Portfolio Management puzzles, solved step by step

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  1. 037Inflation runs at 5% a year in India and 2% in the US, and the exchange rate is 84 rupees to the dollar. Under relative purchasing power parity, where should the rate be in ten years?Currency and global returnsCoreGlobal investingMacro

    Try it first

    Pick the closest answer before you calculate.

    Show the worked solution

    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.

    Relative PPP: the rate drifts up at the inflation gap, not in a straight line8090100110120TodayYr 2Yr 4Yr 6Yr 8Yr 10Rupees per dollar84 todayabout 112.2short-run swings can sit anywhere inand beyond this band for yearsYearly drift1.05 / 1.02+2.94%not a forecast
    Relative PPP takes the rate from 84 to about 112.2 over ten years by compounding a drift of about 2.94% a year. Actual rates can sit well away from that path for years, so the line describes a long-run tendency, not a forecast.

    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.

    The relationship
    S10=S0(1+πIN1+πUS)10=84×(1.051.02)10≈112.2S_{10} = S_0 \left(\frac{1 + \pi_{IN}}{1 + \pi_{US}}\right)^{10} = 84 \times \left(\frac{1.05}{1.02}\right)^{10} \approx 112.2
    S_0today's rate, 84 rupees per dollar
    \pi_{IN}Indian inflation, 5%
    \pi_{US}US inflation, 2%
    What it says in wordsThe rate rises each year by the ratio of the two price levels' growth.

    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.

    Where candidates lose it

    Candidates get the direction right and then apply the 3% gap once, answering 86.5 or 87. Others compound India's 5% alone and land near 137, forgetting that the dollar is losing value too.

    Say the drift is the ratio of the two inflation rates, compound it ten times, and offer the simple 3% version as a check. Then add one sentence on why the short-run rate need not follow.

    What the interviewer asks next

    • If Indian one-year rates are 7% and US rates 4.5%, what forward rate does interest parity imply for one year out?
    • Why can a currency stay far from PPP for several years?
    • How should a rupee-based investor think about hedging a ten-year US equity holding?
  2. 063You borrow yen at 0.5% for a year, convert to rupees and invest at 7%. At the end of the year you convert back and repay. How far can the rupee fall against the yen before the trade loses money?Currency and global returnsCoreGlobal investingHedge funds

    Try it first

    What rupee fall wipes out the trade exactly?

    Show the worked solution

    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.

    Carry trade in yen terms: the rate gap is the only cushionVertical axis starts at 96 to make the small moves visible.100+7.0-6.5-0.50Borrow,invest 100RupeeinterestRupeefalls 6.1%Repayyen loanProfitzeroBreak-even fall d:107 x (1 - d) = 100.5d = 1 - 100.5 / 107d = 6.07%Not 6.5%: the fall hitsthe interest as well asthe principal
    On 100 of borrowed yen, the rupee investment grows to 107 and the loan to 100.5, so a rupee fall of 6.07% turns 107 back into exactly 100.5 and leaves nothing, slightly less than the 6.5-point rate gap.
    The relationship
    1.07 (1−d)=1.005  ⇒  d=1−1.0051.07=6.07%1.07\,(1-d) = 1.005 \;\Rightarrow\; d = 1 - \frac{1.005}{1.07} = 6.07\%
    dthe fall in the rupee against the yen over the year
    1.07the rupee pot after a year at 7%
    1.005the yen owed after a year at 0.5%
    What it says in wordsThe trade breaks even when the grown rupee pot, converted at the new rate, just repays the grown yen loan.

    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.

    Where candidates lose it

    The trap is answering 6.5%, the rate gap, and stopping. It is close, but it misses that the rupee fall applies to the interest as well as the principal, and a sharp interviewer will ask why 6.5% on 107 still loses money.

    The bigger miss is presenting carry as a sure profit. Say that the cushion is small compared with how far currencies can move, and that a forward hedge would remove the gain entirely.

    What the interviewer asks next

    • What one-year forward rate would make the hedged trade earn nothing?
    • Why do carry trades tend to lose money all at once in a market panic?
    • How would you size this trade if the rupee's annual volatility against the yen were 10%?
  3. 092The dollar-rupee spot rate is 84, one-year rupee interest rates are 6.5% and one-year dollar rates are 4.5%. What is the one-year forward rate, and what does hedging the currency do to an Indian investor's return on a dollar deposit?Currency and global returnsCoreGlobal investingFixed income

    Try it first

    For the Indian investor holding dollars, is the hedge a cost or a gain?

    Show the worked solution

    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.

    Two riskless routes to the same place must pay the sameTodayRs 84In one yearRs 89.46Route 1: keep rupees, earn 6.5%84 x 1.065 = Rs 89.46Convert at 84$1.000Earn 4.5%$1.045Sell forward at FRs 89.46F = 89.46 / 1.045 = 85.61, a 1.9% premium
    Rs 84 invested at 6.5% grows to Rs 89.46, and converting to one dollar, earning 4.5% and selling the $1.045 forward must reach the same Rs 89.46, which fixes the forward at 85.61, a 1.9% premium to spot.
    The relationship
    F=S×1+rRs1+r$=84×1.0651.045≈85.61F = S \times \frac{1 + r_{Rs}}{1 + r_{\$}} = 84 \times \frac{1.065}{1.045} \approx 85.61
    Sspot rupees per dollar, 84
    r_Rsone-year rupee rate, 6.5%
    r_$one-year dollar rate, 4.5%
    What it says in wordsThe forward premium on the dollar equals, roughly, the gap between rupee and dollar interest rates.

    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.

    Where candidates lose it

    The common slip is calling the 1.9% a cost for the Indian investor. The sign depends on which currency you are selling forward, and the interviewer asks this way round precisely to catch it.

    The second slip is treating the forward as the market's forecast of the rupee. It is an arbitrage price set by interest rates, and saying so separates you from candidates who memorised the formula.

    What the interviewer asks next

    • Rupee rates rise to 7.5%. What happens to the forward, and to the hedged return?
    • Why do foreign investors in Indian bonds often leave the currency unhedged?
    • What would you do if the quoted forward were 86.5?