# What is the unit of DTS (and why)?

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 :)

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.

• 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 $dPV$ Nov 22, 2023 at 7:20
• I've added the dimensions. Nov 22, 2023 at 7:45