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PnL Explained - FAQ

Question 1) What is MTM?

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

PnL =  MTM today – MTM Prior Day.

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 daily change to the value of their trading positions.   The general formula for PnL is PnL = Value today minus value yesterday.

So if you are a trader and your positions were worth \$100 yesterday and today they are worth \$105, then your PnL for the day was \$5.  It is a profit of 5.

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

The value of something is also known as the Mark-to-Market (MTM) which is defined in this FAQ as the present value of all current (meaning today) and  expected future cash flows (or physical flows for physically settling deals).  It is common practice to add back in any cash flow from the prior day when computing PnL, otherwise PnL would be misrepresented by the amount of the cash that is paid/received.

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

Example

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

 Undiscounted MTM (current and future cash flows) PnL (MTM today -MTM prior + prior day's cash flows) Current Day Cash Flows (payments/ receipts) March 22 \$0 \$0 March 23 \$150 \$150 March 24 \$150 \$0 March 25 \$150 \$0 \$50 March 26 \$100 \$0 March 27 \$100 \$0 \$100 March 28 \$0 \$0

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

If we add in the effects of discounting / 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 a dollar in the bank today and earn interest on it and have \$1 in the back in the future.  The ratio of PV to FV is called the discount factor (DF) and you have these formulas:

DF = PV / FV

or

PV = DF * FV

e.g., if you have \$100 and put is in 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 between 0.00 and 1.00.

 Undiscounted MTM (current and future cash flows) PnL (MTM today -MTM prior + prior day's cash flows) Current Day Cash Flows (payments/ receipts) March 22 \$0.00 \$0.00 March 23 \$149.50 \$149.50 March 24 \$149.60 \$0.10 March 25 \$149.72 \$0.12 \$50 March 26 \$99.83 \$0.11 March 27 \$100.00 \$0.17 \$100 March 28 \$0.00 \$0.00

Note that the PnL from March 24 to March 25 is \$0.12 and that comes from the change in the discount factors which can come from two sources: a) The fact that there is one day fewer to the ultimate payment and b) changes to interest rates.   In other words, both interest rates and time (time to payment date) play a role in determining the discount factor used to future payments and present value them.

Question 3) What is PnL Explained?

Answer 3)  PnL Explained is the 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 or Profit 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/buckets typically appear as columns in a PnL Explained report.

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

The below table takes the above example and buckets the PnL into the three sources applicable for this example.  Note that the numbers are just examples… the breakdown between Impact of Time and Impact of Interest Rates is just for this example and not something you could calculate with just the information given so far.

Sample PnL Explained Reports over several days.

 PnL (MTM today -MTM prior + prior day's cash flows) Impact of New Trades Impact of Time Impact of Interest Rates March 22 \$0.00 March 23 \$149.50 \$149.50 March 24 \$0.10 \$0.03 \$0.07 March 25 \$0.12 \$0.02 \$0.10 March 26 \$0.11 \$0.01 \$0.10 March 27 \$0.17 \$0.01 \$0.16 March 28 \$0.00

See that the sum of the three explanatory 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 PnL Explained report, the report will typically include a column summing the explanatory columns called 'PnL Explained' and another column showing the difference between the 'PnL' column and the 'PnL Explained' column called 'PnL Unexplained'.  For example, for the March 25 example values:

 PnL PnL Explained PnL Unexplained Impact of New Trades Impact of Time Impact of Interest Rates \$0.10 \$0.10 \$0 \$0 \$0.03 \$0.07

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

 PnL PnL Explained PnL Unexplained Impact of New Trades Impact of Time Impact of Interest Rates \$0.10 \$0.10 \$0.02 \$0 \$0.03 \$0.05

PnL Unexplained is bad and should be avoided, meaning to be minimized or reduced to zero.  Depending on the methodology used, it may not be 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.

The Sensitivities Method involves first calculating option sensitivities known as the greeks because of the common practice of representing the sensitivities using Greek letters. For example, the delta of an option is the value an option changes due to a \$0.01 move in the underlying commodity or equity/stock. To calculate 'Impact of Prices' the formula is

Impact of Prices = Option Delta * Price Move

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

The Revaluation Method recalculates the value of a trade based on the current and the prior day's prices. The formula for Impact of Prices using the Revaluation Method is

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

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

 Pros Cons The Sensitivities Method 1) Since this method uses the greeks (delta, gamma, vega, theta, etc) and since many trading systems already calculate the greeks, this method can be easier to implement than the revaluation method. 1) The sensitivity method is inherently incapable of explaining P&L unless all first, second, and higher order sensitivities are calculated as well as all cross effects.  However, calculating all sensitivities is not usually practical from a performance point of view. The Revaluation Method 1) Can be fully accurate, meaning there can be no explained since the revaluation method isn't subject to the limitations in accuracy of the sensitivities method as it is typically implemented. 1) Does not allow for PnL to be attributed to second order effects.

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

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

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

 Underlying Price \$50 Delta \$10

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

You get something like this for various market prices

 Futures Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 Delta \$10 \$10 \$10 \$10 \$10

In other words, whatever the market price (and I just show it for the unchanged prices of \$50 plus and minute a couple of cents… the delta is the same.

Now suppose we are talking about an option 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 the underlying goes to \$50.01.

We can put that in a table like this:

 Option Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 Delta \$8 \$9 \$10 \$11 \$12

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

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

 Option Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 Delta \$8 \$9 \$10 \$11 \$12 Gamma \$1 \$1 \$1 \$1 \$1

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

The option could be anything… none of the 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 example clearer… let's assume the option is…

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

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

 Option Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 Delta \$8 \$9 \$10 \$11 \$12 Gamma \$1 \$1 \$1 \$1 \$1 Option Price ???? ??? \$200 ??? ???

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 take into account the delta… The delta is by definition how much the option price will go up if the underlying goes up \$0.01… since the delta is \$10, the option price is \$210 (which is \$200, the unchanged value, plus \$10, the delta).

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

 Option Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 Delta \$10 Gamma \$1 Option Price - Just taking into account the delta with the price unchanged (at \$50.00) \$180? \$190? \$200 \$210? \$220?

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

 Option Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 Delta \$8 \$9 \$10 \$11 \$12 Gamma \$1 \$1 \$1 \$1 \$1 Option Price - Just taking into account the delta… i.e.., ignoring the gamma for now \$183? \$191? \$200 \$211? \$223?

That is also not quite right… the best answer is to realize that the option delta is gradually changing from \$10 (with crude 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…

 Option Trade Price Unchanged Underlying Price \$50 \$50.001 \$50.002 \$50.003 \$50.004 \$50.005 \$50.006 \$50.007 \$50.008 \$50.009 \$50.01 Delta \$10 \$10.1 \$10.2 \$10.3 \$10.4 \$10.5 \$10.6 \$10.7 \$10.8 \$10.9 \$11

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

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

 Option Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 \$50.03 \$50.04 \$50.05 \$50.06 \$50.07 \$50.08 Delta \$8 \$9 \$10 \$11 \$12 \$13 \$14 \$15 \$16 \$17 \$18 Gamma \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 Option Price - Taking into account the delta and the gamma \$187.50 \$191.50 \$200.00 \$210.50 \$222.00 \$234.50 \$248.00 \$262.50 \$278.00 \$294.50 \$312.00

Note that the way we did this is as follow…

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

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

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

 Option Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 \$50.03 \$50.04 \$50.05 \$50.06 \$50.07 \$50.08 Delta \$8 \$9 \$10 \$11 \$12 \$13 \$14 \$15 \$16 \$17 \$18 Gamma \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 Option Price - Just taking into account the current delta (i.e., \$10) \$180.00 \$190.00 \$200.00 \$210.00 \$220.00 \$230.00 \$240.00 \$250.00 \$260.00 \$270.00 \$280.00 Option Price - Taking into account the delta and the gamma \$182.00 \$190.50 \$200.00 \$210.50 \$222.00 \$234.50 \$248.00 \$262.50 \$278.00 \$294.50 \$312.00

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

 Option Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 \$50.03 \$50.04 \$50.05 \$50.06 \$50.07 \$50.08 Delta \$8 \$9 \$10 \$11 \$12 \$13 \$14 \$15 \$16 \$17 \$18 Gamma \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 Option Price Change - Just taking into account the current delta (i.e., \$10) -\$20.00 -\$10.00 \$0.00 \$10.00 \$20.00 \$30.00 \$40.00 \$50.00 \$60.00 \$70.00 \$80.00 Option Price Change - Taking into account the delta and the gamma -\$18.00 -\$9.50 \$0.00 \$10.50 \$22.00 \$34.50 \$48.00 \$62.50 \$78.00 \$94.50 \$112.00

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

 Option Trade Price Unchanged Underlying Price \$49.98 \$49.99 \$50 \$50.01 \$50.02 \$50.03 \$50.04 \$50.05 \$50.06 \$50.07 \$50.08 Delta \$8 \$9 \$10 \$11 \$12 \$13 \$14 \$15 \$16 \$17 \$18 Gamma \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 \$1 Impact of Delta -\$20.00 -\$10.00 \$0.00 \$10.00 \$20.00 \$30.00 \$40.00 \$50.00 \$60.00 \$70.00 \$80.00 Impact of Gamma \$2.00 \$0.50 \$0.00 \$0.50 \$2.00 \$4.50 \$8.00 \$12.50 \$18.00 \$24.50 \$32.00

Notes:

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

2) Note that in order to explain PnL for price moves (i.e., the price of the underlying moving)… you need to add up both Impact of Delta and Impact of Gamma to get the full PnL predicted amount.

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

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

The delta shift is \$0.01.. which is sometimes 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 has to take into account both that it is the average of the high/low delta and that the deltas change over time by the gamma… the formula is:

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

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

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

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 to a) new trades and b) trade amendments?

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 column could be named ‘new trade pnl’ or ‘impact of new trades’.  If a finer granularity is desired, you could put 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 trade amendments is also typically shown in a PnL Explained report in a single column.  As with the ‘impact of new trades’, there is no one right way to show causes of PnL, i.e., more than one right number of columns/buckets.  For example, you could have one column for PnL due to amendments in trade volume and 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 a PnL Explained report as a percent?

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

a) From one day to another your option 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 are short the option, so you want the option to expire worthless:  +\$500

c) The implied volatility of the options as valued using market prices goes down… making it less likely that the option 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 PnL has not been unexplained.

The percent contribution of each item 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 PnL as 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 accepted way to derive percentages in this case, though you are welcome to use whatever method is helpful for you.

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

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

Also… unexplained could be a negative value.  For example, we could have \$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, but that may not make sense or offer value.

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

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