# Convert spreads to prices for bonds via duration

I have the price of a bond and would like to convert it to spreads. Is this possible by just having dollar duration?

Secondly, if I just care about the relative spreads of multiple bonds, is it enough to consider their prices and dollar durations? Would I simply divide the price by the dollar duration to get relative spreads?

Unless I misread your question, NO: Spread calculation (calibration) is a root-finding exercise. In the simplest case (flat rate curves), given market price $$P$$ and reference yield $$r$$ (e.g. treasury rate), which spread level $$s$$ ensures that

$$s:PV(r,s,t,T)\equiv\sum_{i}c_ie^{-(r+s)(T-t_i)}\stackrel{!}{=}P$$

In root-finding, we can employ Newton's method and iterate towards the correct spread level $$s$$ given some initial guess $$s_0$$

$$s_{n+1}=s_n-\frac{PV(r,s_n,t,T)-P}{\left.\frac{\partial PV(r,s,t,T)}{\partial s}\right|_{s=s_n}}=s_n+\frac{PV(r,s_n,t,T)-P}{\mathrm{Dollar\ Duration(DV01)}}$$

The last equality is true as:

$$dPV/dy = dPV/dr = dPV/ds = -\sum_{i}(T-t_i)c_ie^{-(r+s)(T-t_i)}$$

i.e. $$\mathrm{DV01=CS01}$$. Yet, given only the bond price and its duration is not sufficient - We also need the reference rate level, $$r$$. But if we know $$r$$, we have $$s=y-r$$ and we do not need the value or the duration in the first place.

HTH?

• What if I don’t care about the absolute level of spread; but just want to compare a relative level of spreads? Would it be enough to have prices and dollar duration? Mar 22, 2022 at 7:47
• Assuming equal times-to-maturity and coupons across bonds then, yes, price levels are sufficient to rank spreads: Lowest price = largest spread. I'd say the case with different maturities and coupons require some more analysis. Mar 22, 2022 at 7:58
• Would the duration be enough to compensate for time to maturity as it’s a proxy for that? Mar 22, 2022 at 8:17
• Hi, at this point I'd say: Please update your initial post so that it better reflects your ideas and questions, ok? Mar 22, 2022 at 8:24
• Added second part to the question Mar 22, 2022 at 8:26