Lectures 2 & 3: Portfolio Balance

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

• Currency risk (Lecture 2).

• Country risk (Lecture 3).

• Key parameters: – Risk-aversion, ρ

– Variance of returns, V

– Covariances among returns, Cov.

Each investor at time t allocates shares of 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, and assume uniform preferences among relevant investors.

Lecture 2 assumption (most relevant for rich countries): exchange risk is the important risk.

We will also consider risk in equity markets.

Lecture 3 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 ft+1 – r dt+1) ;

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

« For now, assume foreign assets all denominated in $ (and/or €, ¥, etc.),

and domestic assets all denominated in dirham (domestic currency);

Then portfolio share xi ≡ S Fi / 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).

dtountCurrentAccF

t

t

)(

dtcitBudgetDefiD

t

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 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 risky ½ & 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

Certain assumptions => same problem as Mean-Variance optimization:

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

2

1dddd 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)],()([21

$

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-stochastic and 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 .

(if s+1 is distributed normally, as in the resolution of the Siegal paradox mentioned in an appendix 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 allocates a substantial share abroad, because the min-variance portfolio does.

A foolishly under- diversified American

The most risk-averse

Moderately risk-averse Very risk-tolerant

● x=0

x=.3 ●

x=.75 ●

x ≥ 1.0 ●

Who holds what portfolio?

Appendix: Home bias in portfolio holdings

Macroeconomic Policy Analysis II, Professor Jeffrey Frankel,

• In practice, investors in each country hold relatively more of their own country’s stocks & bonds than the optimal-diversification model seems to say they should.

• Statistics show that home bias, though high, is declining slowly.

• Implications for the portfolio balance model?

• The “small-country” model assumes extreme home bias:

• Foreigners hold none of the domestic country’s assets.

• Most finance models go to the opposite extreme:

• all investors have the same portfolio preferences.

• The realistic case, e.g., the 2-country model, assumes foreigners have a relatively greater preference for their own assets than do domestic residents.

In practice, most equities are held by domestic residents but this “home bias” is slowly declining.

Home bias in equity holdings has slowly declined

The 2-country portfolio-balance model

Foreign residents are in the market for domestic vs. foreign assets, alongside home residents, with weights wH vs. wF.

Now aggregate: . i

i

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.