Portfolio Management puzzles, solved step by step

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  1. 051A two-year bond pays an 8% annual coupon and repays 100 at maturity. Market yields for this bond are 7%. What is its price, and why is it above par?Bond mathsWarm upJ.P. Morgancloumbus · 2026

    Try it first

    Before you discount anything: where does the price land?

    Show the worked solution

    About 101.81. Discount each cash flow at 7%: the year 1 coupon of 8 is worth 7.48 today and the year 2 payment of 108 is worth 94.33, which add to 101.81. The bond sits above par because it pays 8 when the market only asks for 7, and a buyer pays up for that extra coupon until the return on the price paid falls back to 7%.

    What does pricing a bond actually mean?

    Think of a friend who promises you Rs 8 next year and Rs 108 the year after. What would you hand over today? If you can earn 7% elsewhere, each promised rupee is worth less the further away it sits. A bond's price is every promised cash flow divided by one plus the yield, once for each year you wait, and then added up. Here that is 8 divided by 1.07, which is 7.48, plus 108 divided by 1.07 twice, which is 94.33. The total is 101.81.

    The relationship
    P=81.07+1081.072=7.48+94.33=101.81P = \frac{8}{1.07} + \frac{108}{1.07^2} = 7.48 + 94.33 = 101.81
    8the annual coupon on 100 of face value
    108the final coupon plus the principal
    1.07one plus the market yield
    What it says in wordsDiscount each payment by the yield for as many years as you wait, then add.
    Two cash flows, discounted and stacked: the price lands above parYear 0Year 1Year 21088 coupon + 100894.337.48 from year 1from year 2101.81par 100divide by 1.07 twicedivide by 1.07Where the premium comes fromA 7% bond would trade at exactly 100.This one pays 1 a year more, twice.Year 1: 1 / 1.070.93Year 2: 1 / 1.07 / 1.070.87Premium over par1.81Coupon above the market yieldPrice 100 + 1.81 = 101.81
    Discounted at 7%, the year 1 coupon is worth 7.48 and the year 2 payment of 108 is worth 94.33, stacking to 101.81; the 1.81 above par is exactly the extra 1 a year of coupon over a 7% bond, valued today.

    Why must a bond with a high coupon trade above par?

    Suppose it traded at 100. A buyer would earn 8% on a bond when the market pays 7% for the same risk, so everyone would want it and the price would rise. The price climbs until the return on the price paid equals the market yield, and that happens at a premium over par. The premium is easy to see directly: compared with a 7% bond at 100, this one pays an extra rupee each year for two years, worth 0.93 plus 0.87, which is 1.81. That is a check on the long method, and it is quicker to say in the room.

    The same logic runs the other way. A coupon below the market yield means a discount to par, and a coupon equal to the yield means exactly 100. If you are asked how you would price a bond in today's market, say that the yield comes from comparable bonds of the same credit and maturity; the arithmetic above is the easy part. A premium bondA bond whose price is above its face value, because its coupon is higher than the yield the market currently demands. also pulls back towards 100 as it nears maturity, so a buyer at 101.81 loses the premium slowly while collecting the fat coupon.

    Where candidates lose it

    The common slip is adding the extra coupons without discounting them and answering 102. The extra rupee in year 2 is worth only 0.87 today, and saying 102 tells the interviewer you know the direction but not the method.

    The second loss is getting 101.81 without saying why it must be above 100. Give the one-line reason: the coupon beats the market yield, so buyers bid the price up until the return on the price paid is 7%.

    What the interviewer asks next

    • What would the price be if yields were 9% instead?
    • Why does the premium on this bond shrink as it approaches maturity?
    • Where would you find the right yield to price a bond like this in practice?

    Asked at J.P. Morgan, Generalist, cloumbus, 2026 (Wall Street Oasis): How would you price a bond in today's market

  2. 076A Rs 100 crore bond portfolio has a modified duration of 5. What is its DV01, and roughly what does a 25 basis point rise in yields cost?Bond mathsWarm upFixed incomeRisk management

    Try it first

    Say the DV01 before you work it through.

    Show the worked solution

    DV01 is Rs 5 lakh, so a 25 basis point rise costs about Rs 1.25 crore. Modified duration of 5 means the value moves about 5% for each percentage point of yield, or 0.05% per basis point. 0.05% of Rs 100 crore is Rs 5 lakh, and 25 basis points is 25 times that.

    Why turn duration into rupees at all?

    A shopkeeper who says the rent went up 4% has told you something; one who says it went up Rs 2,000 a month has told you what to do about it. Duration is the percentage version. DV01, the rupee change for a one basis point move, is the version a desk can add across positions, compare with a limit and put in a risk report. Two portfolios with the same duration but different sizes carry very different rupee risk, and only DV01 shows it.

    Duration becomes rupees one basis point at a timePortfolio valueRs 100 croreModified durationx 5One basis pointx 0.0001DV01Rs 5 lakhA 25 basis point rise, notch by notch: each notch is another Rs 5 lakh lost0 bp05 bpRs 25 lakh10 bpRs 50 lakh15 bpRs 75 lakh20 bpRs 1.00 crore25 bpRs 1.25 croreLoss at 25 bp: 25 x Rs 5 lakh = Rs 1.25 crore, about 1.25% of the portfolio
    Rs 100 crore times a modified duration of 5 times 0.0001 is Rs 5 lakh per basis point, so a 25 basis point rise in yields takes Rs 1.25 crore off the portfolio, one Rs 5 lakh notch at a time.
    The relationship
    DV01=V×Dmod×0.0001=100×5×0.0001=0.05 crore\text{DV01} = V \times D_{mod} \times 0.0001 = 100 \times 5 \times 0.0001 = 0.05 \text{ crore}
    Vportfolio value, Rs 100 crore
    D_modmodified duration, 5
    0.0001one basis point written as a decimal
    What it says in wordsThe rupee move for one basis point is the value times the duration times one hundredth of one per cent.

    Where does the straight-line answer stop being right?

    The modified durationThe percentage change in a bond price for a one percentage point change in its yield, taken at the current yield. estimate is a tangent: it treats the price and yield relationship as a straight line. For 25 basis points the straight line is close enough, but for a 200 basis point shock the curve bends away from it and the true loss is smaller than DV01 times 200. That bend is convexity. Mention it in one sentence; the interviewer will often ask for it next.

    Say also that DV01 assumes every yield in the portfolio moves by the same amount. If short rates rise and long rates stay still, a single DV01 number misses it, which is why desks also keep DV01 by maturity bucket.

    Where candidates lose it

    The common slip is out by a factor of 100: treating duration 5 as 5% per basis point and saying Rs 5 crore. The interviewer hears that you have not held a real rate position, where the size of one basis point is the first thing you learn.

    The second loss is stopping at the number. Say that DV01 is linear, so it overstates the loss for large rises, and that it assumes a parallel move.

    What the interviewer asks next

    • The portfolio doubles in size and duration falls to 2.5. What is DV01 now?
    • How would you cut DV01 by half without selling any bonds?
    • Why might a desk set a limit in DV01 rather than in duration?