ASSET MANAGEMENT PART I:

STRATEGIC ASSET ALLOCATIONMartijn Boons - Nova SBE

TODAY

Strategic asset allocation: How to allocate wealth

optimally across broad asset classes?

What is an asset class?

Intuition from answer to How much to invest in

risky versus riskless asset? easily generalizes

Asset allocation for short investment horizons Risky: aggregate stock market portfolio, e.g., S&P500

Riskless: short-term Treasury bill

Asset allocation for long investment horizons

Advantages of being a long-term investor

Horizon

Buy and hold versus rebalancing

When returns are not versus are predictable

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RECALL FROM INVESTMENTS

Investors should control the risk (= variance) of their portfolio not

by re-allocating among risky assets, but through the split between

risky and risk-free

Optimal portfolio of risky assets: market portfolio

Held by the aggregate market, and in CAPM equilibrium

optimal for all investors

If not exactly optimal, at least well-diversified and attractive

Although theory suggests market portfolio contains all risky

assets, aggregate stock market index usually used as proxy

Uncertain market return equals capital gain + dividend:

=

with () = and () =

Combine with short-term Treasury bills according to risk aversion

(Nominally) Risk-free one-period return is known at time

and equals =,

,

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THE SINGLE-PERIOD PROBLEM

Investor chooses fraction of wealth invested in risky

asset to maximize mean-variance utility over portfolio

return

max(,)

(,), with ,= +(1-)

(,) = +(1 ) and (,) =

Assumptions

Wealth is 1$, to abstract from wealth effects

Ascertains investor has constant relative risk

aversion (CRRA): dollar investment in risky asset

increases in wealth, but the share of wealth

invested in risky asset remains constant

No practical constraints: positivity or no short-sales

(x 0), no leverage (0 x 1)

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SOLUTION

Unconstrained solution from FOC

=

!"

#$

increasing in , and decreasing in % and

Example: = 8%, = 2%, = 20%, and %=3

=1

3

6%

(20%)= 0.50

Traditional portfolio advice: put 50% of wealth in

stocks

Note, Sharpe ratio of optimal portfolio is independent of %:

. =

=( )

The excess return per unit of risk offered by the market

portfolio

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THE CAPITAL ALLOCATION LINE

/01

=2%

Slope=0.3

0.5*20%=

10%

% = 3

2%+0.5*

6%=5%

1. CAL plots expected

return versus standard

deviation of optimal

portfolios

2. Indifference curves

determine which portfolio

choice is optimal for

each% (with utility

increasing in northwest

direction)

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=8%

= 20%

Question: for which % is 100% in the risky asset optimal?

% = 1.5

THE DYNAMIC PORTFOLIO CHOICE

PROBLEM

1. Horizon of one period is unreasonable for most investors

Pension funds (horizon of liabilities 20 years)

Saving for a house, college, new campus etc.

2. Investor may want to change portfolio weights every period

over this horizon

What is a period? Year (individuals), quarter (institutions),

, second (high-frequency traders)?

Why change?

1. Time-varying investment opportunities (e.g.,

predictable returns and volatilities as the investor

passes through economic recessions / expansions),

2. when approaching the horizon,

3. or when risk aversion varies over time.

Main insight from dynamic portfolio choice: optimal

weight depends on horizon or time 3, or both

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SETUP OF THE DYNAMIC PROBLEM (I)

Consider the portfolio choice at time for an investor with horizon at time 4 > + 1

Wealth dynamics: 7 = 7 1 + ,

Wealth varies from to + 1 due to portfolio return, which is a function of the portfolio choice

Suppose T=5

The sequence of weights over time, {}, is called a dynamic trading strategy. Dynamic because weights can change over time.

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SETUP OF THE DYNAMIC PROBLEM (II)

Maximize expected utility of wealth at horizon T by choosing a dynamic trading strategy: max

{9}(;(7

DYNAMIC PROGRAMMING (I)

The solution to this problem is easily found working backwards

Final wealth is the product of current wealth and uncertain one-

period returns: 7> = 1 + , (1 + ,>)

Assuming again current wealth 7 = 1

Given uncertain wealth at t+4, choose portfolio weights that

maximize expected utility at t+5(=T):

max9@

(;(7>)) = max9@

(;(1 + ,>(A)))

Solution to this single-period problem, assuming mean-variance

utility:

A =

1

%

A ,A

A

Conditional moments, as state of economy at t+4 unknown

Indirect utility: the maximum utility obtained at t+4,

A = (;(1 + ,> A )

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DYNAMIC PROGRAMMING (II)

Now, solve the two-period problem: Given wealth at + 3, choose

B and Ato maximize expected utility at t+5 (=T):

max9C,9@

(;(7>))

The optimal strategy conditional on any outcome at + 3 is

already known: A .

Thus, re-write this problem as a one-period problem:

max9C

(;(1 + ,A(B))A)

The first part is identical to what a single-period investor

would do at + 3.

The A-term captures the advantage of being a long-term

investor. The utility derived from this A-term depends on

uncertain wealth and investment opportunities at + 4

Working back to , we are solving such a one-period problem at

each point in time..

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GRAPHICAL REPRESENTATION

Note how different this approach is

from buy and hold, which solves

one five-period problem!

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LESSONS

Dynamic portfolio choice over long horizons is

first and foremost about solving one-period

portfolio choice problems!

This view destroys two widely held

misconceptions:

1. Long-term investors are fundamentally different

from short-term investors

2. Buy-and-hold is optimal

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1. LONG-TERM INVESTING IS NOT SO

DIFFERENT FROM SHORT-TERM INVESTING

Dynamic programming shows that long-run investors do

everything that short-run investors do!

However, long-run investors can do more, because they have the

advantage of a long horizon.

The horizon effect enters through the indirect utility (VFG) in

each one-period problem

For instance, suppose at t you know that stock market returns

will be high from t+1 to t+2. How might this affect the optimal

portfolio choice xF?

Some will invest more risky, because future return can

compensate if returns are low from t to t+1

Some will invest less risky, to ensure that they have

sufficient money to invest when it is most attractive at t+1

Exact solution depends, among other things, on

Covt(,, ,) with mean-variance utility, and smoothing

preferences with other utility functions

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2. A LONG-RUN INVESTOR SHOULD NOT

BUY AND HOLD!

Buy and hold solves a single, long-horizon problem

Special case of dynamic investing where the investors

optimal choice is to do nothing

Thus, dynamic portfolios can do everything buy and

hold portfolios do, but also much more!

In practice, optimal long-horizon investing is not to buy

and hold; long-horizon investing is a continual process of

buying and selling.

Suppose you calculate the optimal weight in stocks for

the next ten years is 50%.

You need to rebalance each period!

If not, over a sufficiently long period of time, you will have

100% in risky assets. That is not what you wanted, right?

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REBALANCE WHEN RETURNS ARE NOT

PREDICTABLE!

Suppose returns are not predictable (i.i.d.) and the risk-free

rate is fixed

Returns are in fact hard to predict!

In this case, the dynamic strategy is a series of identical

one-period strategies

Intuition: we can take A out of the maximization

max9C

(;(1 + ,A(B)))A

Long-run weight (t) = Short-run weight (t) =

!"

#$

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THE CASE FOR REBALANCING

For the two asset case (stock market and risk-free asset),

rebalancing is countercyclical:

Buy (sell) stocks after low (high) returns

Portfolio rebalancing ensures wealth remains to be

allocated optimally (in line with risk preferences) over

time, and is also advantageous if returns are

mean-reverting / predictable: prices drop when

expected future returns increase (more on

predictability later)

Example: Great depression

Investors rebalance infrequently and incompletely

Partial explanations include: inertia, natural tendency

to invest more in assets that do well than in assets

doing poorly, transaction costs (rebalancing bands)17

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COUNTERCYCLICAL REBALANCING IN THE

GREAT DEPRESSIONA

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With rebalancing: sell some stocks before they are hit hard in 1930,

and buy some stocks before they rebound in 1932

Reduces variance

and increases returns

Similar evidence obtains in recent financial crisis!

THE DIVERSIFICATION RETURN (SEE ERB

AND HARVEY, 2006)

Portfolio diversification decreases portfolio variance

without reducing portfolio arithmetic return

This benefit is obtained in a single period, but dies out if

you do not rebalance (weight in risky asset 100%)

However, what is less well understood is that

diversification does increase portfolio geometric

return

This diversification return exists for a long-term investor

and is collected by rebalancing (also known as rebalancing

return or variance reduction)

Which two consecutive returns do you prefer: (90%,-50%) or

(10%,-10%)?

Excel example

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OPPORTUNISTIC STRATEGIES WHEN

RETURNS ARE PREDICTABLE (I)

If returns are predictable (not i.i.d.): additional benefits

from long-term horizon

Long-term weight (t) =

1. Long-run myopic weight +

2. (Short-run weight (t) Long-run myopic weight) +

3. Opportunistic weight (t)

Strategic asset allocation is the sum of these three

components!

1. Long-run fixed weight determined by long-run

average return and volatility 1

%

I

This is the constant rebalancing weight in the i.i.d. case!

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OPPORTUNISTIC STRATEGIES WHEN

RETURNS ARE PREDICTABLE (II)

2. Tactical asset allocation: the response of both

short- and long-run investors to changing means

and volatilities

!",

#$

!I"

#$, where () = , () =

If market Sharpe ratio is temporarily high: both

short-and long-term investors can benefit.

3. Captures how long-term investor can take

advantage of time-varying, predictable returns in

ways short-run investors cannot.

The knowledge that market Sharpe ratio is going

to be high in the future: only the long-term

investor can benefit. 21

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CHARACTERIZING THE OPPORTUNISTIC

WEIGHT

Difficult, but two broad determinants

1. Investor-specific: risk tolerance (like in one-period portfolio) and horizon

2. Asset-specific: how do returns vary over time?

Interaction between horizon and time-variation is crucial:

An asset with low returns (high volatility) today, but high returns (low volatility) in the (long-run) future is not attractive for short-term investors, but long-term investors might want to invest in them.

We can obtain more insight into the opportunistic weight (or intertemporal hedge demand, as it was coined in Merton (1973)), thinking about optimal buy-and-hold portfolios as in Campbell and Viceira (CV, 1999, 2002, 2005)

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THE CV-APPROACH

CV set out to find the optimal portfolio choice for buy-and-

hold investors with investment horizon K

If returns are predictable, long-term portfolio choice is

(among other things) determined by the conditional

expectation and conditional variance of the risky assets

returns over the investors horizon K

More formally, (L) ((

(L)), (

L)), where (

(L)) is

the average expected per period return over horizon K

((L) defined analogously)

Empirically, forecasts of returns and variances follow from

a regression of stock and t-bill returns on a set of predictors

(e.g., the dividend yield)

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OPTIMAL BUY-AND-HOLD PORTFOLIOS (I)

To get the necessary intuition, let us think about a

case where ,is time-varying, but the market risk

premiumM = , is constant

Optimal investment in risky asset for investors with a

K-period horizon are of the following approximate

form:

(L)

1

%

(M L

)

(M L

)+ (

1

% 1)

/OP(M L

, ,L)

(M L

)

where /OP(M L

, ,L

) is the average per period

covariance of risky assets return with risk-free return

over horizon K

1. Myopic demand of a K-period investor (as before)

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OPTIMAL BUY-AND-HOLD PORTFOLIOS (II)

2. Intertemporal hedge demand defined over

/OP(M L

, ,L

)

Suppose the covariance is positive: on average, excess

market returns are high whenever expected future returns

on both market and t-bills are high (,L)

Risk averse investor (%>1) will under-weight market portfolio

(relative to myopic demand), because it pays off exactly when

he does not need the money, i.e., when the future is bright

Risk loving investor (%

OPPORTUNISTIC HEDGING DEMANDS IN

PRACTICE

Although elegant and intuitive, the optimal size of

hedging demands is debated heavily

CV estimate is large: risk-averse, long-run investor

over-weights stocks dramatically. Why?

Mean-reversion captured by the fact that dividend

yield predicts stock returns with a positive sign in-

sample

When future expected stock returns increase,

current prices decrease and dividend yield

increases (/OP(M L

, ,L

)

CV: STOCKS ARE LESS RISKY IN THE LONG

RUN

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ON MEAN REVERSAL AND PREDICTABILITY

IN STOCK RETURNS

Evidence in CV consistent with in-sample stock return

predictability by a range of variables:

Cash-flow based (dividend yield, earnings yield); Business

cycle indicators (term spread); Technical (momentum)

Recent research questions extent and exploitability of this

predictability out-of-sample, i.e., for an investor that is making

his investment decisions in real-time

Goyal and Welch (2008): coefficients unstable in sub-samples;

in-sample R2 small; out-of-sample R2 tiny

State-of-the-Art recommendation by Ang

the evidence for predictability is weak, so I recommend that both

the tactical and opportunistic portfolio weights be small in practice.

Opportunistic hedging demands become much smaller once investors

have to learn about return predictability or when they take into

account estimation error.

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EXTENSION: LIABILITY HEDGING

Few investors are without liabilities.

With liabilities the optimal long-run portfolio advice becomes:

First, meet the liabilities and then invest wealth, in excess of the

present value of liabilities, as before

Long-run weight (t) = Liability hedge (t) + Short-run weight

(t) + Opportunistic weight (t)

Liability hedge: portfolio that best ensures investor meets

liabilities

Invests in assets with large covariance with liabilities

Intuition: if stock market return is high when liabilities

increase in value, it is attractive as a hedge. The more risk

averse the investor (%), the more weight he will place on

liability hedging.

Q

!",

#$ (

1)

RST(","U,)

V(")

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EXAMPLES OF LIABILITY HEDGE

PORTFOLIOS

1. Cash flow matching: Match a portfolio of fixed, future outflows

with bonds of appropriate maturities

2. Duration matching: Match duration of interest rate bearing

liabilities with a portfolio of Treasury bonds

3. Assetliability matching: Match liability characteristics besides

just duration

For most pension funds

horizon exceeds longest available maturity of liquidly

traded Treasury bonds

liabilities denominated in real, not nominal, terms, whereas

real, long-maturity bonds have only been traded in recent

years, but not in every country

Additional risks that need to be matched: liquidity risk,

longevity risk, economic growth, and credit risk.

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THREE-WAY SEPARATION IS POPULAR

Practitioners frameworks usually consist of three buckets

1. Protective portfolio, which covers personal risk. The portfolio is designed to minimize downside risk and is a form of safety first.

2. Market portfolio, which is a balance of risk and return to attain market-level performance from a broadly diversified portfolio and is exposed to market risk.

3. Aspirational portfolio, which is designed to take measured risk to achieve significant return enhancement. Aspirational risk is a property of an investors utility function and is a desire to grow wealth opportunistically to reach the next desired wealth target.

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CONCLUSIONS

Rebalancing is the foundation of any long-term

investment strategy!

Under i.i.d. returns the optimal policy is to

rebalance to constant weights for both short- and

long-term investors.

When returns are predictable, the optimal short-

run portfolio changes over time, and the long-run

investor has additional opportunistic strategies

Liabilities need to be adequately matched before

the investment portfolio is constructed.

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LITERATURE

Books

Ang, Asset Management, Ch. 2-4

Campbell and Viceira, Strategic Asset Allocation, Ch. 2-3

Articles

Erb and Harvey, 2006, The Tactical and Strategic Value of

Commodity Futures, Financial Analysts Journal.

Merton, 1973, An Intertemporal Capital Asset Pricing Model,

Econometrica

Campbell and Viceira, 2005, The term structure of the risk-return

trade-off, NBER Working Paper

Campbell and Viceira, 2005, The term structure of the risk-return

trade-off, Financial Analysts Journal

Goyal and Welch, 2008, A Comprehensive Look at The Empirical

Performance of Equity Premium Prediction, Review of Financial

Studies

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