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

Why being a long-term investor matters?

Time-varying investment opportunities

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 (Capital Market Line)

Even 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 T-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 ascertain investor has constant

relative risk aversion (CRRA): share of wealth in

risky asset is constant, but dollar amount is

increasing in wealth

No practical constraints: no short-sales (x 0), no

leverage (0 x 1)

More relevant for N>2

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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 :

0123 =

=( )

=

The excess return per unit of risk offered by the market

portfolio

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THE CAPITAL MARKET LINE

456

=2%

Slope=0.3

0.5*20%=

10%

= 3

2%+0.5*

6%=5%

1. CML 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 house, college, new campus etc.

2. Investor can 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),

2. when approaching the horizon,

3. time-varying risk preferences (not today).

Main insight from dynamic portfolio choice: optimal

weight depends on horizon or time 8, 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 9 > + 1

Wealth dynamics:

SETUP OF THE DYNAMIC PROBLEM (II)

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

{}(?(

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:

DYNAMIC PROGRAMMING (II)

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

choose F and Eto maximize expected utility at t+5 (=T):

maxG,D

(?(

GRAPHICAL REPRESENTATION

Working back to , we are

solving such a one-period

problem at each point in time..

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 clashes with two widely held

convictions:

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 (VJK) in

each one-period problem

For instance, suppose at t you predict that stock market

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

optimal portfolio choice xJ?

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(,, ,) and intertemporal 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 ( = , = , , = for all t)

Aggregate market 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 E out of the maximizationmaxG

(?(1 + ,E(F)))E

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)

Intuition from DDM (price P when constant future

dividends are discounted at rate R):

P =

%

Example: Great depression 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

Excel example: 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 geometricreturn This diversification return exists for a long-term investor

and is collected by rebalancing

Also known as variance reduction return:

geometric mean arithmetic mean *variance

Which two consecutive returns do you prefer?

90%,-50% or

10%,-10%

Theory and exact formulas in 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 besides rebalancing

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

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

!

$%,

&'

!

$N%

&',

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.

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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, though.

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BUY AND HOLD PORTFOLIOS

We have made a case for trading each period to rebalance, but

ample evidence that 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)

Average holding period of a stock in institutional portfolios is

3.5 years (Lan, Moneta, Wermers (2015))!

Investors might trade a small part of their portfolio often

(rebalancing), but infrequently trade a much larger part

Pension funds, Warren Buffett

Therefore, it is not meaningless to think about the question: what

is optimal portfolio over a multi-period horizon, without

rebalancing?

Campbell and Viceira (CV, 1999, 2002, 2005) answer this

question in a series of papers.

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THE CV-APPROACH

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

hold investors with investment horizon P

Their solution method is beyond the scope of this course,

but the intuition is very relevant!

If returns are predictable, long-term portfolio choice is

(among other things) determined by the conditional

expectation and conditional variance of the risky assets log returns over the investors horizon P

More formally, (Q) ((

(Q)), (

Q)), where (

(Q))

is the vector of average expected per period returns over

horizon P (the covariance matrix (Q) is defined

analogously)

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OPTIMAL BUY-AND-HOLD PORTFOLIOS (I)

Empirically, forecasts of returns and variances follow from

a regression of stock market and T-bill returns on a set of

predictors (e.g., dividend yield, term spread, default

spread)

Innovations in the predictors measure news about

future returns

Optimal investment in market for investors with a K-

period horizon are of the following approximate form:

(Q)

1

(S Q

)

(S Q

)+ (

1

1)

4TU(S Q

, V3WXQ)

(S Q

)

where 4TU . measures the covariance of excess market

returns with the news about future returns (on the T-bill and

thus also the market) over horizon K

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OPTIMAL BUY-AND-HOLD PORTFOLIOS (II)

Two parts:

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

2. Intertemporal hedge demand

Suppose 4TU(S Q

, V3WXQ)>0: excess market returns are

high when news about future returns is positive

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

Risk loving investor (

OPPORTUNISTIC HEDGING DEMANDS IN

PRACTICE

Although the many extensions and applications covered in

CV are 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

DDM: When future expected stock returns increase,

current prices decrease and dividend yield increases

(r , P , D/P )

In this way, current stock returns will be a hedge

against future stock returns

Example: the Tangency portfolio for stocks and a 5-year

bond (CV 2005)

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CV: STOCKS ARE LESS RISKY IN THE LONG

RUN

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Investment in stocks is increasing in horizon. Investors with a

10-year horizon will even want to short bonds to invest >100% in

stocks!

This seems rather extreme

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 R-squared small; out-of-sample R-squared 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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CONCLUSIONS

Rebalancing is the foundation of any long-term investment strategy! Interesting piece on the stabilizing role of rebalancing in

financial markets: FT article 1

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

Practical strategic asset allocation advice: pick reasonable weights on a few easily investible asset classes and rebalance. E.g., 50% in S&P500 index tracker and 50% in long-term

government bond performs very well compared to many industry alternatives that are far more costly to implement (see, e.g., FT article 2)

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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

FT1: http://www.ft.com/intl/cms/s/0/5b349240-6817-11e5-97d0-

1456a776a4f5.html#axzz3xDaDdWFt

FT2: http://www.ft.com/intl/cms/s/0/73ba77b2-c1dc-11e4-bd24-

00144feab7de.html#ixzz3TXVB7qFy31

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