Parallel and Key-Rate PV01 from Bootstrapped Yield Curves

# PV01: bumping all tenors along the curve or only a tenor at a time?


# PV01: bumping all tenors along the curve or only a tenor at a time?












Item 22 from this ISDA SIMM 2.3 document gives the following definition for the PV01 of an instrument $i$ with respect to the tenor $t$:

$$s(i, r_t) = V_i(r_t+1bp, cs_t) - V_i(r_t, cs_t)$$

where $r_t$ and $cs_t$ are the interest rate and credit spread for that tenor.

Following this definition, I understand that a curve has been built from market prices, and only its tenor $t$ is being bumped by 1bp. How to deal with the resulting discontinuity in the curve? Should the market prices and the bumped tenor be interpolated?

This contrasts with this answer, where PV01 is defined as the change in present value from a 1bp parallel shift in the yield curve.

## Answer by Dimitri Vulis (score 5, accepted)

https://quant.stackexchange.com/a/77213

The linked ISDA document is ISDA Standard Initial Margin Model (SIMM) Methodology version 2.3. Here is version 2.5 https://www.isda.org/a/Pf2gE/ISDA-SIMM-v2.5.pdf , although 22 is unchanged.

Item 22 says (adding bold):

> For Interest Rate risk factors, the sensitivity is defined as the PV01. The PV01 of an instrument $i$ with respect to tenor $t$ of the risk-free curve $r$ (ie the sensitivity of instrument $i$ with respect to the risk factor $r_t$) is defined as...

so we have some instrument $i$, such as, for example an interest rate swap being marked to some pricing model, or a cash bond, whose price is observable, and some model that predicts the price impact of interest rate changes.

The risk-free interest rate curve $r$ for some currency is usually built or bootstrapped from a series of observable market rates, which the ISDA document denotes by $r_t$, e.g. 2-year, 3-year, 5y, etc swap rates, and perhaps some interest rate futures too. E.g., for most developed countries' currencies and the 30-year tenor, we can usually observe a 30-year swap rate and use some kind of interpolation model to price, e.g., a 28-year interest rate swap.

Most people tasked with calculating interest rate risk consider the following two kinds of risk scenarios, and sometimes many other additional scenarios:

- all the market rates shifted by 1 basis point together - the impact of this parallel-shift scenario gives you the same information as "duration"; and

- each market rate in turn is shifted by 1 basis point, while other rates remain unchanged - the impact of each such scenario gives you the same information as "key rate duration". (If we're perturbing an interest rate future, we should try to perturb its rate, rather than the future's price.)

For many simple instruments, these two may suffice, because:

1 the sum of the impacts of 1bp shifts in each instrument is close enough to the impact of a 1bp parallel shift

2 the impact of a $n$bp move for a small enough $n$ is close enough to $n\times$ the impact of 1bp shift. This generally doesn't work for large $n$, so most people also consider "stress" scenarios, where rates move by hundreds of basis points.

(For instruments whose interest rate sensitivity is too non-linear, people have to look at more risk scenarios, for example, the impact of 1 or 2 standard deviations of the first 3 historical principal components of the interest rates curve.)

So I think your question is two-fold:

1 do we calculate the impact of only parallel-shift risk scenarios or tenor buckets as well? In my very humble opinion, there's little excuse not to calculate both, given the abundance of readily available free tools. But for a quick-and-dirty approximation, one might, for example, calculate only the parallel-shift impact of a portfolio of bonds, and then place them into tenor buckets by maturity.

2 How do we know that perturbing just 1 rate by 1bp, and leaving others unchanged, won't break something? This is a very good question because, in general we don't! Indeed, when stress-testing a interest rate curve model, it's useful to try how the implementation would behave if we first overconstaint it by unputting too many closely spaced rates, and try to perturb the inputs in ways that would impy very negative forward rates, admit arbitrage, and generally give rise to pertured curves that "can't happen".

But in practice, suppose for concreteness that we've observed a 10 years swap rate = 4.12%, and then a 11 year swap rate = 4.75%, and we are tasked with figuring out a discount factor for a cash flow in 10 years and 6 months. As you said, we use some interpolation, often quite complicated. We don't just linearly interpolate the 10.5 year swap rate to exactly 4.435%, although in practice it's likely to be close. There's lots of literature out there on, basically, how to interpolate interest rate curves to achieve some desired behavior.

If we next perturb the 10 year rate to 4.12%+1bp=4.13%, and solve again for the interpolation parameters, we may change slightly the discount factors for every date, even those far away from 10 years, but hopefully we'll only see "numerical noise" sized impact outside of the 9 year -- 11 year range. In particular, the 10.5 year swap rate in the perturbed curve will be pretty close to to 4.44%.

Edit: an additional point from comments. What can you when you need a sensitivity to a market rate that is not observable, or that was not used to build the interest rate curve for some other reason?

Example: in 20 years you have a cash flow in an emerging markets currency that does not have 15 year and 30 year swap rate quotes, and yet you need to calculate the sensitivities to those tenors. Most interest rate curve models let you simply interpolate or extrapolate the swap rates at the additional tenors, then build the curve adding these new tenors as inputs, and not materially change any discount factors. Some models need a little tweaking to ensure that adding instruments changes nothing marerially. Now you can perturb the new tenors and reprice.

Example: you build an interest rate curve using interest rate futures for tenors up to 5 years, and swap rates afterwards. You use futures because they're more liquid and you'd use them, rather than swaps, to hedge interest rate risk in those tenors. You calculate the sensitivities of your instrument to the futures. Yet you also need to calculate the sensitivities to the swap retes in tenors before 5 years. You can calculate the matrix of sensitivities of the swap rates to the futures, invert it, and multiply by the vector of sensitivities of the instrument to the futures to get the sensitivities of the intruments to the swap rates. However this introduces some numerical noise.

It is a good industry practice, becoming more widespread, to tag the resulting risk numbers with metadata noting all these transformations, so that any risk report can be footnoted to clearly show, e.g. that the 30-year rate war rather aggressivly extrapolated from shorter tenor market observables.