Mark-to-Market P&L and Risk Factor Attribution Across Instruments

# How is PnL calculated


# How is PnL calculated












In Fixed Income, I know that bonds PnL are evaluated depending on where the price lies on price/yield curve at the end of the day, compared to where it started from at beginning of the day. The portfolio of bonds will have a specific DV01, which will be used to compute the PnL.

Can someone tell me if this is right or is there something more? For equities it should be just a simple sum of stock prices at the end of day vs beginning of day? Is this right?

## Answer by Helin (score 8)

https://quant.stackexchange.com/a/14077

You can use DV01 * (change in yields) to calculate the approximated P&L, but you really shouldn't do it.

The exact PnL calculation depends on the instruments you're trading. If it's exchange-traded (e.g., futures, futures options), then its price is readily available from the exchange, and the daily change in price should be used for marking to market.

For a bond in general, the daily PnL is (today's clean price + today's accrued interest + coupon payments received today if any) – (yesterday's clean price + yesterday's accrued interest). Oftentimes, government bond trades are financed by repo, so you need to subtract the financing cost from the quantity above, which is roughly the original dirty price (clean price + accrued interest) * repo rate * 1 / 360 (assuming holding period of 1 day and day count convention of Act/360).

For swaps, you'll need to calculate its new market value using the new swap curve. Swaptions are similar – you'll also need to reprice it using the new swap curve & vol cube.

## Answer by Kiwiakos (score 6)

https://quant.stackexchange.com/a/14112

Assuming that you are working for a bank, there are three different P&Ls depending on the function/ usage:

- Actual P&L calculated by Finance/ Product Control and is based on the actual price of the instrument in the market (or the corresponding model if a market does not exist). This reflects the true P&L if the position is closed at market prices. In many cases (like bonds in your case) these prices are observed and unambiguous, this is 'marking to market'; in other cases (where you might hold an illiquid exotic, like a PRDC for example) this price is estimated by the Front Office pricer, this is 'marking to model'.

- Clean P&L which is the Actual minus fees, commissions, bid-offer spreads, intraday trading, and other elements like reserve P&L applied to capture marking to model risk. This is also calculated by Finance/ Product Control. Note that this depends on the local regulation, therefore the same position can potentially have different Clean P&L if booked in books that are subject to different regulators. Clean P&L is used for backtesting VaR models for regulatory capital.

- Hypothetical (hypo) P&L which is based on some model (risk or front office) that is perturbed by the actual movements of the factors involved. This might as simple as 'PV01 * shock', reading off a stylized yield curve as you describe, or it might be more involved. Typically this is calculated by some Risk function. The empirical relation between Hypo and Clean P&L is used to produce diagnostics under the forthcoming FRTB regulation.

## Answer by Matt B. (score 4)

https://quant.stackexchange.com/a/14082

The pnl calculation is done in 2 steps. By definition, you value your portfolio as of today, you value your portfolio as of yesterday, and the difference will be your pnl.

Now that's an important number (that gets reported, etc.) but that doesn't give you a lot of information on what generated that pnl.

The second step is to move every variable that could affect your pnl to measure the contribution that a change in this variable has on the total pnl.

For a Zero coupon bond for instance, which has a TV of $\exp(-r(T-t)/365)$ (very generic, you need to account for repo etc.). You will report the pnl as follows: PnL(1-0)=$(\exp(-r_0((T-t-1)/365)) - \exp(-r_0((T-t)/365))) + (\exp(-r_1((T-t)/365)) - \exp(-r_0((T-t)/365))) + \varepsilon$.

The first them is your carry, your $\theta$, ie the money you make because your bond is pulled to par (if there was a coupon it would be included in this). The second term is due to your change in interest rate. $\varepsilon$ is simply what you can't explain. If everything is neat, your $\varepsilon$ should not be too high. You can also see that this is very close to a Taylor expansion when everything is linear, which is why you can use your duration as an approximation for the 2nd term.

On the equity side, if you sold an option for instance, it is the same process but with more variables (volatility, spot price).

So this number is used for earnings (profit or loss) but also to monitor traders and their limits (a huge hit in one category would mean something is wrong).

Hope this helps.