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  1. 061A bond has a modified duration of 7. Yields rise by 50 basis points. Roughly how much does its price change?Rates, risk and optionsWarm upDebt capital marketsSales and trading

    Try it first

    Pick the move.

    Show the worked solution

    The price falls by about 3.5%. Modified duration says how many per cent the price moves for a one-percentage-point move in yield, in the opposite direction. A 50 basis point rise is half a point, so 7 x 0.5 = 3.5% down. On a Rs 100 crore holding that is a loss of about Rs 3.5 crore. The bend in the price curve makes the true fall slightly smaller.

    Why do bond prices fall when yields rise?

    A bond pays fixed cheques. If new bonds start paying more, nobody will pay the old price for the old, smaller cheques, so the old bond's price drops until its yield matches the market. Think of a flat let at a fixed rent when rents nearby jump: a buyer now pays less for it. Price and yield move in opposite directions, and duration measures how hard the price reacts. A longer bond reacts more, because more of its cash arrives far in the future, where a change in the discount rate bites hardest.

    Duration is the tangent: almost exact for small moves, too gloomy for big ones2%4%6%8%10%80100120140YieldPrice, 100 todaytoday: 6%, price 100+50 bp: line -3.5%, curve -3.43%curve -18.7%line -21.0%dashed red line: the tangent,its slope set by duration
    For a bond with modified duration 7, the straight tangent predicts a 3.5% fall for a 50 basis point rise and the true curve gives 3.43%, almost identical, but for a 300 basis point rise the line says 21.0% while the curve falls only 18.7%.

    How accurate is the duration answer?

    Duration draws a straight line, the tangent, along a price curve that actually bends. For small moves the line and the curve are almost the same, and for large moves the curve sits above the line, so the true loss is smaller than duration says. Take a zero-coupon bond yielding 6% with 7.42 years to run, which has a modified duration of exactly 7. A 50 basis point rise cuts its price by 3.43%, against 3.5% from the shortcut. A 300 basis point rise cuts it by 18.7%, against 21.0% from the line.

    The relationship
    ΔPP≈−Dmod×Δy=−7×0.005=−3.5%\frac{\Delta P}{P} \approx -D_{mod} \times \Delta y = -7 \times 0.005 = -3.5\%
    D_modmodified duration, here 7
    Delta ythe change in yield, 50 basis points or 0.005
    Delta P / Pthe percentage change in price
    What it says in wordsThe percentage price change is roughly minus duration times the change in yield.

    Add one sentence that marks you out. The bend is convexityHow much the price curve of a bond bends, which is the change in duration as yields move., and for an ordinary bond it works in the holder's favour both ways: the price falls less than duration predicts when yields rise and gains more than predicted when they fall. For a 50 basis point move the effect is a few hundredths of a per cent, so 3.5% is the answer to give.

    Where candidates lose it

    The usual slip is an order-of-magnitude error, 0.35% or 35%, from mixing up basis points and percentage points. Say that 50 basis points is half a percentage point before you multiply.

    The second is getting the sign wrong. Yields up means prices down; say the direction first, then the number.

    What the interviewer asks next

    • What is the modified duration of a 5-year zero-coupon bond yielding 6%?
    • Two bonds have the same duration but different convexity. Which would you rather hold, and when does it matter?
    • How would you hedge the rate risk on Rs 100 crore of this bond?
  2. 083A stock trades at Rs 100 and pays no dividend. A one-year European call struck at Rs 100 costs Rs 10, and the one-year interest rate is 5%. What should a put with the same strike and expiry cost?Rates, risk and optionsHardDebt capital marketsSales and trading

    Try it first

    Pick the put price before you work it.

    Show the worked solution

    About Rs 5.24. A call plus cash worth the strike's present value pays the larger of the share price and 100 at expiry, and so does a put plus the share. Identical payoffs must cost the same today: 10 + 100/1.05 = P + 100. The present value of 100 is 95.24, so the put is 5.24. No pricing model or volatility estimate is needed.

    Why can you price the put without knowing volatility?

    If two combo meals contain exactly the same food, a restaurant cannot charge different prices for long: everyone would buy the cheaper one. Options work the same way. Two portfolios that pay the same in every outcome must cost the same today, so if you know the price of one, the other follows without any view on where the share goes. Volatility matters for what a call is worth, but the call's price already contains it. The put inherits it through the parity relationship.

    Build the two portfolios. A: buy the call and put aside 95.24 at 5%, which grows to exactly 100. B: buy the put and buy one share. If the share ends at 130, A exercises the call for 30 and adds its 100 cash, B holds a share worth 130 and lets the put expire: both have 130. If the share ends at 70, A lets the call expire and holds 100 cash, B exercises the put to sell the share for 100: both have 100.

    Two portfolios, one payoff line, so one priceA: call + cash of 95.24601001401000callcash, grows to 100total: larger of S and 100Share price at expiryB: put + one share601001401000putsharetotal: larger of S and 100Share price at expirySame payoff, same price today: C + PV(K) = P + S10 + 95.24 = P + 100, so P = 5.24
    A call plus cash of 95.24 and a put plus one share both pay the larger of the share price and 100 at expiry, so their prices today must match, which pins the put at 5.24.

    What does the formula say, and what if the put traded at Rs 7?

    The relationship
    C+K1+r=P+S⇒P=10+1001.05−100≈5.24C + \frac{K}{1+r} = P + S \quad\Rightarrow\quad P = 10 + \frac{100}{1.05} - 100 \approx 5.24
    Ccall price, Rs 10
    Pput price, the unknown
    Sshare price today, Rs 100
    K/(1+r)the strike discounted one year at 5%, 95.24
    What it says in wordsA call plus the discounted strike is worth the same as a put plus the share.

    If the put traded at 7, portfolio B would cost 107 against 105.24 for A, so you would sell B and buy A, pocketing 1.76 today with payoffs that cancel at expiry. Selling B means writing the put and short selling the share. Traders arbitraging that gap is what holds parity in place, which is why quoted put and call prices sit close to it in practice.

    Say the assumptions: European options, so no early exercise; no dividend before expiry; and annual compounding. With continuous compounding the strike's present value is 100 x e to the minus 0.05, about 95.12, and the put comes out near 5.12. Give the convention, not just the number.

    Where candidates lose it

    The instinctive answer is Rs 10, on the idea that at-the-money calls and puts must cost the same. They do not, because the call holder defers paying the strike for a year, and that deferral is worth something when rates are positive.

    The second loss is remembering parity but flipping a sign and answering 4.76. Build the two portfolios out loud before writing the formula and the signs take care of themselves.

    What the interviewer asks next

    • If the stock paid a Rs 2 dividend before expiry, how would the put price change?
    • Why does parity hold exactly for European options but only as a bound for American ones?
    • What position replicates a long put using only the call, the share and borrowing?
  3. 093The rupee trades at 83 per dollar. One-year interest rates are 7% in India and 5% in the US. What should the one-year forward rate be, and what could you do if the quoted forward were far from it?Rates, risk and optionsCoreDebt capital marketsSales and trading

    Try it first

    Where should the one-year forward sit?

    Show the worked solution

    About 84.58 rupees per dollar. Rs 83 invested at 7% grows to 88.81. Converted to 1 dollar and invested at 5% it grows to 1.05 dollars. Both routes are riskless once the forward is agreed, so they must end equal: 1.05 x F = 88.81, giving F of about 84.58. If the quoted forward sat far above that, you could borrow rupees, invest in dollars and sell them forward to lock in a gain.

    Why is the forward set by interest rates and not by a forecast?

    Imagine two banks next door, one paying 7% on rupee deposits and one paying 5% on dollar deposits, and a counter that promises today to swap your dollars back into rupees in a year at a fixed rate. If that promised rate were 83, everyone would ignore the dollar bank. The forward rate must exactly cancel the interest difference, otherwise one route earns more than the other with no risk, and the gap would be traded away. That rule is called covered interest parity, covered because the forward removes the currency risk.

    Two ways to hold Rs 83 for a year must end in the same placeTodayIn one yearRupeesDollarsRs 83.00Rs 88.81$1.00$1.05invest at 7% in Indiainvest at 5% in the USconvert at spot 83convert back at forward FEqual endings: 1.05 x F = 88.81, so F = 84.58 rupees per dollarQuoted at 86 instead? Borrow Rs 83 (owe 88.81), buy $1, earn 5%, sell $1.05 forward at 86 = Rs 90.30.Locked-in gain Rs 1.49 per dollar with no view on the rupee, until the quote moves back.
    Investing Rs 83 at 7% gives Rs 88.81 in a year, and converting to 1 dollar at 5% gives 1.05 dollars, so the forward must be about 84.58 for the two routes to match, and a quote of 86 would let a trader lock in Rs 1.49 per dollar.

    What would you actually do if the forward were quoted at 86?

    At 86, dollars are expensive in the forward market relative to parity, so sell dollars forward and build the dollars cheaply today. Borrow Rs 83 at 7%, owing 88.81 in a year. Convert to 1 dollar at spot and deposit it at 5%, which grows to 1.05 dollars. Sell those 1.05 dollars forward at 86 for Rs 90.30. After repaying the rupee loan you keep about Rs 1.49 per dollar, without taking any view on where the rupee goes. Traders doing exactly this pushes the quote back toward 84.58.

    The relationship
    F=S×1+rINR1+rUSD=83×1.071.05≈84.58F = S \times \frac{1 + r_{\text{INR}}}{1 + r_{\text{USD}}} = 83 \times \frac{1.07}{1.05} \approx 84.58
    Fthe one-year forward, rupees per dollar
    Sspot, 83
    r_INR, r_USDone-year rates in India and the US, 7% and 5%
    What it says in wordsThe forward equals spot scaled by the ratio of the two currencies' interest growth.

    The forward rupee is about 1.9% weaker than spot, roughly the 2 point rate difference. A quick approximation is spot times one plus the difference, 83 x 1.02 = 84.66, close enough to sanity-check. The limitation: in practice, capital controls, transaction costs and limits on who can borrow in each currency let small gaps persist, so the arbitrage is cleaner on paper than on a dealing desk.

    Where candidates lose it

    The common answer is that the forward is the market's forecast of the rupee, so it should be 83 or whatever you expect. Interviewers on rates and FX desks ask this precisely to hear you say it is an arbitrage relationship, not a forecast.

    The second slip is flipping the ratio and getting a stronger forward rupee. Check the direction: the higher-yielding currency trades weaker forward.

    What the interviewer asks next

    • What is the six-month forward on the same numbers?
    • An importer must pay 1 million dollars in a year. How does the forward help, and what does it cost?
    • Why might the quoted forward differ from parity for an extended period?