Debt Capital Markets puzzles, solved step by step

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All topicsLeverage, coverage and cash flow9Mental maths and numeracy8Estimation and market sizing7Logic and brainteasers8Cost of capital and valuation riddles7Bond pricing and yield7Compounding, PIK and fees6Issuance and refinancing arithmetic8Credit spreads and default probability8Duration and convexity8Capital structure and recovery8Probability and expected value10Yield curve and forward rates6
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  1. 071You hold Rs 100 crore face of a 5-year corporate bond with a DV01 of 0.043 per Rs 100 of face. You want to hedge its interest rate risk by shorting a 10-year government bond whose DV01 is 0.068 per Rs 100. How much face value of the government bond do you short?Duration and convexityCoreFixed income asset managementSyndicate desks

    Try it first

    How much of the government bond do you short?

    Show the worked solution

    Short about Rs 63.2 crore face of the government bond. The corporate position moves Rs 100 crore x 0.043%, Rs 4.3 lakh, for each basis point. Each Rs 1 crore of the 10-year government bond moves Rs 6,800 per basis point. Matching the two takes 4.3 lakh over 6,800, about 63.2 lots of Rs 1 crore. The hedge ratio is the ratio of the DV01s, 0.043 over 0.068, not one for one on face value.

    What exactly are you trying to balance?

    Two children on a seesaw balance when weight times distance from the pivot matches on both sides, not when they weigh the same. A rate hedge balances when the rupee change per basis point matches on both sides, so you hedge DV01 against DV01, not face value against face value. The corporate bond's DV01The change in a bond position value for a one basis point change in yield, in rupees. is 0.043 per Rs 100, so Rs 100 crore of it gains or loses Rs 4.3 lakh for every basis point. That is the number the short has to offset.

    Balance rupees per basis point, not face valueRs 100 croreRs 63.2 crLong 5-year corporateDV01 0.043: Rs 4.3 lakh a bpShort 10-year governmentDV01 0.068: Rs 4.3 lakh a bparm 0.043arm 0.068DV01 hedge: short Rs 63.2 crore63.2 x 0.068% = Rs 4.3 lakh a bp: balancedFace-value hedge: short Rs 100 croreRs 6.8 lakh a bp: over-hedged by Rs 2.5 lakh
    Rs 100 crore of the corporate bond at a DV01 of 0.043 balances Rs 63.2 crore of the government bond at a DV01 of 0.068, both carrying Rs 4.3 lakh per basis point, while shorting a matching Rs 100 crore would over-hedge by Rs 2.5 lakh per basis point.
    The relationship
    Fhedge=F×DV01corpDV01gov=100×0.0430.068=63.2 croreF_{hedge} = F \times \frac{DV01_{corp}}{DV01_{gov}} = 100 \times \frac{0.043}{0.068} = 63.2\ \text{crore}
    Fthe face value you hold, Rs 100 crore
    DV01_corpthe corporate bond's price change per basis point, per Rs 100
    DV01_govthe government bond's price change per basis point, per Rs 100
    What it says in wordsScale the hedge so that its rupee move per basis point equals the position's.

    Why do you need less of the government bond than you hold?

    The 10-year government bond has a longer duration, so each rupee of it moves more when rates move: 0.068 per Rs 100 against 0.043. A smaller position in the more sensitive bond carries the same rupee risk. Check the balance: Rs 63.2 crore x 0.068% is Rs 4.3 lakh a basis point, the same as the corporate position. If rates fall 20 basis points in parallel, the corporate bond gains about Rs 86 lakh and the short loses about Rs 86 lakh.

    Then say what the hedge does not cover, because that is the follow-up. A DV01 hedge removes the risk of a parallel move in rates, but it leaves you exposed to the curve twisting between 5 and 10 years and to the corporate bond's credit spread moving. If 5-year yields rise while 10-year yields stay put, the hedge does nothing for you. Desks often hedge with a bond or swap closer in maturity for exactly this reason, and rebalance as DV01s drift with time and rates.

    Where candidates lose it

    The instinctive answer is Rs 100 crore, shorting the same face value you hold. With a longer, more sensitive hedging bond that over-hedges by more than half and turns a long rate position into a short one.

    The second slip is inverting the ratio and shorting Rs 158 crore. Check the direction with intuition: the hedging bond is more sensitive, so you need less of it, not more.

    What the interviewer asks next

    • Rates fall 20 basis points in a parallel move. What is the profit or loss on each leg?
    • The 5-year yield rises 10 basis points and the 10-year is unchanged. What happens to the hedged position?
    • Why might you hedge with a 5-year interest rate swap instead?
  2. 078A 3-year bond pays a 10% annual coupon and yields 10%, so it prices at par. Work out its Macaulay duration from a table of present values, and explain why the answer is less than 3 years.Duration and convexityCoreFixed income asset management

    Try it first

    Before building the table: roughly where does the duration land?

    Show the worked solution

    The Macaulay duration is about 2.74 years. Discount each cash flow at 10%: 9.09 at year 1, 8.26 at year 2 and 82.64 at year 3, adding to the price of 100. Weight each year by its share of the price: 1 x 9.09 plus 2 x 8.26 plus 3 x 82.64 is 273.55, over 100. It is below 3 because the coupons are paid before maturity and pull the average in.

    What is duration actually averaging?

    Suppose a friend owes you money and pays a little next month, a little the month after, and most of it in the third month. Asked when, on average, you got paid, you would not say the third month: some of the money came earlier. Macaulay duration is the average time until you are paid, with each payment weighted by its present value as a share of the bond's price. Present values rather than face amounts, because a rupee that arrives later is worth less today and should carry less weight.

    YearCash flowDiscount factor at 10%Present valueShare of priceYear x share
    1100.90919.099.09%0.0909
    2100.82648.268.26%0.1653
    31100.751382.6482.64%2.4793
    Total130100.00100.00%2.7355
    The present values add to the price of 100, and the year-weighted shares add to a duration of 2.7355 years.
    Duration is where the present values balance on a time line9.09Year 18.26Year 282.64Year 3TodayBalance point 2.74 yearsmaturity 3.00Year x present value1 x 9.099.092 x 8.2616.533 x 82.64247.93273.55 / price 100 = 2.74 years
    Placed as weights on a time line, the present values 9.09, 8.26 and 82.64 balance at 2.74 years, a little inside the 3 year maturity, because the two early coupons pull the balance point towards today.
    The relationship
    DMac=∑tt PVtP=1(9.09)+2(8.26)+3(82.64)100=2.74D_{Mac} = \sum_t t\,\frac{PV_t}{P} = \frac{1(9.09) + 2(8.26) + 3(82.64)}{100} = 2.74
    tthe year a cash flow arrives
    PV_tthe present value of that cash flow at the 10% yield
    Pthe bond price, the sum of the present values
    What it says in wordsWeight each payment date by how much of today's price arrives on that date.

    Why is it always below maturity for a coupon bond?

    Any payment before the final date pulls the balance point towards today, so only a zero-coupon bond has a duration equal to its maturity. Raise the coupon and the pull grows: the same 3-year bond with a 15% coupon, still discounted at 10%, has a duration of 2.65 years. Divide by one plus the yield and you get the modified durationMacaulay duration divided by one plus the yield. It gives the approximate percentage price change for a one point change in yield., 2.49: a 1 point rise in yield should take about 2.49% off the price. The exact price at 11% is 97.56, a fall of 2.44%, close to the estimate.

    Where candidates lose it

    The common error is weighting the years by the raw cash flows, 10, 10 and 110, instead of their present values. On this bond it gives 2.77 years: close enough to look right and wrong enough to fail the first follow-up, because it ignores that later money is worth less.

    The other loss is saying duration equals maturity for any bond. It does only for a zero. Say that once, with the reason, and you have answered the why before it is asked.

    What the interviewer asks next

    • What is the modified duration, and what price change does it predict for a 50 basis point rise in yield?
    • Without calculating, is the duration of a 3-year zero-coupon bond higher or lower than this one?
    • What happens to this bond's duration if its yield rises to 15%, and why?