Portfolio Management puzzles, solved step by step

About 19.9% at plus 0.3 and 15.6% at minus 0.3. The unhedged return is roughly the local return plus the currency return, so the variances add with a correlation term: 16 squared plus 8 squared, plus or minus 2 x 0.3 x 16 x 8. That is 396.8 or 243.2, with square roots of 19.9% and 15.6%. With negative correlation the unhedged position is less volatile than the hedged one at 16%.

How can adding a second risk reduce the total?

Think of a shop that sells umbrellas and sunglasses. Each product's sales swing a lot with the weather, but in opposite directions, so the till is steadier than either product alone. When two sources of return tend to move against each other, combining them lowers risk even though each one is volatile on its own. For an Indian investor holding foreign shares, the currency is the second source. If the foreign currency tends to strengthen against the rupee when that market falls, it cushions the loss, and the unhedged position is steadier: 15.6% instead of 16%.

So should a global portfolio hedge its currency?

The puzzle gives the risk side of the answer, not the whole decision. A hedge removes the currency's volatility, but it also removes the currency's correlation with the market, and when that correlation is negative the hedge adds risk rather than cutting it. The rest of the decision is cost, which depends on the interest rate gap between the two currencies, and the investor's own liabilities in rupees. Say the limitation: correlations are measured on history and tend to shift in a crisis, so the minus 0.3 that makes the unhedged position look safer is the number least likely to hold when it matters. The formula also ignores the small cross term from multiplying the two returns.

About 112 rupees to the dollar. Relative PPP says the higher-inflation currency loses value at the inflation gap. Each year the rate rises by 1.05 over 1.02, about 2.94%. Over ten years that compounds to a factor of 1.336, and 84 times that is about 112.2. The simple 3% shortcut gives 112.9, close enough in the room.

Why should higher inflation weaken a currency?

If a cup of tea costs Rs 20 this year and Rs 21 next year while the same cup abroad goes from 1 dollar to 1.02, the rupee has lost more buying power than the dollar. If money is to buy roughly the same basket in both countries over time, the exchange rate has to move by the gap between the two inflation rates. That is relative purchasing power parity: it does not say what the rate should be today, only how it should drift.

What does this mean for an investor holding assets abroad?

For a rupee-based investor, a US asset earns its local return plus the rupee's depreciation. A US bond yielding 3 points less than an Indian bond is not worse on this reasoning, because the expected currency drift gives back roughly the same gap. Measured the other way, the rupee loses about 25% of its dollar value over the ten years, which is what a dollar-based investor in Indian assets must earn back through higher local returns.

Say the limitation plainly. PPP is a weak guide over one or two years: capital flows, interest rate moves and risk appetite can push the rate far from this path and hold it there. Over a decade the evidence for the drift is better, but it is a tendency, not a rule. The inflation rates themselves are assumptions; for a real view, confirm current figures and consider what each central bank is targeting.

About 6.1%. On 100 of borrowed yen, the rupee investment grows to 107 and the loan grows to 100.5. The trade breaks even if 107 rupee-units convert back to exactly 100.5 yen-units, which means a fall of 1 minus 100.5 over 107, or 6.07%. It is a little less than the 6.5-point rate gap because the fall also eats into the interest earned.

What is the trade actually betting on?

Picture borrowing from a relative who charges almost nothing and lending the money to a friend who pays well, except the friend repays in a different currency. You pocket the gap in rates as long as the friend's currency holds its value. A carry trade earns the interest rate gap and loses whatever the high-yielding currency falls, so the gap is the only cushion. Here the cushion is 7% less 0.5%, 6.5 points a year.

Why is 6.5% not quite right, and what else should you say?

Because the fall hits the whole pot, principal and interest, and a 6.5% fall on 107 costs almost 7. The exact break-even is 6.07%, and the difference matters only for precision. The bigger point is that a currency can lose 6% in days, while the carry is earned slowly over a year. That lopsided pattern, small steady gains and occasional sharp losses, is what makes carry trades dangerous to size.

One more thing is worth one sentence. If you hedged the currency with a one-year forward, the forward rate would already build in a rupee fall of about the rate gap, and the profit would disappear. The unhedged carry trade is a bet that the rupee does better than the forward price implies. That is a view on the currency, not free money.

About 13.4% in rupees. The stock turns each dollar into 1.08 dollars, and each dollar now buys 5% more rupees, so each rupee invested becomes 1.08 x 1.05, or 1.134 rupees. The extra 0.4% over simple addition is the currency gain earned on the stock gain.

Does a weaker rupee help or hurt this investor?

A student in India whose parents send a fixed dollar allowance from abroad is better off when the rupee weakens: the same dollars convert into more rupees. An Indian investor holding dollar assets is in the student's position, so a weaker rupee adds to the return. Say the direction first, because half the candidates who get this wrong get the sign wrong, not the arithmetic.

Where does the extra 0.4% come from?

Follow Rs 100. At the start it buys a dollar amount; a year later that amount has grown 8%. The 5% currency gain is earned on the grown amount, not the original one, so the currency also earns 5% on the 8% profit, which is 0.4%. For small moves the cross term hardly matters; for a 30% stock gain with a 10% currency move it is 3 points.

Run it the other way as a check. If the rupee had strengthened 5% instead, the return would be 1.08 x 0.95, less one, which is 2.6%: a good stock year mostly erased by currency. That is why global allocations report returns in both currencies.

The forward is about 85.61, a 1.9% premium to spot, and hedging lifts the dollar deposit's return to about 6.5% in rupees. Converting, earning 4.5% and selling forward must give the same rupees as simply earning 6.5% at home. For an Indian investor holding dollars the hedge earns the 1.9% premium; it is a cost only for someone on the other side.

Why is the forward rate set by interest rates rather than forecasts?

If two shops sell the same phone at different prices with free delivery between them, someone will buy in one and sell in the other until the gap closes. Investing rupees at home and converting to dollars, investing and selling forward are both riskless routes from the same starting sum, so they must end at the same number of rupees; the forward rate is whatever makes that true. This is covered interest parity.

So who pays for hedging, and what does it tell the investor?

The hedged dollar deposit earns 1.045 dollars sold at 85.61, which is 6.5% in rupees: exactly the rupee rate. A fully hedged foreign bond earns roughly the home interest rate plus its own spread, so hedging removes the currency bet but cannot manufacture a higher rate. The same 1.9% is a genuine cost for a dollar investor hedging rupee bonds, which is why hedged returns on Indian debt look lower to foreign funds.

One honest limit: the forward is not a forecast. It builds in a 1.9% rupee depreciation only because of the rate gap. The unhedged investor earns 4.5% plus whatever the rupee actually does, which may be more or less than the forward implied.