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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

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    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?