Funding Value Adjustments and Discount Rates in the Valuation of Derivatives John Hull Marie Curie Conference, Konstanz April 11, 2013 1
Question to be Considered Should funding costs be taken into account when derivatives are valued? Background Paper: Should a Derivatives Dealer Make a Funding Value Adjustment? By John Hull and Alan White on www.rotman.utoronto.ca/~hull 2
Steps for Dealer in Valuing Uncollateralized Portfolio of Derivatives with a Counterparty Value the portfolio assuming that neither side will default Adjust for the possibility that the counterparty will default (CVA) Adjust for the possibility that dealer will default (DVA) Final Value = No-default value CVA + DVA The Question: Should an FVA adjustment be made to the no-default value to reflect the dealer s funding costs? 3
The Traditional Risk-Neutral Valuation Approach Project market variables in a risk-neutral world and discount expected payoffs at the risk-free rate This produces a no-default value for the derivative. A risk-free rate is necessary to define the discount rate and to define risk-neutral growth rates 4
The Risk-Free Rate Before the crisis of 2008, practitioners calculated the risk-free zero curve from LIBOR and LIBOR-swap rates Following the crisis the overnight indexed swap (OIS) rate has emerged as a candidate for the risk-free rate Current practice is to use OIS for collateralized transactions and LIBOR for non-collateralized transactions 5
Theory vs. Practice To practitioners, the risk-free rate has always been an estimate of the average funding cost This explains the FVA adjustment FVA can be defined as the difference between valuing a portfolio of uncollateralized transactions using the assumed risk-free rate and valuing it using the bank s average funding cost If the trader buys or sells at the FVA-adjusted price, delta hedges, and the average funding cost applies to funds used or funds generated, the trader should break even. 6
But FVA flies in the face of finance theory. The discount rate for an investment should reflect the risk of the investment s cash flow, not the company s average funding costs 7
The Motivation for FVA Finance theory has a valuation focus. Projects with a positive NPV are accepted and those with a negative NPV are declined In banking the performance of different activities (including derivatives trading) is calculated by a return on capital measure. Funds are charged at the bank s average funding cost and capital is regulatory or economic capital 8
Application to Lending For a particular type of lending, the interest charged is 6%, the average cost of funds is 3%, admin. costs are 0.8% per year, loan losses are 1% per year, and capital required is 5% of loan principal Pre-tax return on capital is 24% Similar calculation is used to determine the performance of the derivatives desk ex post 9
Example (Rates are compounded annually) Trader sells a one-year European call option on a non-dividend paying stock at the beginning of a year and perfectly delta hedges it to the end of the year The strike price is 100, the stock price is 120, volatility is 30% The risk-free interest rate in the bank systems is 2% and bank s average funding cost is 4% If the trader sells for the price given by the bank s systems (26.79) and pays 2% for funding, the trader breaks even 10
Simple Example continued But if the trader is charged interest at 4%, a loss of 1.34 will be incurred. This is because the value of the derivative increases from 26.79 to 28.13 when the discount rate is increased to 4% Not surprisingly the trader argues that the bank systems should discount at 4% 11
Fair Value The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between market participants at the measurement date The fair value is not in general the same as the FVA-adjusted price 12
Determining Fair Value The role of a model is nearly always to calculate the fair value of a derivative that is not actively traded consistently with other derivatives that are actively traded The procedure is to imply calibrating parameters such as implied forward prices, implied volatilities, and implied correlations from actively traded derivatives Interpolation procedures are then used to estimate these parameters for the derivative of interest 13
Determining Fair Market Value continued For fair market values to be accurate, it does not matter whether the interest rate is OIS or LIBOR or LIBOR plus a spread But, if volatilities and other implied parameters are communicated from one dealer or broker to another, errors will creep into the calculation if they are not all using the same interest rate 14
Can We Match Both Funding Costs and Market Prices? For options the answer is yes In the earlier example we can choose a discount rate of 4% and change the implied volatility from 30% to 25.66% to match a market price of 26.79 But if market prices correspond to an interest rate of 2%, this is what the trader will earn if the trader s model is calibrated to market prices 15
There Is Only One Fair Value The fair value of a derivative (or any other asset) is the price that balances supply and demand The funding costs of individual dealers may play a role in determining supply and demand, but this does not mean that they should be used to determine fair market value 16
Agreement on Fair Value If they have the same models and values for market variables, two parties, A and B, should agree on the value of a derivatives portfolio after CVA, DVA, and CRA adjustments have been made This is because B s no-default value is equal and opposite to A s no-default value A s CVA is B s DVA and vice versa A s CRA is equal and opposite to B s CRA 17
FVA Creates Disagreement If A and B have different borrowing rates, their FVA adjustments will not be equal and opposite This means that if they use FVA-adjusted prices as fair values they will not agree 18
Theoretical Arguments The incremental funding costs of a risk-free investment should be the risk-free rate Using the same discount rate for all investments is potentially dysfunctional There are two DVAs: DVA1 (impact of default on derivatives) and DVA2 (impact of default on derivatives funding) DVA1 is included in valuation FVA is equal in magnitude and opposite in sign to DVA2 19
Unintended Consequences of FVA Low-funding-cost dealers will tend to be most competitive when they are a) shorting forwards, b) selling uncollateralized call options, and c) buying uncollateralized put options from end users. High-funding-cost dealers will tend to be competitive when they do the reverse Using collateralized derivatives to hedge uncollateralized derivatives will not be done optimally 20
To Summarize: The Key Problem We need models that produce prices consistent with fair values But the incentives provided for traders lead to them to want to use a different models from these for trading 21
Possible Solutions Use the trader model for calculating fair value (Accountants are unlikely to agree to this) Match market prices with trader model by adjusting volatility (Does not solve trader s problem) Use two models, one for fair value and one for trading (Trader s performance measure for a period may be out of line with the bank s reported performance) Change the way the performance of derivatives trading is assessed. (Charge funding to derivatives desk at rate assumed by the market and give Treasury department credit for DVA2) 22