1

ASSET ALLOCATION

M A X I M I Z E E X P E C T E D U T I L I T Y

HwwwUE '2

1')(

i s t h e v e c t o r o f e x p e c t e d r e t u r n sw i s t h e v e c t o r o f p o r t f o l i o w e i g h t sH i s t h e v a r i a n c e c o v a r i a n c e m a t r i x i s t h e c o e f f i c i e n t o f r i s k a v e r s i o n

2

With Riskless Asset

HwwrwwMax

wwts'

2' 00

1'.. 0

)1

01 rHwra

3

Mean Variance Relative to a Benchmark

000 '

2)'( wwHwwwwMax

10

1 Hww

4

Hedging First Asset

•

• If lambda is large or mu is small, then this is least squares

HwwwMax

wts'

2'

1.. 1

1,0,...,0 is 'e where,'1

1'

11

11

11

1

11

1

He

eHe

eHHw

5

With No Riskless Asset

TH E O PTIM AL ALLO CATIO NEQ UATIO N:

w VV

VV

1

11

1

1

11

'

'

where is a vector of ones.

6

Allocation subject to VaR

HwwrwwMax

Hwwts'

2'1' 0

'.. 2

212

01

0

1

~'~1

~ where,~'~2

1

and binding,not is constraint if ,~1

H

rHrU

Hw

7

But if constraint is binding:

• The constraint reduces expected utility

• Higher generalized Sharpe ratios still lead to improved utility

~'~2/

,~~'~

120

11

HrEU

HH

w

8

STATIC ALLOCATIONS

ASSET ALLOCATION ACROSSSTOCKS AND BONDS:

DATA FROM 1982-1996(July)DAILY S&P500 AND TREASURYBOND FUTURES

STOCK VOLATILITY = 15.4 %MEAN RETURN = 11.63%

BOND VOLATILITY = 11.4%MEAN RETURN = 7.27 %

CORRELATION = .407

9

Static Allocations

TABLE 1

PORTFOLIO SHARES

Risk ( ) 2 4 6 8 16 % Equity 123 74 58 50 38 26

10

Derivatives of Optimal Allocation

TABLE 2EQUITY WEIGHTS (%)

Risk () 2 4 6 8 16 dw

drr 2.5 1.3 0.8 0.6 0.3 0.0

dw

d-2.1 -1.4 -1.2 -1.1 -0.8 -0.69

dw

d0.47 0.16 0.05 .002 -0.08 -0.15

11

Derivatives of Portfolio Risk

TABLE 3PORTFOLIO RISK

Risk() 2 4 6 8 16 1.29 .66 .55 .51 .47 .45dv

drr 4.38 1.100.490.270.07 0.0

dv

d-1.00-.08 0.090.150.210.23

dv

d0.64 0.240.170.140.120.11

12

Dynamic Asset Allocation

• Maximize the Expected Utility of Terminal Wealth

• Ignore hedging demands and simply chose portfolios to be myoptically optimal

• For all t choose portfolio weights so that

ttttt

wwHwwMax

t11 '

2'

13

Measure of Success

• If one dollar is invested, what should expected utility be?

• A good model for mean and variance should achieve a higher level of realized utility

2

11

1

ˆ2

ˆˆ

2

1

rU

rVrEEU

rA

T

tt

T

tt

T

ttT

14

Active Portfolio Strategies

• Typically involve efforts to estimate expected returns

• Covariance matrix is taken as constant • Constraints imposed on short positions or

ranges of asset shares• Tilts toward some factor may be done as an

overlay• Rebalancing is periodic

15

Volatility Based Asset Allocation

• Keep expected returns constant to isolate the effects of volatility

• Volatilities can be estimated at high frequencies so this gives potential for fast acting asset allocation

• Risk management can be implemented in this way so that the trade-off between risk and return is explicitly considered

16

Results for Stock Bond Problem

• Use Multivariate GARCH model of component form

• Rebalance every day in response to new volatility and correlation information

• Keep expected returns equal to their realized values to isolate volatility effects

17

50

100

150

200250

0.5

1.0

1.52.0

500 1000 1500 2000 2500 3000 3500

SVOL SBCOR1 WMU6

18

RETURNS FROM DAILY ACTIVE ASSET ALLOCATION

RETURN RMU4 RMU6 RMU8 RMV R6040 RFIXMVMean 0.0537 0.0481 0.0454 0.0370 0.0400 0.0342Max 5.05 4.66 4.46 3.93 6.66 4.92Min -8.08 -5.85 -4.73 -3.78 -14.23 -6.85

Std.Dev 0.872 0.753 0.706 0.643 0.745 0.672Skew -0.24 -0.11 -0.05 0.04 -1.86 -0.20

Kurtosis 6.8 6.4 6.3 6.1 43.5 8.6UBAR6 1.03 1.04 1.01 .82 .69 .78

Where UBAR6= average realized utility for risk aversion 6, U=200

62rt rt when portfolio

returns, r, are in daily percentages.

19

Strategic Rebalancing

• To reduce transaction costs and trading, only rebalance when the expected gain exceeds a threshold

• Rebalance on day t if Expected Utility at t+1 with optimal weights exceeds Expected Utility with existing weights by more than a fixed number

20

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

500 1000 1500 2000 2500 3000 3500

WMU8 W0MU8