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. 024A 5-year bond yields 7.40%, the 4-year point on the same curve is 7.10%, and you fund the position at 6.50%. If the curve does not move for a year, what are the carry and the roll-down, and what is the rough total return over funding? Assume a modified duration of about 3.5 after a year.Yield curve and forward ratesCoreFixed income asset managementSyndicate desks

    Try it first

    Roughly what excess return over funding does the position earn if nothing moves?

    Show the worked solution

    Carry of 0.90%, roll-down of about 1.05%, so roughly 1.95% over funding. Carry is the bond's 7.40% yield less the 6.50% cost of funding it. Roll-down comes from the bond ageing into a 4 year bond priced at 7.10%: its yield falls 30 basis points, which at a duration of about 3.5 is a 1.05% price gain. On an unchanged steep curve, standing still earns both.

    What is carry, and what is roll-down?

    Carry is the simple part: you earn the bond's yield and pay your funding rate, like renting out a flat for more than its loan EMI. Roll-down is the price gain that comes purely from time passing on an upward sloping curve: the bond gets shorter, and shorter maturities are priced at lower yields. Nothing about the market has to change for either to happen; that is why desks call them the return from standing still.

    Standing still on a steep curve earns carry and roll-down6.5%7.0%7.5%3 yr4 yr5 yrfunding 6.50%5 yr 7.40% today4 yr 7.10% in a year0.90%Carry+1.05%Roll-down1.95%Excess return7.40 - 6.500.30 x 3.5
    Over a year the 5 year bond ages into a 4 year bond and is repriced from 7.40% to the curve's 7.10%, a 30 basis point fall worth 1.05% at duration 3.5; added to 0.90% of carry over 6.50% funding, the position earns about 1.95% if the curve does not move.

    How do you turn 30 basis points into a return?

    Use duration. A yield change converts to a price change of roughly duration times the change, so a 30 basis point fall at a duration of 3.5 is a gain of about 1.05%. Use the duration the bond will have after the year, when it is a 4 year bond, because that is the bond being repriced; the question gave 3.5 for exactly that reason. Then total return is the yield earned, 7.40%, plus the roll gain, about 8.45% before funding.

    The relationship
    excess≈(y5−f)+D×(y5−y4)=(7.40−6.50)+3.5×0.30=0.90+1.05=1.95%\text{excess} \approx (y_5 - f) + D \times (y_5 - y_4) = (7.40 - 6.50) + 3.5 \times 0.30 = 0.90 + 1.05 = 1.95\%
    y_5, y_4today's 5 year and 4 year yields on the curve
    ffunding rate, 6.50%
    Dmodified duration after one year, about 3.5
    What it says in wordsExcess return is carry plus the duration-weighted fall in yield as the bond rolls down the curve.

    How much can the market move before the trade loses money?

    Divide the cushion by the duration. A 1.95% return cushion at duration 3.5 absorbs a rise of about 56 basis points in the 4 year yield by year end before the position breaks even. That breakeven is the number a portfolio manager actually trades on. Say the limits too: the curve rarely stands still, the roll-down depends on the curve's shape between the two points, and the duration figure is an approximation that ignores convexity.

    Where candidates lose it

    The common miss is quoting carry alone, 0.90%, and forgetting that on a steep curve the bond also gains in price as it ages. The question gave the 4 year yield specifically to see whether you use it.

    The second slip is applying the 30 basis points without duration, calling the roll-down 0.30%. A yield move is not a return until it is multiplied by duration.

    What the interviewer asks next

    • The curve between 4 and 5 years flattens so both yield 7.40%. What is the excess return now?
    • Funding rises to 7.50% the day after you buy. Is the trade still attractive?
    • Why is roll-down usually largest at the steepest part of the curve?
  2. 039One-year money yields 7% and two-year money yields 7.6%, both annually compounded. What one-year rate, one year from now, does the curve imply, and why is it above both spot rates?Yield curve and forward ratesCoreFixed income asset management

    Try it first

    Roughly where is the one-year rate, one year forward?

    Show the worked solution

    The implied one-year rate a year from now is about 8.20%. Rs 100 at 7.6% for two years grows to 115.78. Rs 100 at 7% for one year grows to 107, so the second year must turn 107 into 115.78: a rate of 8.20%. It is above both spot rates because the two-year rate is an average, and the second year has to be high enough to pull the 7% first year up to 7.6%.

    What does a forward rate actually mean?

    You can reach the next town by the highway in one go or by a side road with a stop halfway. If both leave at the same time and arrive at the same time, the second leg of the side road must make up whatever speed the first leg lost. A forward rate is the rate for the second leg that makes lending for two years in one go pay exactly the same as lending for one year and rolling over. If the two routes paid differently, everyone would take the better one until they did not.

    The relationship
    (1.076)2=(1.07)(1+f)  ⇒  f=1.1577761.07−1≈8.20%(1.076)^2 = (1.07)(1 + f) \;\Rightarrow\; f = \frac{1.157776}{1.07} - 1 \approx 8.20\%
    1.076^2growth of 1 rupee over two years at the 2-year spot rate
    1.07growth over the first year at the 1-year spot rate
    fthe one-year rate one year from now implied by the curve
    What it says in wordsTwo years at the long rate must equal one year at the short rate followed by one year at the forward.
    Two roads to year 2 must end in the same placeTodayYear 1Year 2Rs 100107.60107.00115.787.6% for year 17.6% for year 27.0% for year 1forward 8.20%the unknown
    Rs 100 at 7.6% for two years reaches 115.78, and Rs 100 at 7% for one year reaches 107.00, so the second year must earn 8.20% for the two routes to end at the same value.

    Why must the forward sit above both spot rates?

    Because the two-year rate is roughly the average of the two one-year legs. If the first leg earns 7% and the average is 7.6%, the second leg must earn about 7.6% plus the 0.6 the first leg fell short, roughly 8.2%. The quick approximation is two times 7.6 less 7, which is 8.2%; compounding adds the last few hundredths to give 8.20%. On an upward sloping curve, forwards always sit above the spot rates, and the steeper the curve, the further above.

    Say what the forward is not. It is not a forecast of where one-year rates will be; it is the break-even rate that makes the two routes equal today. If investors demand extra yield for locking money up longer, a term premium, the forward will sit above what the market truly expects. A desk uses it to price a forward starting loan or to judge whether rolling short or locking long is cheaper.

    Where candidates lose it

    The common wrong answer is 7.3%, the midpoint of the two spot rates. It treats the forward as a point between them rather than as the missing piece that makes the average work.

    The second trap is presenting the forward as a prediction of future rates. State that it is the rate that equalises the two routes, and mention the term premium if the interviewer pushes.

    What the interviewer asks next

    • The three-year rate is 8.0%. What is the one-year rate two years forward?
    • If the curve were inverted, 7.6% at one year and 7% at two, where would the forward sit?
    • How would you lock in the forward rate today using only spot borrowing and lending?