Bond Pricing: Roll PnL

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
        Roll PnL

Holding bonds leads to several sources of PnL. Bonds pay coupons periodically, and this must be set against your cost of financing (generally a cost if you are borrowing to buy, and a benefit if you sell short and take cash – although shorts will incur a borrowing cost). As we have seen before, interest rate and credit spread moves in the market will cause positional PnL according to your DV01. There are several other factors to think about (I’ll do a full ‘PnL Explain’ post soon), but one of the most interesting is what dealers call ‘Roll PnL’, since it comes from a position rolling down the yield curve (in this piece I discuss Roll PnL for bonds, but exactly the same principle applies for CDS contracts, which are effectively just a portfolio that is long a credit-risky bond and short the corresponding risk-free bond).

Holding a position with an upward-sloping yield curve causes market yield of the asset to fall, realising a cash PnL

Consider the yield curve above. The YtM for 3y is 3%, and the YtM for 2y is 2.2%. If we buy a bond with 3y to maturity and hold it for a year, its maturity will of course fall to 2y – so assuming the yield curve stays fixed, the yield of the bond set by the market will now be 2.2%, and as we’ve seen before a fall in yield means an increase in price.

Concretely, considering for simplicity a ZCB, ets imagine we borrow $100 at the 3y risk-free rate of 3% and buy a risk-free ZCB, which we hold for a year. The PnL over the year will be

(1)   \begin{align*} = & \frac{100}{(1 + 0.022)^2} - \frac{100}{(1 + 0.03)^3} - 3 \nonumber \\ = & \Bigl( \frac{100}{(1 + 0.022)^2} - \frac{100}{(1 + 0.03)^2} \Bigr) + \Bigl( \frac{100}{(1 + 0.03)^2} - \frac{100}{(1 + 0.03)^3} \Bigr) - 3 \nonumber \\ = & \Bigl( \frac{100}{(1 + 0.022)^2} - \frac{100}{(1 + 0.03)^2} \Bigr) + \Bigl( \frac{3}{(1 + 0.03)^3} \Bigr) - 3 \nonumber \\ \simeq & 100 * 2 * (0.03 - 0.022) \nonumber \end{align*}

leading to a Roll PnL of about $1.50 (after financing costs – on the tenuous assumption that we can borrow at the risk-free rate!)

Mathematically, increasing value with decreasing time means that a bond has a positive theta. We can calculate this using the chain rule

(2)   \begin{align*} \Theta = -\frac{\delta C(t)}{\delta t} & = \frac{\delta}{\delta t} \sum_{i=1}^N \frac{CF(t_i) }{(1 + r_y)^{t_i}} \nonumber \\ & = -\sum_{i=1}^N CF(t_i) \cdot \frac{\delta}{\delta t} \frac{1}{(1 + r_y)^{t_i}} \nonumber \\ & = -\sum_{i=1}^N CF(t_i) \cdot \Bigl( \frac{-\ln{(1 + r_y)}}{(1 + r_y)^{t_i}} - \frac{t_i}{1+r_y} \cdot \frac{\delta r_y}{\delta t_i} \Bigr) \nonumber \\ & \simeq \sum_{i=1}^N \frac{CF(t_i) \cdot r_y}{(1 + r_y)^{t_i}} + \sum_{i=1}^N \frac{CF(t_i) \cdot t_i}{1+r_y} \cdot \frac{\delta r_y}{\delta t_i}  \end{align*}

and from our concrete example in Eq. 1, we recognise the first of these terms as the ‘financing cost’ of the portfolio, and the second term as the Roll PnL (ie. proportional to the local derivative in the yield curve for each payment)

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