VRM 12: Applying Duration, Convexity, and DV01

DV01 answers a blunt question: how much does this portfolio gain or lose if every interest rate moves by one basis point? With the small parallel shift in basis points and the resulting change in value:

The price to rate relationship is not linear, so the exact object is the derivative of price with respect to yield, negated. DV01 is also called dollar duration when the rate change is a decimal.

Take a Treasury bond maturing in three years, face value USD 1 million, paying a 10% per annum coupon semi-annually. Five coupons of USD 50,000 arrive at six-month intervals and USD 1,050,000 at the end, each discounted at the matching spot rate below.

Source: spot rates from Table 12.1 of the chapter. The third column is derived.

Nothing forces the shock to be one basis point. Price is close to linear over a small interval, so a five-basis-point shift divided by five gives almost the same answer.

Source: Table 12.2 of the chapter, transposed and with the decrease shown first.

Averaging those two estimates returns 265.72 once more. They disagree only because price is not exactly linear in interest rates, and measuring that curvature is what convexity does.

So far every spot rate was nudged by one basis point. Shocking a different quantity gives a slightly different number, and the common alternative is the bond yield, the single discount rate that reproduces the observed price.

The solution is approximately 8.5404%. Raising the yield one basis point, to 8.5504%, cuts the price by USD 265.92, and lowering it to 8.5304% raises the price by USD 266.00. Averaging gives a yield-based DV01 of 265.96, a shade from the 265.72 obtained by moving all spot rates. A third possibility raises every forward rate by one basis point, landing near the other two.

Source: definitions from the chapter. The third column collects results computed earlier in this lesson.

The convention from here on is the middle row: DV01 means the price change from a one-basis-point parallel shift in the spot rate curve. Quoting the convention alongside a hedge ratio avoids confusion. On USD 1 million of face value the three definitions differ by less than a quarter of one United States dollar per basis point.

DV01 can be computed for anything whose value depends on interest rates, which is what makes it useful for hedging. Suppose a bank measures the DV01 of its existing position at -463 United States dollars per basis point. The negative sign says the position behaves opposite to a long position in a bond: a one-basis-point rise in all rates adds USD 463, and a fall takes USD 463 away. A bank paying fixed and receiving floating on swaps shows this profile.

Neutralising the exposure needs an offsetting position with a DV01 of +463, so the two cancel under any small parallel shift. The 10% coupon bond is a candidate, but at a face value of USD 1 million its DV01 is only 265.72, so the holding is scaled up in proportion.

That holding leaves the book insensitive to small parallel shifts: a rise in the term structure gains on the existing position and loses on the bond, and a fall does the reverse. The chapter describes 1,742,436 as the value of the position to be established, and since the DV01 of 265.72 belongs to a face value of USD 1 million, the figure is a face amount rather than a market value.

What the hedge does not cover

Two assumptions are buried in the arithmetic: that the bank’s rates and the rates driving the hedging bond move in lock step, and that the shift is small enough for the linear DV01 relationship to hold. Break either and the hedge leaks. A steepening moves short and long rates by different amounts, which the calculation cannot anticipate.

DV01 reports a change in money. Effective duration reports the same sensitivity as a percentage of the value of the position, so it is comparable across holdings of different sizes. Writing effective duration as D and the parallel change in all rates as a change in r:

The link to DV01 falls out at once: with the rate change in basis points, effective duration is DV01 divided by the price. For the three-year 10% coupon bond, priced at USD 1,037,922.03 with a change per basis point of -265.72, the duration is 265.72 divided by 1,037,922.03, or 0.000256.

Why one duration gets quoted as three different numbers

That 0.000256 is the proportional price change for a one-basis-point move, equivalently 0.0256%. Convention usually reports duration for a 100-basis-point change, giving 2.56% per 100 basis points. Measuring rates as decimals makes the quote per 10,000 basis points, since one basis point is 0.0001, and the duration of this bond is 2.56.

These are scaling conventions and nothing more. Nobody revalues a bond after a 10,000-basis-point move, since a shift that size would break the linear approximation. The calculation uses a small shock and scales it: a five-basis-point revaluation is multiplied by 20 for a 100-basis-point quote, or by 2,000 for the decimal quote.

A callable bond gives the issuing company the right to buy the bond back at a pre-determined price at certain future dates, and governments and special purpose vehicles issue them too. The right gets exercised when interest rates have fallen far enough for the issuer to refinance more cheaply, so the call is a live threat precisely when bond prices are high.

Take a five-year bond callable after three years and at no other time. Setting the call aside and treating the security as an ordinary five-year bond is wrong, and the error is predictable in direction: the call shortens the expected life of the bond, so it reduces duration.

Two approaches, one of which is still wrong

A more considered attempt fixes the probability of the call. Suppose that at today’s rates there is a 40% chance the bond will be called. One could then blend the three-year bond’s effective duration, the duration if called, at a weight of 0.4, with the five-year figure, the duration if not called, at a weight of 0.6.

The flaw is that the call probability is not a constant. When interest rates increase, refinancing becomes less attractive and the probability of a call falls, so freezing that probability at 40% discards part of the sensitivity being measured.

The correct procedure keeps the option alive at every step. Value the bond today with the embedded option properly priced, then value it again assuming all interest rates rise by one basis point, letting that feed through to the call probability as well as the discounting. Read effective duration off the resulting percentage change in price. Binomial trees are the standard machinery for bonds with embedded options.

A puttable bond, where the holder rather than the issuer can demand early repayment, is treated the same way. Only the direction of the link between rates and exercise differs: the probability of the put being exercised increases as interest rates increase.

Both measures answer the same question and differ in their units, so the choice is whether the answer is wanted in currency or in percentage terms. The sharpest practical difference shows up when a position is resized: DV01 is proportional to the size of the holding, so doubling the position doubles it, while effective duration is unchanged, and so is convexity.

Source: properties drawn from the chapter discussion of the two measures.

Bond investors think in returns, which are quoted as percentages, so effective duration is usually the more informative measure. It ranks a five-year holding and a twenty-year holding on one scale regardless of how much money sits in each.

When a percentage measure has nothing to divide by

Sometimes effective duration cannot be computed at all. Picture a bank moments after it enters an interest rate swap. The change in value for a one-basis-point move is well defined, but effective duration divides by the value of the position, and a swap at inception is worth zero or close to it. DV01 is therefore the natural measure for swaps and for interest rate futures, both of which carry substantial exposure with negligible initial value.

Effective duration is the slope of the price to rate relationship at current rates, and a slope describes a curve exactly at one point only. Effective convexity measures how quickly that slope changes with the level of interest rates.

The numerator is the amount by which the two revalued prices, added together, exceed twice the current price. On a straight line that is zero, so it isolates curvature, approximating the second derivative of price against yield.

Apply it to the same three-year bond. A five-basis-point shift means a rate change of 0.0005, and both revalued prices are already to hand: USD 1,039,251.68 after the decrease and USD 1,036,594.52 after the increase, against USD 1,037,922.03 today. That gives an effective convexity of 8.246.

Source: Tables 12.1 and 12.3 of the chapter, placed side by side.

The picture also settles the direction of the error. The true percentage price change always exceeds what duration alone predicts, so duration overstates price falls and understates price rises, and convexity estimates that gap. Positive convexity means a parallel shift treats the portfolio better than duration suggested, and negative convexity means worse.

Hedging on effective duration follows the logic of DV01 hedging, expressed proportionally. Let the investment have value V and duration D sub V, and the hedging bond value P and duration D sub P. Each changes by minus its value multiplied by its duration and by the rate change, so the exposure vanishes when the two duration-weighted values sum to zero.

Suppose the position is worth USD 2 million with an effective duration of 4.0 and the hedging bond has an effective duration of 5.0. The required position is minus the product of 2 and 4 divided by 5, which is -1.6, so USD 1.6 million of the bond is sold short. The offset checks out: the investment changes by -8 times the rate change and the short bond position by +8 times it.

Neutralising convexity as well needs a second bond

A duration hedge discards the quadratic term, so it protects only against small parallel shifts. Removing convexity exposure adds a second condition, and two conditions require two instruments. Write the value, duration and convexity of the two bonds as P sub 1, D sub 1, C sub 1 and P sub 2, D sub 2, C sub 2. Setting the linear and quadratic totals to zero gives two equations.

Each measure buys a different amount of protection. DV01 hedging and effective duration hedging both defend against small parallel shifts, and adding a convexity condition extends that to larger ones. None of these measures defends against a twist or a steepening, because none carries information about how different maturities move relative to one another.

Everything so far has shocked the whole term structure. The yield-based family, introduced with the interest rate material in Financial Markets and Products, shocks a single yield instead, and it explains where the word duration came from. Continuous compounding gives the cleanest formula, so start there.

Differentiating with respect to the yield gives the price change as minus the sum of each cash flow multiplied by its time to receipt and its discount factor, times the change in yield. Dividing by the price defines yield-based duration.

Read that as an average rather than a derivative and its meaning changes. A discounted cash flow divided by the price is the proportion of the bond’s value arriving at that date, and those proportions add to one. Duration is therefore an average of the payment times, weighted by the share of value at each. It measures how long the investor waits to be paid, which is why the word was chosen.

Macaulay, modified and the compounding adjustment

The continuous-compounding version is Macaulay’s duration, after Frederick Macaulay, who proposed it in 1938. Yields are normally quoted with semi-annual compounding, and switching convention means dividing by one plus half the yield, giving modified duration.

Convexity follows the same pattern with the times squared, so yield-based convexity averages the squared time to maturity under those identical weights. Dividing by the square of one plus half the yield gives modified convexity.

How far apart are the two families? Not far. The DV01 from a one-basis-point yield change was 265.96 against 265.72 from a change in all spot rates, and the pattern holds: modified duration sits close to effective duration, and modified convexity close to effective convexity. Using yield-based measures for two bonds in one hedge assumes their yields move in lock step.

For the ten-year zero-coupon bond of Example 2 the measures line up. Macaulay’s duration is exactly ten years, since the only cash flow arrives at the ten-year point. Modified duration divides that by one plus 0.04 over 2, giving 9.804, and effective duration came out at 9.804 as well. For a zero-coupon bond the two must agree.

A portfolio measure is assembled from the measures for its parts, and the aggregation rule differs between DV01 and the other two because of units.

DV01 is an amount of money, so portfolio DV01 is the plain sum of the component DV01s. Take three positions with DV01s, in thousands of United States dollars, of 30, 40 and 50; adding them gives a portfolio DV01 of 120. Short positions enter with a negative sign and reduce the total, which makes the rule convenient for a desk sizing an overall hedge.

Duration is a proportion, so it cannot be added. Portfolio duration is instead a value-weighted average of the individual durations, using market values as the weights.

Source: holding values and durations from the chapter. The weight and contribution columns are derived.

Weights come from dividing each holding by the combined value of 50, and each weight multiplied by its duration gives the contributions in the last column. Those add to a portfolio effective duration of 9.1, the number to use when estimating what a small parallel shift does to the book. Convexity aggregates identically.

The same rule for yield-based measures

Value weighting also works for yield-based duration, though the shift is now in the yields on the bonds rather than in the spot curve. Consider a portfolio holding one bond yielding 3% and another yielding 4%. The yield-based duration answers what happens when the first yield moves to 3.01% and the second to 4.01% at the same moment.

Two portfolios can carry the same value and duration while carrying quite different convexities, and the classic illustration is the bullet against the barbell. Three bonds set up the example. One matures in 5 years and pays a 2% coupon, one matures in 10 years paying 4%, and one matures in 20 years paying 6%. The term structure is flat at 4% with semi-annual compounding, so every bond yields 4% and the yield curve is a horizontal line.

Source: Table 12.4 of the chapter, computed with interest rates measured in decimals, reordered longest maturity first. The fourth column is derived.

Convexity rises much faster than duration as maturity extends, which is why mixing a short bond with a long one delivers more convexity than a middle-maturity bond of the same duration.

An investor wanting an effective duration of 8.1758 can simply buy the ten-year 4% coupon bond, which is a bullet investment: a single security. The alternative builds the same duration from the five-year and 20-year bonds only, a barbell investment, since the holdings sit at the two ends of the maturity range.

Why the barbell does not actually dominate

Higher convexity improves the outcome whenever the curve shifts in parallel, and the improvement grows with the shift. Since both strategies yield 4% and share a duration of 8.1758, the barbell appears to beat the bullet under every parallel shift, which looks like free money: buy the barbell and short the same amount of the bullet.

The trade fails for two reasons. First, shifts are not always parallel, and the bullet performs better under many non-parallel ones. Should the curve turn upward-sloping over the following year, leaving yields of 3%, 4% and 5% on the five-, ten- and 20-year bonds, the bullet is unchanged in value while the barbell falls by about 2%.

Second, the flat 4% curve is artificial. With a flat curve that does not move, all three bonds return 4% and the strategies tie. Real term structures are usually upward-sloping and convex, and on such a curve the bullet yields more than the barbell, so if yields stay put the bullet wins by that differential. The barbell buys convexity and pays for it in yield.

A modelling lesson sits underneath. Term structure models aim to be no-arbitrage models, and one where the curve is always flat and shifts only in parallel would hand this barbell arbitrage to anyone who noticed it.