PnL Explained FAQ (2024)

PnL Explained - FAQ

Question 1) What is MTM?

Answer 1) MTM is short forMark-to-Market and in the context of trading means the value of something,i.e., a trade. This concept is alsocalled ‘Present Value’. See below andsee that the general formula for trading PnL can be expressed as:

PnL = MTM today – MTM Prior Day.

Click here for more information about MTM

Question 2) What is PnL?

Answer 2) PnL stands for Profit and Loss. The 'and' usually gets written as a 'n' or'N' or '&' (as in 'PnL', 'PNL' or 'P&L). PnL is the way traders refer to the dailychange to the value of their trading positions. The general formula for PnL is PnL = Valuetoday minus value yesterday.

So if you are a trader and yourpositions were worth $100 yesterday and today they are worth $105, then yourPnL for the day was $5. It is a profitof 5.

So if you are a trader and yourpositions were worth $111 yesterday and today they are worth $105, then yourPnL for the day was -$6 and it was a loss.

The value of something is alsoknown as the Mark-to-Market (MTM) which is defined in this FAQ as the presentvalue of all current (meaning today) andexpected future cash flows (or physical flows for physically settlingdeals). It is common practice to addback in any cash flow from the prior day when computing PnL, otherwise PnLwould be misrepresented by the amount of the cash that is paid/received.

Note that 'deal' and 'trade' areused interchangeably and mean the same thing in this FAQ.

Example

On March 23 a trader does a trade thatmeans his firm will receive 2 cash payments, one for $50 on March 25 and onefor $100 on March 27. This table showsthe MTM, current day's payments and PnL… all assuming values arenon-discounted, meaning that interest rates are zero or not counted.

The important point the above tableshows is that after the first day, the PnL is zero because making payments (or receivingcash) doesn't change the trading PnL.

If we add in the effects ofdiscounting / present value / time value of money / interest rates. The concept of present value (PV) and future value(FV) is that a dollar today is worth more than a dollar in the future. That is because you can put less than adollar in the bank today and earn interest on it and have $1 in the back in thefuture. The ratio of PV to FV is calledthe discount factor (DF) and you have these formulas:

DF = PV / FV

or

PV = DF * FV

e.g., if you have $100 and put isin the back for a year and have $102 in one year then your present value is$100, your future value is $102 and your discount factor is 0.980392157. Note that discount factors are always between0.00 and 1.00.

Note that the PnL from March 24 to March25 is $0.12 and that comes from the change in the discount factors which cancome from two sources: a) The fact that there is one day fewer to the ultimatepayment and b) changes to interest rates.In other words, both interest rates and time (time to payment date) playa role in determining the discount factor used to future payments and presentvalue them.

Question 3) What is PnL Explained?

Answer 3) PnL Explained isthe practice of attributing the changes in the daily value (i.e., PnL) into categories. It is sometimes called 'PnL Attribution'which means the same thing (or P&L Explained or P&L Attribution orProfit and Loss Explained / Profit and Loss Attribution). Sometimes the categories are called'buckets' so the act of attributing PnL into categories is sometimes called'bucketing'. The categories/bucketstypically appear as columns in a PnL Explained report.

There are three sources of PnL inthe above example. The PnL comes fromnew trades, changes in time, and changes to interest rates. The sources/categories/buckets of PnL changesare often labeled something like 'Change in MTM value due to changes in time'or, more commonly, 'Impact of Time'.

The below table takes the aboveexample and buckets the PnL into the three sources applicable for thisexample. Note that the numbers are justexamples… the breakdown between Impact of Time and Impact of Interest Rates isjust for this example and not something you could calculate with just the informationgiven so far.

Sample PnL Explained Reports overseveral days.

See that the sum of the threeexplanatory columns adds up to the PnL?That is the ideal case for a PnL Explained report. In order to help out the reader of a PnLExplained report, the report will typically include a column summing theexplanatory columns called 'PnL Explained' and another column showing thedifference between the 'PnL' column and the 'PnL Explained' column called 'PnLUnexplained'. For example, for the March25 example values:

If for some reason, the formula forPnL due to changes in interest rates was off and calculated $0.05 instead of$0.07 then the report would look like this

PnL Unexplained is bad and shouldbe avoided, meaning to be minimized or reduced to zero. Depending on the methodology used, it may notbe possible to eliminate all PnL Unexplained.

Question 4) What are the methodologies for calculating PnL Explained?

Answer 4) There are two methodologies for calculating Pnl Explained, the 'sensitivities'method and the 'revaluation' method.

TheSensitivities Method involves first calculating option sensitivities knownas the greeks because of the common practice of representing the sensitivitiesusing Greek letters. For example, the delta of an option is the value an optionchanges due to a $0.01 move in the underlying commodity or equity/stock. Tocalculate 'Impact of Prices' the formula is

Impact of Prices = Option Delta *Price Move

so if the price moves $0.05 and theoption's delta is $100 then the 'Impact of Prices' is $500.

TheRevaluation Method recalculates the value of a trade based on the currentand the prior day's prices. The formula for Impact of Prices using theRevaluation Method is

Impact of Prices = (Trade Valueusing Today's Prices) - (Trade Value using Prior Day's Prices)

Question 5) What are the pros and cons of the Sensitivities Methodversus the Revaluation Method?

Answer 5)

Question 6) How do you calculate 'Impact of Gamma' (aka Gamma PnL),i.e., changes in PnL due to option gamma?

Answer 6) First… some definitions…

For example, the delta of an optionis the value an option changes due to a $0.01 move in the underlying commodityor equity or bond. The gamma is how muchthe delta changes for a $0.01 move.

For example… suppose you have acommodity trading at $50. Suppose thedelta of your position is currently $10. In other words.. you make $10 if theprice of the underlying goes up $0.01 to $50.01You could put that in a table like this:

With a non-option trade…. such as afutures or a swap… the delta won't change… it remains the same… so they'll callthis a linear (meaning in this case unchanging in a straight line) trade… orcall it a linear instrument….

You get something like this forvarious market prices

In other words, whatever the marketprice (and I just show it for the unchanged prices of $50 plus and minute acouple of cents… the delta is the same.

Now suppose we are talking about anoption trade…. and suppose the gamma of the trade is $1. That means that if the delta of an option is$10 now (i.e., with the underlying trading at $50… then the new delta will be$11 (i.e., old delta of $10 plus the gamma of $1) if the market price of theunderlying goes to $50.01.

We can put that in a table likethis:

Notice that the delta goes up from $10to $11 when the market price goes up $0.01 and it goes down to $9 if the marketprice goes down $0.01. Notice also thatthe rate of change of the delta isn't changing… the gamma is staying at $1….that is not realistic. In reality thegamma would also be changing… however for the simplicity of this example I keptthe gamma at $1 for each $0.01 move in the underlying price.

The gamma at each price of theunderlying would be shown like this:

Now let's add in the value of theoption trade… and let's assume that the value is $200.

The option could be anything… none ofthe trade details matter except for the value of the option (which is $200),the delta (which is $10) and the gamma (which is $1). However… in order to make the exampleclearer… let's assume the option is…

The right to buy 100 barrels ofcrude oil at a strike price of $50 when crude oil is trading at $50/barrel andthe option expires in three month. I.e., the underlying is crude oil and it isan at-the-money call option.

So what would the value be atdifferent market prices (i.e., different prices of the underlying crudeoil)? We can create a table like this:

We filled in the option price of$200, which is what it is based on our assumption for this example. So what would it be if the price moves up$0.01 to $50.01? First let's just takeinto account the delta… The delta is by definition how much the option pricewill go up if the underlying goes up $0.01… since the delta is $10, the optionprice is $210 (which is $200, the unchanged value, plus $10, the delta).

This table has the new optionvalues as calculated just taking into account the delta of the option as it isright now… which is $10.

The above is close, but not quiteright… because while the delta is $10 now (crude oil at $50/barrel) it goes upto $11 when crude oil goes up $0.01 to $50.01.So should the table look like this?

That is also not quite right… thebest answer is to realize that the option delta is gradually changing from $10 (withcrude oil at $50) to $11 (with crude oil at $50.01) and take the average… i.e.,you get $10.5 which is ($10 + $11) / 2.

With smaller price increments… of$0.001 instead of $0.01, you see the option delta changing…

See that how on average, the deltais $10.50 as the crude oil price goes from $50 to $50.01 (and the delta goesfrom $10 to $11)

Now we are ready to populate thefull table of option prices taking into account both the delta and the gamma.

Note that the way we did this is asfollow…

Step 1) Use the gamma (i.e., theoriginal gamma from when the price of the underlying is $50) to calculate thedelta for different prices… in this case a range of prices from $49.98 to$50.08).

Step 2) Now that we have calculatedthe deltas (i.e., the delta for each $0.01 increment…. we calculate the newmarket prices by taking the original market price and adding (or subtracting)the average of the deltas. E.g., From anunderlying price of $50.05 to $50.06 the price of the option goes up by theaverage of the deltas, i.e., the market price goes up by $15.50.

Now we can look at a comparison ofthe two approaches…. in one case we just look at the change in the option priceif we assume that the current delta, which is $10… isn't changing… and theother case we'll use the correctly calculated option prices.

Now we can look at the change inthe option price (i.e., the PnL) compared to the unchanged value (i.e., $200)and get this table:

Now we can figure out the extraimpact that taking into account the gamma of an option has versus just looking atthe (original) delta… we'll just subtract the two rows above… i.e., the impactof delta and gamma (bottom row) minus the impact of delta row (second frombottom).

Notes:

1) Note that the impact of gamma isalways a positive number (in this example) while the impact of delta can bepositive or negative.

2) Note that in order to explainPnL for price moves (i.e., the price of the underlying moving)… you need to addup both Impact of Delta and Impact of Gamma to get the full PnL predictedamount.

Now that we worked out the stepsand produced the table above the long way, i.e., each value by hand, we areready to condense that work into formulas.The formula for Impact of Delta is:

Impact of Delta = Delta (from theprior day) * [ (today's price - prior day's price) / delta shift ]

The delta shift is $0.01.. which issometimes called the tick size.

For example, if today's price is$50.02 and yesterday's price is $50.00 then the Impact of Delta is:

$20 = $10 * [ ($50.02 - $50.00) / 0.01]

The formula for impact of gamma hasto take into account both that it is the average of the high/low delta and thatthe deltas change over time by the gamma… the formula is:

Impact of Gamma = Gamma (from theprior day) * [ (((today's price - prior day's price) / delta shift)^2) / 2 ]

You'll notice that the formula forImpact of Gamma is like the Impact of Delta formula with the added

a) squared (i.e., the ^2 meanssquared)

and

b) the divide by 2.

For example, if today's price is$50.04 and yesterday's price is $50.00 then the Impact of Gamma is:

$8 = $1 * [ ((($50.04 - $50.00) / 0.01) ^2) / 2]

$8 = $1 * [ ((($0.04) / 0.01) ^2) / 2]

$8 = $1 * [ (((4) ^2)/ 2]

$8 = $1 * [ (16/ 2]

$8 = $1 * [ 8]

Question 7) In a PnL Explained report, how would you report PnL due toa) new trades and b) trade amendments?

Answer 7)

New Trades – PnL due to new trades,i.e., trades done on the current date, is typically put in its own column(a.k.a. its own ‘bucket’). The columncould be named ‘new trade pnl’ or ‘impact of new trades’. If a finer granularity is desired, you couldput the PnL from new trades into multiple columns (a.k.a. ‘buckets’). For example, you could split it by deal type,e.g., ‘new options PnL’ vs. ‘new non-options PnL’.

Amendments – PnL due to tradeamendments is also typically shown in a PnL Explained report in a singlecolumn. As with the ‘impact of newtrades’, there is no one right way to show causes of PnL, i.e., more than oneright number of columns/buckets. Forexample, you could have one column for PnL due to amendments in trade volumeand a separate column for amendments for other (i.e., not volume) changes.

Question 8) Can you express source of PnL, i.e., buckets/columns in aPnL Explained report as a percent?

Answer 8)

By way of an example… suppose youare short one call option… and you have PnL of +$1,000 due to these causes:

a) From one day to another youroption value drops to decreasing time to expiration, also known as‘theta’: +$300

b) The market price moves down,making it less likely that option will be exercised (in this example, you areshort the option, so you want the option to expire worthless: +$500

c) The implied volatility of theoptions as valued using market prices goes down… making it less likely that theoption will be exercised: +$200

To recap:

Impact of Time (a.k.a. ‘theta’) : +$300

Impact of Prices (i.e., price change): +$500

Impact of Volatility (changes): +$200

and a total PnL of $1000… so no PnLhas not been unexplained.

The percent contribution of eachitem would be:

Impact of Time (a.k.a. ‘theta’) : 30%

Impact of Prices (i.e., price change): 50%

Impact of Volatility (changes): 20%

which totals to 100%

So far, so good. However, what if we have some sources of PnLas positive numbers and some as negative numbers?

For example:

Impact of Time (a.k.a. ‘theta’) :+$300

Impact of Prices (i.e., price change): -$100

Impact of Volatility (changes): +$100

for a total PnL of -$300

There isn’t a universally acceptedway to derive percentages in this case, though you are welcome to use whatevermethod is helpful for you.

Now let’s assume that theexplanatory factors are not perfect… and that there is some unexplained.

Support we have $1000 of PnL ofwhich $900 is explained, i.e., we are able to attribute it to known causes and$100 is still unexplained. We could saywe have 10% of the PnL unexplained.However, is that useful?Possibly not. For example, theMTM, i.e., the value of the original deal could be $100,000,000 on one day andup to $100,001,000 the next day for our total PnL of $1000. Now a $100 of unexplained might be 10% of thePnL change, but it is a negligible percent of the overall value of thetrade.

Also… unexplained could be anegative value. For example, we couldhave $1000 in PnL and yet when we calculate our ‘explained’ formula… we get$1100 (instead of $900 for the previous example. So unexplained could be -$100. You could say that is -10% unexplained, butthat may not make sense or offer value.

Also… you could have the situationwhere PnL is actually $0.00 (zero) and yet your formula comes up with anexplained of $100. So now you would have$100 / 0 unexplained so either an infinite percentage or an undefinedpercentage depending on how you look at it.

To recap: Looking at PnL Explainedattributions can be done and because there is no one universally sensible andaccepted way to do it, therefore it is up to you to decide on a way that makessense and has meaning for you.