Bonds: yield, duration and convexity

TVM · Chapter 313 min readAsked at Jane Street, DRW, Citadel, Optiver

Assumes Discounting: compounding conventions and present value.

After this lesson you should be able to

  • Explain the price–yield relationship and why it is convex.
  • Compute a duration and use DV01 to size a hedge.
  • Bootstrap a forward rate from two zero rates.

A bond is a fixed set of cash flows, so its price is entirely determined by the curve used to discount them. Duration is the first derivative of that relationship and convexity the second — and the whole of rates risk management is the observation that a first-order hedge leaves a second-order exposure that is always in your favour when you are long.

Proposition 3.1

Price and yield move opposite ways

Higher required yield means the same cash flows are discounted harder, so the price falls. The relationship is not linear: it is convex, falling steeply at low yields and flattening at high ones, because each further rise in yield discounts an already smaller price.

Holds when

  • A bond trades at par when the coupon equals the yield, below par when the coupon is lower, above when higher.
  • Yield to maturity is the single rate that reprices the bond — a summary of the curve, not a forecast.
  • It assumes reinvestment of coupons at the same yield, which is why realised return rarely equals YTM.
024681080100120Five-year bond, 4% annual couponYield (%)Price per 100 face
Figure 3.2 · The price–yield curve bends rather than falls in a line. The 4% coupon bond is at par when yield is 4%. Its price is the five coupon present values plus the discounted 100 principal. The slope is duration; the visible bend is convexity, so an equal fall in yield adds more than an equal rise removes.

Equation 3.3

Duration

Modified duration is the percentage price change per unit change in yield. Macaulay duration is the weighted average time to the cash flows, and the two differ by a factor of (1+y/m)(1 + y/m).

Dmod=−1PdPdy,ΔPP≈−Dmod ΔyD_{\text{mod}} = -\frac{1}{P}\frac{dP}{dy}, \qquad \frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y
DMacD_{\text{Mac}}
Weighted average maturity of the cash flows, in years.
Dmod=DMac/(1+y/m)D_{\text{mod}} = D_{\text{Mac}}/(1 + y/m)
The price-sensitivity version.
DV01=Dmod×P×10−4\text{DV01} = D_{\text{mod}} \times P \times 10^{-4}
Money change per basis point — what a desk actually uses.

Duration is a centre of mass. Picture the cash flows as weights placed along a time axis, each weight being its present value. Macaulay duration is the balance point. That picture answers most duration questions without arithmetic: a zero-coupon bond has all its weight at the end, so its duration equals its maturity; a coupon bond has weight pulled forward, so duration is less than maturity; and a higher coupon pulls the balance point further forward, shortening duration.

Example 3.4

Sizing a hedge

You are long $10\$10m of a bond with modified duration 77 and want to hedge with a future whose underlying has duration 44 and price $100\$100. How many contracts, each on $100,000\$100{,}000 face?

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Worked solution

  1. Formula
    contracts=Dport×VportDfut×Vfut\text{contracts} = \frac{D_{\text{port}} \times V_{\text{port}}}{D_{\text{fut}} \times V_{\text{fut}}}
  2. Substitute
    =7×10,000,0004×100,000= \frac{7 \times 10{,}000{,}000}{4 \times 100{,}000}
  3. Solve
    =70,000,000400,000= \frac{70{,}000{,}000}{400{,}000}
  4. Answer
    sell 175 contracts\text{sell } 175 \text{ contracts}

Sanity check. Check via DV01: the portfolio moves 7×10m×10−4=$7,0007 \times 10\text{m} \times 10^{-4} = \$7{,}000 per basis point, and each contract moves 4×100,000×10−4=$404 \times 100{,}000 \times 10^{-4} = \$40. Dividing gives 175, as it must.

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