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Transcript of Lectures 2 & 3: Portfolio Balance - Harvard University 119/API-119 slides/L2... · Lectures 2 & 3:...
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.
“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.
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.