L24 & L25: PORTFOLIO BALANCE

• Motivation– How can we allow for effects of risk?

• Currency risk (Lecture 24).• Country risk (Lecture 25).

• Key parameters:– Risk-aversion, ρ– Variance of returns, V– Covariances among returns, Cov.

Each investor at time t allocates sharesof his or her portfolio to a menu of assets,

as a function of expected return & risk:

Sum across investors i to get the aggregate demand for assets, which must equal supply in the market.

Invert the function to determine what Etrt+1 must be, for supplies xt to be willingly held.

xi, t = βi (Et rt+1 , risk ) .

The general portfolio balance case: Tobin (1958, 1969)

lots of assets (M, Bonds, Equities), with different attributes & lots of investors with different preferences.But we will focus more on one-period bonds, andassume uniform preferences among relevant investors.

Lecture 24 assumption (most relevant for rich countries):exchange risk is the important risk.

Lecture 25 assumption (most relevant for developing countries): default risk is important.

Portfolio Diversification

Motivating questions for Portfolio Balance Model:

Starting point: Most investors care not just about expected returns, but also about risk. => rp ≠ 0 => UIP fails.

« How do we think about effects of:• Current account deficits, • Budget deficits, and• (sterilized) forex intervention, which had no effects in monetary models?

« What determines the risk premium? How large is it?

« How can we bring more information to bear on the structure of investors’ asset demands?

« How should you manage a portfolio, e.g., a Sovereign Wealth Fund?

Open-Economy Portfolio Balance Model

Demand for foreign bonds by investor i: x i, t = Ai + Bi Et (r f

t+1 – r dt+1) ;

where x is the share of the portfolio allocated to foreign assets, vs. domestic.

« For this lecture, assume foreign assets all denominated in $ (and/or €, ¥, etc.), and domestic assets all denominated in dirham (domestic currency);

Then portfolio share xi ≡ SFi / Wi ,

« Assume, for now, no default risk. Then expected real return differential = exchange risk premium rpt ≡ i

$t – i

d t + Et ∆s t+1

where Wi ≡ Di + S Fi ≡ total wealth held;

Di ≡ domestic assets held, Fi ≡ foreign assets held, and S = exchange rate.

So x i, t = Ai + Bi rpt .

Financial market equilibrium: assets held = assets supplied….

« where aggregate portfolio share xt ≡ St Ft / Wt ,

« W ≡ D + SF ≡ total wealth held,

« F ≡ total foreign ($) assets held, &« D ≡ total domestic assets held.

Sum asset demands across all investors in the marketplace:

Total demand for foreign assets ≡ xt ≡ Σ [ x i, t ]= Σ [Ai + Bi rpt ]

xt = A + B rpt

For now assume investors to have identical parameters Ai=A and Bi=B:

« In general, x foreigners > x local residents (Home bias).

dtountCurrentAccFt

t

)(

dtcitBudgetDefiDt

t

)(

How do asset supplies get into the market?

« Domestic debt is issued by the government:

In extreme “small-country case,” xforeigners = 1 => only local residents’ holdings are relevant.

Then aggregate supply of foreign assets given by:

Note: forex intervention, even if sterilized, would subtract from D & add to F.

Now invert: rp t = B-1 x t - B-1 A . We see that asset supplies are a determinant of the risk premium.

To repeat, xt = A + B rpt .

A large x forces up the expected return that portfolio holders must be paid.

Special case :

| B-1 | = 0 =>

• perfect

substitutability ( |B|=∞ ),

• no risk premium (rpt = 0).

Now assume investors diversify optimally

Tobin: “Don’t put all your eggs in one basket.”

“MANAGEMENT OF COMMODITY REVENUES – BOTSWANA’S CASE” by Linah Mohohlo, Governor, Bank of Botswana

Allocation of Portfolio between Bonds & Equity in the Pula Fund

very safe very risky½ & ½

Efficient Frontier: Allocation of Portolio between Bonds (“Fixed Income”) & Equity

“MANAGEMENT OF COMMODITY REVENUES – BOTSWANA’S CASE” by Linah Mohohlo, Governor, Bank of Botswana

very safe

very risky

½ & ½

Optimally Diversified Portfolios

xt = A + B rpt

Under certain assumptions,

=> same problem as to maximize Φ [E(W+1), V(W+1)], Φ1>0, Φ2<0.

End-of-period wealth W+1

)])(()1[(1

$

11

dd rrxrW

)]()()1[(1

$

111

dd rrExErWEW ),(2)()()( 1

$111

$1

21

21

dddd rrrxCovrrVxrVWWV

Problem: Choose xt to maximize Et [ U (Wt+1 ) ]

)1)(1()1( 1$1

drxWrWx

[

= Minimum-variance + Speculative portfolio portfolio

Optimal diversification

)(1

$

11

drrWE 0)],()([2 1

$

111

$

1

2

2 ddd rrrCovxrrVW

Define

, RRA ≡ , W21

2

& V V( r$+1 – rd

+1).

)(1

$

1

drrErp

)],([1

$

11

dd rrrCovVxrp

This matches

for the optimal-diversification case B-1 = ρ V

and .

ABxBrp 11

),(1

$

11

1 dd rrrCovVA

dx

dV

dx

dE

dx

d ()()21

}

First-order condition: = 0 .

Then .

For example, if goods prices are non-stochasticand s+1 is the only source of uncertainty,

then V = Var(s+1)

Also, depending how rp is defined, rp may differ from i - i* - Es by a convexity term = (α – ½) V .

(See resolution of Siegal paradox, in latter part of Addendum to the forward bias lecture.)

and A = α , the share of foreign goods in consumption basket.E.g., if all consumption is domestic (A=α = 0), domestic bonds are safe; very risk-averse investors do not venture abroad (because Cov (rd, r$-rd ) = 0).

A is the minimum-variance portfolio(in x = A + [ρV]

-1 rp):

It’s what an investor holds if risk-aversion ρ = ∞.

Equities: Whatever is risk-aversion ρ , the optimal portfolio allocatesa substantial share abroad, because the min-variance portfolio does.

Who holds what portfolio?

A foolishly under-diversified American

The most risk-averse

Moderately risk-averseVery risk-tolerant

● x=0

x=.3 ●

x=.75 ●

x ≥ 1.0 ●

But in practice: Home bias in equity holdings.Most equities are held by domestic residents.

Home bias in equity holdings has slowly declinedthough more so among Advanced Markets than EMs

Appendix: Beyond the small-country modelForeign residents are in the market for domestic vs. foreign assets, alongside home residents, with weights wH vs. wF.

Now aggregate: .ii

i AwBxBrp 11

A difference in consumption preferences, H < F, for home vs. foreign residents => some preference for local assets, AH < AF (home bias).

If the domestic country runs a CA surplus=> Its share of world wealth, wH, rises over time, and foreigners’ share falls. => Domestic preference, AH , receives increasing weight in total global demand. => Global demand for domestic assets rises. => Required expected return falls.

In practice,

Americans hold most of their portfolio in US securities,

Japanese hold theirs in Japanese securities,

Etc.

Home Bias in International Allocation of Equity Portfolios

Further evidence on home bias in USEquity shares are from 1997 comprehensive survey of US residents’ holdings of foreign securities.

Bias column ≡ 1 minus (foreign equity share / world market share). If US investors held foreign securities in proportions equal to those in the world equity market benchmark, bias would = 0.

From: G.Baekert & R.Hodrick, Intl. Fin.Management, 2004, Table 14.16, Panel A Source: Based on Table 1, in Ahearne, Griever & Warnock (2002).

Country Share in U.S. Bias Equity Portfolio

Spain 0.21 0.83Australia 0.26 0.79Hong Kong 0.23 0.87Mexico 0.29 0.56Brazil 0.26 0.76India 0.05 0.91China 0.02 0.98Taiwan 0.04 0.97Russia 0.07 0.87South Africa 0.08 0.92

Country Share in U.S. Bias Equity Portfolio US 89.90UK 1.82 0.79Japan 1.14 0.88France 0.71 0.75Canada 0.59 0.75Germany 0.54 0.85Italy 0.35 0.76Netherlands 0.89 0.55Switzerland 0.52 0.79Sweden 0.32 0.72

International diversification has risenProportion of foreign bonds + equities in total equity +

bond portfolios of residents in the reported countries. From a 2002 UBS Asset Management study. Source: Baekert & Hodrick, op.cit., Table 14.16, Panel B

1991 2000 US 4% 11%

Japan 12% 27%The Netherlands 12% 62%UK 23% 26%Switzerland 11% 21%Australia 14% 19%Sweden 4% 25%

…though since the 2008 GFC, we have seen some signs of a return to home bias.