This document discusses XVA, which represents the cost of running an over-the-counter derivatives operation. It includes credit valuation adjustment (CVA) for counterparty risk, debit valuation adjustment (DVA) for one's own default risk, and funding valuation adjustment (FVA) for the cost of funding collateral in pricing portfolios. The document provides examples of interest rate swaps and uses the LIBOR market model and Monte Carlo simulation to model term structures and calculate XVA.
2. PURPOSE OF XVA
XVA REPRESENTS THE COST OF RUNNING AN OTC DERIVATIVES OPERATIONS
XVA DETERMINE THE AMOUNT OF CAPITAL REQUIRED UNDER BASEL III
RISK FACTORS
COUNTERPARTY RISK (CREDIT VALUATION ADJUSTMENT)
RISK OF ONE’S OWN DEFAULT(DEBIT VALUATION ADJUSTMENT)
FUNDING COLLATERAL IN PRICING TRADING PORTFOLIO(FUNDING
VALUATION ADJUSTMENT)
EXAMPLE - BANK A TRADES WITH BANK B OR INTEL OR MICROSOFT
3. FINANCIAL CRISIS 2008
SUBPRIME
MORTGAGES
MORTGAGES WHERE RISK OF DEFAULT IS HIGHER DUE TO POOR CREDIT HISTORY OR THE RATIO OF LOAN TO VALUE IS HIGH OR BOTH (Hull, 2012)
DEMISE OF BEAR
STEARNS (GLOBAL
INVESTMENT
BANK)
MARCH 12, 2008 – STOCK CLOSES AT $61.58, AVERAGE TARGET PRICE : $98.87
MARCH 14, 2008 – STOCK CLOSES AT $30.85, AVERAGE TARGET PRICE : $93.62
MARCH 16 , 2008 – JP MORGAN AGREES TO BUY BEAR STEARNS $2 A SHARE
MARCH 17, 2008 – STOCK CLOSES AT $4.81, AVERAGE TARGET PRICE : $2
(https://www.reuters.com)
LEHMAN
BROTHERS
SEPTEMBER 15, 2008 –FILED FOR BANKRUPTCY (https://www.financialexpress.com)
4. COMPONENTS OF DERIVATIVES PRICES
(BEFORE AND AFTER THE FINANCIAL CRISIS OF 2007-2009) (Andrew Green, 2015)
PRE-CRISIS
RISK-NEUTRAL PRICE(LIBOR DISCOUNTING)
HEDGING COSTS
CVA
PROFIT
POST-CRISIS
RISK-NEUTRAL(OIS DISCOUNTING)
HEDGING COSTS
CVA & DVA
PROFIT
FVA(INCLUDING COST OF LIQUIDITY BUFFER)
KVA(LIFETIME COST OF CAPITAL)
MVA(MARGIN COST)
TVA(TAX ON PROFITS/LOSSES)
14. THE FUTURE (ANDREW GREEN, 2015)
XVA IS GROWING WITH
MORE ADJUSMENTS BEING
ADDED TO THE FRAMEWORK
XVA IS ABOUT EXPLAINING
COSTS OF DOING
DERIVATIVES BUSINESS
THESE COSTS WERE ALWAYS
PRESENT BUT IGNORED
SOME COSTS ARE NEW AND
ASSOCIATED WITH CHANGES
IN THE MARKET
1. An XVA is a generic term referring collectively to a number of different “Valuation Adjustments” in relation to various tradable assets or derivative instruments held by bank.
2. For my project I focussed on 3 components of XVA i.e. Credit , Debit and Funding Value Adjustment. These adjustment measure the risk for doing trade with particular institution or counterparty.
It’s primary purpose is to hedge for possible losses due to counterparty default. What is counterparty ? Lets just say I am Bank A trading with Bank B or giving a load to company for example Intel or Microsoft as they want to expand their business. So for Bank A , Bank B or Intel or Microsoft is a counterparty. This kind of trading where Bank A trades directly with Bank B is called OTC i.e. Over the counter trading.
Since 2008 financial crisis banks are suppose to maintain certain amount of capital for their businesses (i.e. OTC Derivatives Operations). This process is defined under a regulatory framework known as Basel III.
Possible risk factors are
counterparty credit risk i.e. Lets say Bank B defaults and refuses to pay
One’s own default risk i.e. Lets say Bank A defaults or refuses to pay
OTC stands for Over the counter i.e. one bank could trade directly with other bank. Lets just say I am bank A goes to Bank B and says I would like to do this trade with you.
Basel III is part of banking regulatory framework. This came into after 2008 financial crisis. Main objective is risk management for banks so that they have enough funds to do the trades they are planning to do. FVA is funding cost of uncollateralised derivatives(trades). It represents the costs and benefits of writing a hedge for a client who is not posting collateral, and then hedging that trade with a collateralised one in the interbank market.
Subprime Mortgages – Mortgages where the risk of default is higher due to poor credit history or the ratio of loan to value is high or both.
Bear Stearns – They owned a lot of these subprime mortgages, those mortgage back securities started to decline in 2006 but Moody’s declined there MBS to B or C levels in 11th March 2008. This when the actual demise of Bear Stearns started. There was no DVA or FVA at the time for the banks to check their on credit worthiness.
All these had a lot of cascading effects on other banks to the point that lehman brothers actually filed for bankruptcy later on in the same year.
This financial crisis was the start of XVA initiative. They are like avengers infinity stones of financial markets now. Just like infinity stones you have 6 flavours of value adjustment at the moment.
This is the comparison chart which represents the value adjustments used pre crisis and post crisis. As you can see there are 5 new value adjustments in post crisis.
We have already discussed about DVA and FVA. Now
KVA ensures banks have enough capital to survive unexpected credit, market or operational losses.
MVA ensures banks have enough collateral to manage day-to-day mark-to-market fluctuations.
TVA represents impact of taxation on profits and losses.
Now we are going to look into the mathematical approach towards these valuations models for a fixed income product i.e. Interest Rate swaps.
In a simple term Swap is an OTC agreement between two companies to exchange cashflows in futures. Interest Rate swap is the most common type of swap also know as plain vanilla swap.
Intel is borrowing at LIBOR + 20 BPS which means bank will receive floating cashflows on Notional amount 1 million.
Bank is hedging itself against LIBOR rates with having an offset swap with Microsoft borrowing at LIBOR + 60 BPS.
Intel and Microsoft wont know about these offsetting swap from the bank.
Column 1 represents Tenor structure, for simplicity of calculations I have chosen it to be a year.
Column 2 represents LIBOR rates, in actuality these will libor rates for first year and euro dollar futures for subsequent years. If you notice the values at year 6 , 8 and 9 is missing.
Column 3 represents the zero curve which does linear interpolation using the available libor rates as represented in the formula for zero curve.
Column 4 represents the discount factors which are calculated using these zero curves.
We can use the discount factors to calculate forward rates and swap rates as shown in the formula below and results in the column above.
If we notice in the beginning libor rates is low as oppose to fixed rates so the bank pays cashflows to Intel for first 2 years but for remaining 5 years it receives caslfows.
Last 3 columns represents fixed and floating PV and net value of interest rate swap per year. This net value represents a possible scenario of counterparty credit exposure for this tenor.
We will use LIBOR market models for generating various scenarios of counterparty credit exposure.
There are 2 possible market models which can be used for this purpose.
Lognormal Forward-LIBOR model – which uses market forward rates for pricing derivatives
Lognormal Forward-SWAP model – which uses market swap rates for pricing derivatives
I will focus on LFM model during this presentation.
Generic SDE represent LFM and its solution is under spot measure.
This model is arbitrage free under spot measure instead of risk neutral measure.
Risk neutral measure in pricing derivatives require determining the instantaneous short rate at each point in time. But that would be inconvenient for market models as LIBOR rates are observed quarterly or yearly.
Therefore we used spot measure for fixed maturities.
Drift terms is calculated under spot measure and is a cumulative sum of various libor rates observed or interpolated in the market. Along with observable volatilites.
As we are not using live market data for our modelling we have used functional form for calculating volatility and correlation terms.
It is not possible to apply an analytical formulae to this solution of SDE we are going to use monte carlo technique to calculate net values and counterparty exposure using the solution of SDE for swap’s term structure.
This slides shows 10 scenarios for tenor 1 to 9 using the SDE solution to generate possible forward rate scenarios using initial value given in column 2. This initial value is interpolated using observable LIBOR rates.
Using this values we will simulate 1000 scenarios for forward rates for each tenor period.
This will provide us 1000 simulated floating PV for interest rate swap’s floating leg and its corresponding net value will give us counterparty exposures scenarios.
If we reverse the results , it will give us its own default exposure.
These exposure scenarios will be used for the calculations of value adjustment.
I will elaborate on CVA results during this presentation and discuss about the formulas used for the calculation of DVA and FVA results.
First line represents the formula used for CVA calculations. I will elaborate the formula from right to left.
Vt represent the net value = receiving leg PV – paying leg PV and as we are evaluating counterparty credit exposure it make sense to consider when the net value is +ive i.e. Bank is expected to receive cashflow from counterparty.
This net value is discounted using the discounting factor we have calculated earlier normally we calculate the expected value first before discounting. Once we have cacluated the discounted Expected exposure.
We are going to calculate the probability of default for counterparty. This is normally calculated using the CDS spread available in the market and recovery rate also available in the market.
Finally we take the cumulative sum to and using recovery rate calculate the approximate value for CVA.
These values are used for marking the market value of corresponding derivatives and discount offered by the counterparty due to risk of default.
First figure represents counterparty exposure for the 1000 simulation results.
Second figure represent its corresponding expected exposure of those 1000 scenarios.
Third figure represent the discounted expected exposure along the term structure.
DVA is calculated along with CVA with the formula shown above and its called bilateral CVA. In this case the probability of default calculation are slightly different , clearly if the counterparty defaults the DVA is pointless and vice versa. So the probability of defaults is adjustment this also impact the total cost of risk and usually reduces the value of CVA based on its own default.
Discounted Expected Exposure represent the net value of swap seen from banks perspective but discounted Expected Expsure D is banks exposure seen from counterparty perspective.
FVA is divided into two component adjustments , this represents Funding Cost Adjustment and therefore is a function of bank’s credit quality.
For this we use Counterparty Effective financial rate and banks ZC bonds spreads recovery. All these values are observable in the markets except the net value of derivatives which we have calculated earlier using MC techniques on forward libor model.
Read XVA is growing
As you can see XVA
The development and understand of XVA is certainly been part of this project and as far as derivatives are concerned XVA is one of the major elements of market change.