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)

This last term looks complicated, but all it is is the duration-weighted sum of cash-flows, discounted by the YtM (and divided by , 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 of 1%) move in YtM

(2)

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.*