# What's the intuition behind DTS(duration times spread) in fixed income?

I am having some difficulty grasping the concept of using DTS to measure credit risk. In the equity world, one typical measure of risk is beta, which is quite well-defined as the exposure to a common market factor, say S&P 500. But in credit market, it's not clear to me what the analogous common market factor would be. The original DTS paper says that DTS is the exposure to the relative spread change. However, the relative spread change can be calculated for each bond. Therefore, I have failed to see how DTS can be an exposure to some common factor. Can anyone explain exactly what DTS is measuring?

• have you read Lehman's report "DTS (DURATION TIMES SPREAD) – SCOPE AND APPLICABILITY. EMPIRICAL DURATION AND DEBT SENIORITY"? Sep 30 '14 at 2:30
• No, but I read the subsequent paper by the same authors at SSRN(papers.ssrn.com/sol3/papers.cfm?abstract_id=956825). I guess my question is whether to think about DTS as just a volatility measure or a measure of exposure to some common factor like beta.
– user2163
Sep 30 '14 at 2:42
• Could you shortly describe the definition of DTS that you have found?
– Ric
Sep 30 '14 at 13:47

The DTS paper (Ben Lor, Dynkin et al) describes how DTS (duration times spread) can be used as both an exposure to a common factor, and a measure of specific risk. Assume that relative spread changes for a set of bonds $i\in I$ are described by an exposure to a common risk factor, and a specific risk, that is

$$\frac{\Delta s_i}{s_i} = \frac{\Delta s_I}{s_I} + \frac{\Delta s^{\rm idio}_i}{s_i}$$

Then the change in spread for this bond is

$$\Delta s_i = s_i \left( \frac{\Delta s_I}{s_I} \right) + \Delta s^{\rm idio}_i$$

That is, the spread of bond $i$ can be viewed as the beta of absolute spread changes of bond $i$ to $relative$ spread changes in the sector/market/index represented by $I$.

The return of bond $i$ is given by the spread duration multiplied by the change in spread, that is

$$R_i = -D_i\Delta s_i = -D_is_i \left( \frac{\Delta s_I}{s_I} \right) - D_i \Delta s^{\rm idio}_i$$

The second term could also be written as $D_i s_i \times (\Delta s^{\rm idio}_i / s_i)$, so the DTS can be seen as both a "beta" to relative spread changes in the sector/market/index, as well as a measure of specific risk.

• That's for simplifying things above. A question - Why do you mention that for Plain Vanilla bonds, interest rate duration and Spread duration are same? If so, does any change in market rate affect Rcommon or Sspecific or both? Thanks, Oct 18 '15 at 8:40

Do you know the concept of duration? It is an approximation of how much the price of the bond changes if the interest rate (appropriate for the market in which the bond trades) changes. This is the interest rate is used to discount cash flows. It is common to all the bonds in the same market (e.g. German govis).

For various reasons (liquidity, credit risk, ...) bonds trade at a price that can not be explained by the curve of the market that it used for discounting. Then you need some extra discounting - this number, usually additive, is the spread. It is specific for the bond.

For a zero coupon bond and exponential discounting as a toy exampke you can say that the price is given by $$P = \exp(- r T),$$ where $T$ is the time to maturity and $$r = r_{common} + s_{specific}$$ where $r_{common}$ is the rate for the market and $s_{specific}$ is the specific spread.

If $r_{common}$ changes then you can approximate this by the usual duration if $s_{specific}$ changes then you can call this spreadduation.

For vanilla fixed rate bonds interest rate duration and spread duration are the same. For floaters or bonds with optionalities it is different.

Note that for fixed rate bonds it does not matter whether $r_{common}$ or $s_{specific}$ change. If one of these changes (by a parallel shift) by $x$ the price will by $$P = \exp(- (r+x) T)$$ and the change in values is $$\exp(- (r+x) T)-\exp(- rT).$$ In the case of plain vanilla bonds most of the concepts of duration analysis can be taken over to changes in spread. Essentially this is what they do.

Note that on page 2 they write that the change in price is approx. the change in spread times the duration. This is clear if we know how duration works.

If we change the wording from absolute numbers (spread widens by 10 bps, say from 15 bps to 25bps) to relative numbers (spread widens from a level of 15 bps by $66\%$) then we simply pull out the spread level and look at relative changes and have $$R = -D*s*r_s$$ where $R$ is the relative change of the bond price, D is the duration, s is the spread level and $r_s$ is the relative change of the spread level. Then we arrive at $D*s$ which is duration times spread.

• Thanks for the explanation. Perhaps I didn't phrase my question clearly. In the equity world, risk is measured by beta to a "single common" market factor, say SP500. It seems like DTS is essentially a beta to the relative spread change, but this relative spread change can be calculated for each bond. Therefore, I have failed to see how DTS is a measure of exposure to some "common" market factor. My question is whether the analogy to equity beta is the right way to think about it.
– user2163
Oct 1 '14 at 10:08
• I would say that usual Duration has some analogy to beta but spread duration not. While the former relates to the yield curve of one market, the latter corresponds to specific spreads. The latter could rather be compared to unsystemic risk in one factor equity risk models.
– Ric
Oct 1 '14 at 11:08

if equity beta is the measure of how an individual assets expected return changes wrt a change in the market equity risk premium, then duration measures the change in return to a change in the yield/return of a common market fixed-income risk factor; it is all slightly vague

• and in the above to determine if DTS is idiosyncratic to the bond or a common factor depends on the structure of the data generating process..if bond returns were modelled by a multi-factor process which included the common market bond factor then rspecific would be everything else which would include both a common factor such as industry and specific factors as well Feb 25 '18 at 6:33