

where W Bt+1 is the wealth bequeathed to descendants at death.

In U.S.: unrealized gains are forgiven at death

Other countries: unrealized capital gains taxed at death


Bequest motive; taxation of capital gains at death

20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Age

Wel

fare

gai

n


k=0k=5k=10



Bequest motive; forgiveness of capital gains at death

20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

Age

Wel

fare

gai

n


k=0k=5k=10



Conclusion

Optimal stock-bond allocation over the life cycle is only littleaffected by the realization-based feature of capital gains taxationor by return taxation at all

The realization-based feature of capital gains taxation is of minorimportance to individual investors

Modification: some effect for very old investors with strongbequest motives if capital gains are forgiven at death

Even if you ignore taxation of financial returns completely, youwill only lose relatively little

The limited computational resources are probably better spenton adding other relevant state and decision variables (e.g.,time-varying returns, housing) than on including the complicatedreal-life rules for taxation of financial returns

Documents

Dynamic Asset Allocation...We follow Davis & Norman (1990, Math. Operations Research) and consider following model I a single risky asset (a stock) and a riskfree asset, I proportional