How to quickly calculate PV01? Or quickly calculate notional back given PV01 and duration?

Question

is there a way to quickly calculate the PV01 of for example for a swap EUR fix floating 10 years? And is it possible to calculate the Notional, with a given PV01 and the years of the trades or duration?

Thank you very much in advance!

Answer 1

The proper way to calculate risk measures such as PV01 is:

price the instrument using un-bumped market data

bump the market data (ideally, both up and down)

re-price the instrument using bumped market data

calculate the sensitivities.

Any "quick and dirty" shortcuts will introduce noise that's not necessary because the exact calculation should be quick enough.

if you know the risk measure for some standard notional, say 10 million, and you know how much risk you want, then the notional you need is risk you want / risk for standard notional $\times$ standard notional.

If this isn't homework and you actually work with interest rate swaps, then you should at least be looking at interest rate risk by tenor bucket, not just parallel shift.

Answer 2

An approximation

If you need a "quick & dirty" approximation for the impact of a 1bp parallel shift in all quoted tenors of a 'standard' quoted instrument curve $c$, you have, very roughly, for a payer swap (pay fixed) with notional $N$ (in currency, say $USD$) and maturity $T$ (in years):

$$ \begin{align} PV01\equiv& PV(c+1bp)-PV(c)\quad [USD]\\ \approx& N\times T \times \frac{1}{10,000}\\ =& N\times T\times 1bp\\ & [USD][y][1/y]=[USD] \end{align} $$

Other approximations follow from that:

$$ \begin{align} N&\approx PV01\div T\times10,000\\ T&\approx PV01\div N \times 10,000 \end{align} $$

Background

In a single curve world, a 'textbook' annual-for-annual payer swap (we receive float) is valued as

$$ \begin{align} PV&= PV(\mathrm{float\ leg})-PV(\mathrm{fixed\ leg})\\ &=N\left(1-D_n-c\sum_{i=1}^nD_i\right) \end{align} $$

where $D_i=e^{-rt_i}$ is the discount factor for tenor $t_i$ and $r$ is the (flat) zero rate (or at least a flat spread on top of the zero rate, hence the parallel shift). The derivative w.r.t. $r$ is

$$ \begin{align} \frac{\partial PV}{\partial r}&=N\left(t_nD_n+c\sum_{i=1}^nt_iD_i\right) \end{align} $$ With $r\approx c\approx 0$ and $r,c\ll t_n$ we have $D_i\approx 1$ and thus $$ \begin{align} \frac{\partial PV}{\partial r}&\approx N\times t_n \end{align} $$ This approximation is exact for $r=c=0$.

Finally,

$$ \begin{align} PV01&\equiv PV(r+1bp)-PV(r)\\ &\approx \frac{\partial PV}{\partial r}\times 1bp\\ &\approx N\times t_n \times \frac{1}{10,000} \end{align} $$

We have introduced two approximations here:

1. By the Taylor expansion of the PV01 formula.
2. By approximating the derivative through assuming $c=0,r=0$.