For bonds I've newly seen the measure DTS, spread duration times spread. The pnl is then approximated

$$ V = -DTS \frac{\Delta S}{S}$$

where $S$ the current spread and $\Delta S$ the spread change. My question is, what is the unit of this? Assume a spread duration of 10, and $S=3\%$. My DTS is then $DTS = 10*0.03 = 0.3$. Now using only the spread duration, for a change of $0.1\%$ this yields a value change of $10*0.001=0.01$, hence I lost $1\%$. Note here the spread move is entered in numercis $(0.001)$ and the result is also in numerics! Lets assume for $DTS$ that spread widen $10\%$. My value change $0.3*0.1 = 0.03$. Although I'm almost certain, here the $0.03$ are in $\%$, i.e. in numerics $0.0003$. But I can't see why this is the case mathematically. It must be simple still don't get it :)


1 Answer 1


Assume $[s]$ to be the unit of the rate or rate spread, e.g. percentage points or basis points. Then

$$ \begin{matrix} dPV&\approx \frac{\partial PV}{\partial S} dS & \frac{\\\$}{s}s\\ &=\frac{S}{S}\frac{\partial PV}{\partial S} dS& \frac{s}{s}\frac{\\\$}{s}s\\ &=\partial PV\left/\frac{\partial S}{S}\right.\frac{dS}{S}& \frac{\\\$}{\%}\%\\ &=DTS \frac{dS}{ S} & \frac{\\\$}{\%}\% \end{matrix} $$

so DTS should be in Dollar-per-percent-change.

  • $\begingroup$ many thanks for your answer. What do you exactly mean by "in" dollar-per-percent-change? I do see that $dS/S$ is a percentage change, but what confuses me is, although I translate % into numerics I still get % for $dPV$ $\endgroup$
    – swissy
    Nov 22, 2023 at 7:20
  • $\begingroup$ I've added the dimensions. $\endgroup$ Nov 22, 2023 at 7:45

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