Bond Pricing: Duration/Convexity and DV01

This is a series of short posts examining the bond markets and some of the key pricing, risk and quoting concepts (with EXAMPLES!) –
        The Yield Curve
        Duration/Convexity and DV01
        Spread Quoting
        Z-Spread
        Roll PnL

For dealers who buy and sell bonds, the change of a value of a portfolio as market rates change will be extremely important. As we discussed in the previous post, yields of bonds will typically rise and fall along with market-implied yields coming from risk-free treasuries, and having some quantities that capture the first-order magnitude of this change is useful – such first order measures are especially easy to handle as they sum linearly across a portfolio.

If market yields fall, the price of already-issued bonds will typically rise. To measure this, we define the ‘Duration’ of a bond to be the percentage decrease in a bond’s price as yields rise by 1%. From before, the value of a bond expressed in terms of YtM is

(1)   \begin{align*} C(t) &= \sum_{i=1}^N CF(t_i) \cdot \frac{1}{(1 + r_y)^{t_i}} \nonumber \\ \text{Duration} = \frac{1}{V} \frac{\delta C(t)}{\delta r_y} &= \frac{1}{V} \sum_{i=1}^N CF(t_i) \cdot \frac{\delta}{\delta r_y} \Bigl( \frac{1}{(1 + r_y)^{t_i}} \Bigr) \nonumber \\ & = \frac{1}{V \cdot (1 + r_y)} \sum_{i=1}^N \frac{CF(t_i) \cdot t_i}{(1 + r_y)^{t_i}} \nonumber \end{align*}

This last term looks complicated, but all it is is the duration-weighted sum of cash-flows, discounted by the YtM (and divided by 1+r_y, but this is likely to be very close to 1), so we see why ‘Duration’ is a sensible name for this risk measure!




Very often, rather than the relative change in the bond price due to the yield change, we will be interested in the absolute, dollar-value of the change (ie. the PnL of the portfolio). The measure often used here is DV01 – the dollar value of a 1bp (ie \frac{1}{100} of 1%) move in YtM

(2)   \begin{align*} \text{DV01} &= V \cdot \text{Duration} \cdot \frac{1}{10000} \nonumber \\ & = \frac{1}{10000 \cdot (1 + r_y)} \sum_{i=1}^N \frac{CF(t_i) \cdot t_i}{(1 + r_y)^{t_i}}\nonumber \end{align*}

which is very close to the discounted duration-weighted present value of the cashflows.

If a portfolio is constructed with DV01 close to 0, it will be fairly insensitive to YtM changes. However, note that we haven’t specified where the yield change came from – it could be coming from a change in the yield curve coming from govvie prices changing, or from a widening credit spread for corporate bonds. Changes in just the credit spread, for example, will affect the yields of some bonds more than others and a crudely hedged portfolio might suddenly show PnL.

Example: a portfolio is constructed made up of positions on two bonds A and B, both United States Treasuries paying bi-annual coupons. A has 3.5 years left to run, pays 2.5% annually and is priced at $99 on the market; B has 2 years left to run, pays 3% annually and is priced at $101 on the market. The portfolio is made up of $1MM face value of bond A. How large a short position should the portfolio manager hold in position B for the portfolio to be DV01-flat? What risk remains in the portfolio? (as usual, assume bonds have recently paid a coupon and that’s already reflected in the pricing)

First, we calculate the yield for each bond as described in the previous post (not forgetting that as coupons are bi-annual, they are each half of the annual total). A trades at YtM of 2.82%, and B trades at YtM of 2.5%.

Next we use these YtMs to calculate the unit DV01 of the bonds fromthe equation above. A has a DV01 of 0.00325 $/bp, while bond B has a DV01 of 0.00193. Since the ratio of these is 1.68, a short position of $1.68MM face value of bond B is required to flatten the DV01 of the portfolio.

Because we have used DV01 with bonds of different maturity to flatten the risk, we are immune to parallel shifts in the yield curve – ie. those that increase or decrease the yield of both bonds by the same amount. However, if the yield moves to become steeper or shallower, this will increase/decrease the yield of one bond more than the other, and our simple hedge will not be sufficient to prevent PnL.