Lesson 6 · Portfolio ManagementCFA L1

International Portfolio Management

Going global widens the opportunity set — low cross-border correlations push the efficient frontier outward — but it adds exchange-rate risk. This lesson covers the diversification gain (and why investors under-use it via home bias), exchange-rate arithmetic and triangular arbitrage, how to put a foreign asset's return and risk into your home currency, and how forwards and the interest-rate parities tie spot rates, forward rates and the carry trade together.

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International diversification Home bias Exchange rates Cross rates & arbitrage Foreign asset return Forwards & covered parity Carry trade & uncovered parity

Investors naturally focus on their home market, but assets from all countries can enter the same optimisation. Adding international assets can never hurt (at worst they are ignored), and usually helps — reducing risk, lifting the efficient frontier and improving the Sharpe ratio.

Learning outcomes

Explain the diversification benefit of international assets and the home-bias anomaly.
Perform exchange-rate, cross-rate and triangular-arbitrage calculations.
Compute the return and risk of a foreign asset in the accounting currency.
Apply covered interest rate parity to price a forward exchange rate.
Analyse the carry trade and state uncovered interest rate parity.

International diversification

For an equally-weighted portfolio of $n$ assets with common variance $\sigma^2$ and average pairwise covariance $c$:

Risk of a large equally-weighted portfolio

$$ \operatorname{var}(r_p)=\frac{\sigma^2-c}{n}+c\;\xrightarrow[n\to\infty]{}\;c $$

The firm-specific term vanishes with size, leaving the average covariance $c$ as the residual systematic risk — so the lower the average covariance, the greater the risk reduction. Equity correlations are high within a region (North America, Europe, Asia) but low across regions (though they rise in market distress), so a globally diversified pool has a lower average covariance — pushing out the efficient frontier and creating a steeper, better capital allocation line than a domestic-only investor could reach.

Home bias

A persistent anomaly

Despite the theory, investors heavily overweight domestic assets — the home-bias anomaly. Candidate explanations: exchange-rate risk, ambiguity aversion (unfamiliar foreign stocks), language and cultural barriers, trading costs and frictions, and regulation on foreign holdings.

Exchange rates

An exchange rate is the price of one currency in another. We use the "real-life" convention (banks, traders, media): USD/JPY = 105 means 1 USD = 105 JPY — read the slash as division, so the reciprocal gives the other side: $\text{JPY/USD}=1/105=0.009524$. (Caution: many textbooks and the CFA exam use the opposite convention.) If USD/JPY rises, the USD appreciates against the JPY (and JPY depreciates).

Worked example

Appreciation is not symmetric

USD/JPY rises 5% from 105 to 110.25. By how much does the JPY depreciate against the USD?

Show solution

Before: 1 JPY $=1/105=0.009524$ USD. After: 1 JPY $=1/110.25=0.009070$ USD. Change $=\frac{0.009070-0.009524}{0.009524}=-4.77\%$. A 5% USD appreciation is a 4.77% JPY depreciation — the two percentages differ because they are measured from different bases.

JPY depreciates 4.77% (not 5%)

Cross rates and triangular arbitrage

Treating rates as fractions gives the cross-rate shortcut: $\text{A/C}=\text{A/B}\times\text{B/C}$. E.g. with EUR/CHF = 1.086 and USD/CHF = 0.897, $\text{EUR/USD}=1.086/0.897=1.211$. Any market quote that breaks this relation creates triangular arbitrage.

Worked example

Triangular arbitrage

Fair EUR/USD is 1.211, but a dealer quotes 1.3 (with EUR/CHF 1.086, USD/CHF 0.897). Exploit it.

Show solution

EUR is overpriced vs USD. Borrow 1 EUR → sell for 1.3 USD → convert to CHF ($1.3\times0.897=1.1661$) → convert to EUR ($1.1661\times1.086=1.074$) → repay the 1 EUR. Net profit 0.074 EUR, riskless. Such trades are how the no-arbitrage relation $\text{A/C}=\text{A/B}\times\text{B/C}$ is enforced.

Riskless profit of 0.074 EUR per cycle

Foreign asset return and exchange-rate risk

A foreign asset's native return is not what a home investor earns — the conversion back through the exchange rate matters. With native return $r^A$ and exchange-rate change $r_{A/B}$, the return in the accounting currency $B$ is:

Foreign asset return, mean and variance

$$ r^B=(1+r^A)(1+r_{A/B})-1\approx r^A+r_{A/B} $$

$$ E(r^B)=E(r^A)+E(r_{A/B}), \quad \operatorname{var}(r^B)=\operatorname{var}(r^A)+\operatorname{var}(r_{A/B})+2\operatorname{Cov}(r^A,r_{A/B}) $$

Worked example

A UK equity fund for a US investor

US riskfree 1%; UK fund (native GBP) $\mu=3\%,\sigma=30\%$; GBP/USD $\mu=4\%,\sigma=7\%$; correlation 0.1. Find the fund's USD return/risk, then a 70%-fund/30%-riskfree portfolio.

Show solution

$E(r^{USD})=3\%+4\%=7\%$. $\operatorname{var}=0.3^2+0.07^2+2(0.3)(0.07)(0.1)=0.0991\Rightarrow\sigma=31.48\%$. Portfolio: $\mu_p=0.7(7\%)+0.3(1\%)=5.2\%$; $\sigma_p=0.7\times31.48\%=22.04\%$. The currency contributes both extra return and extra risk.

Fund in USD: 7%, 31.48%; portfolio: 5.2%, 22.04%

Forward rates and covered interest rate parity

A forward exchange rate locks in a future conversion rate today; the forward premium = forward − spot (negative = discount), often quoted in basis points. No-arbitrage pins the forward to the two interest rates — covered interest rate parity (CIP):

Covered interest rate parity

$$ F_{A/B}=X_{A/B}\cdot\frac{1+i_B\frac{\tau}{360}}{1+i_A\frac{\tau}{360}} $$

Worked example

A 6-month GBP/EUR forward

$i_{GBP}=0.1\%$, $i_{EUR}=0\%$, spot GBP/EUR = 1.1563, $\tau=180$. Find the forward and its premium in basis points.

Show solution

$F=1.1563\times\dfrac{1+0\%(180/360)}{1+0.1\%(180/360)}=1.155722$. Premium $=1.155722-1.1563=-0.000578$, i.e. $-0.000578/0.0001=\textbf{−5.78 bp}$ — a forward discount, because GBP carries the higher interest rate. (Higher-yield currencies trade at a forward discount.)

Forward 1.15572; −5.78 bp (a discount)

The carry trade and uncovered interest rate parity

Interest rates differ hugely across currencies (Japan vs Australia), motivating the carry trade: borrow the low-rate currency (JPY), convert, and deposit in the high-rate currency (AUD) to harvest the spread. It is not riskfree — the future exchange rate is unknown:

Carry payoff and uncovered parity

$$ \text{payoff}=\frac{X_1^{AUD/JPY}}{X_0^{AUD/JPY}}(1+i_{AUD})-(1+i_{JPY}) $$

$$ \text{UIP: } E\!\big(X_1^{AUD/JPY}\big)=\frac{1+i_{JPY}}{1+i_{AUD}}\,X_0^{AUD/JPY} $$

If investors were risk-neutral, they would bid the trade until its expected payoff is zero — giving uncovered interest rate parity (UIP): the high-rate currency is expected to depreciate by just enough to wipe out the carry. Combining UIP with CIP gives the unbiased forward hypothesis: $F=E(X_1)$ — the forward rate is an unbiased forecast of the future spot.

Does the carry trade actually work?

UIP says it shouldn't — but UIP assumes risk-neutrality. Empirically, carry trades have been somewhat profitable, yet with excess kurtosis and substantial crash risk (fat-tailed returns). The honest question is not "is it profitable?" but "is the reward justified by the risk?"

Key takeaway

International assets lower the average covariance, improving the frontier — yet investors cling to home bias. Once you account for currency, a foreign return is $r^B\approx r^A+r_{A/B}$, so the exchange rate adds both return and risk. Forwards let you hedge it, priced by covered interest rate parity (the higher-yield currency trades at a forward discount), and uncovered interest rate parity says the expected spot move should erase any interest-rate edge — the theoretical reason the carry trade "shouldn't" work, even though in practice it pays a premium for bearing crash risk.

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