Interpreting the Units of Spread Duration Times Spread

Summary

The note explains how to interpret the units of DTS, defined as spread duration multiplied by the current credit spread. It starts from the approximate price change relation using spread duration and an absolute spread move, then rewrites the change in terms of the spread’s proportional move. This shows that DTS scales the relative spread change to estimate a monetary price change; its units are currency per percentage change when price is measured in currency.

The example uses a spread duration of 10 and a spread of 3%, and distinguishes an absolute move of 0.1 percentage points from a 10% relative widening. The key is to keep the spread change and spread in consistent units in the ratio. The answer provides a dimensional analysis rather than empirical evidence, and the result is an approximation based on spread sensitivity; it does not account for nonlinear effects or other bond price drivers.

Key ideas

• DTS is spread duration multiplied by the current spread.
• The price change approximation can be expressed using the proportional change in spread.
• When price is measured in currency, DTS has units of currency per percentage change.
• Keep the spread and its change in consistent units when calculating their ratio.

Tags

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


# 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 :)

## Answer by Kermittfrog (score 3, accepted)

https://quant.stackexchange.com/a/77444

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.