On funding costs and the valuation of derivatives

Transcription

1 On funding costs and the valuation of derivatives Bert-Jan Nauta Double Effect September 9, 202 Abstract This paper contrasts two assumptions regarding funding costs of a bank in the context of the valuation of derivatives. One assumption, mostly used in the literature, is that funding costs are fixed. This leads to derivatives values that, in general, depend on the funding rate of the bank. The other, newly introduced, assumption is that funding costs immediately adjust after each new transaction to reflect the new asset composition. It is shown that under this assumption, in the Black-Scholes model, the funding costs of the bank do not affect the value of derivatives. It is argued that the latter assumption is more appropriate for the valuation of derivatives. Introduction This paper investigates the impact of funding costs on the valuation of derivatives. unding costs come into play since banks (or other financial actors) cannot simply borrow cash from the market at the risk free rate. Instead the bank pays a so-called funding rate for borrowing. The funding rate incorporates the premium for the default risk of the bank. In the valuation of derivatives this potentially has an impact, since the valuation methodology is based on the replication of pay-offs, that is cash flows. This replication involves borrowing and lending of the bank and potentially involves funding costs. A number of papers in recent years aim to include funding cost in the valuation of derivatives [, 2, 3, 4, 5]. E. g. Piterbarg [] derives the Black-Scholes valuation model including funding costs and collateral agreements, and Pallavicini, Perini, and Brigo [2] aim to include funding costs in a comprehensive pricing framework. bert-jan.nauta@doubleeffect.nl

2 Most of these papers assume the funding rate is fixed. However Dermine argues in [6] in the context of unds Transfer Pricing that the feedback effect of a transaction on the funding cost is important. In particular, Dermine argues that a bank with rating A should be able to invest in a AAA bond, even though seemingly its funding costs make it a loss generating deal. The argument is that the AAA bond increases the quality of the bank s assets which should be reflected in lower funding costs overall. The purpose of this paper is to include this feedback effect in the valuation of derivatives. To this end, I will contrast the assumption of fixed funding costs with the assumption that funding costs immediately reflect the new transaction. It is shown, in a Black-Scholes model, that under the latter assumption the funding costs of the bank have no impact on the value of the derivative. 2 Two cases In this section, I consider two cases based on two different assumptions. or both cases complications given by tax benefits of issued debt or other tax benefits are neglected. Specifically I will neglect the specific funding composition and assume funding to consist completely of equity. Under Modigliani-Miller conditions the results can be extended to more general funding compositions. Also I assume that the risk-free interest rate is equal to zero, since this allows for some notational simplicity. 2. Inelastic case This section reviews the derivation of the value of a zero-coupon (ZC) bond under the so-called inelastic assumption. This assumption states that the funding costs of a bank are fixed and a new transaction will not change these funding costs. Inelastic unding assumption: A new transaction does not affect the funding cost of a bank. As an example consider an uncollateralized zero coupon bond with notional m and maturity T. In the inelastic case the fair rate, s ZC, for this zero coupon bond is determined by considering the following strategy. At t = 0 bank invests m and receives s ZC bank borrows m at funding rate s At t = T if counterparty is not in default bank receives m exp(s ZC T ) if counterparty is in default bank receives 0 (under the assumption of zero recovery) 2

3 The fair rate is the rate, s ZC, that does not cause a profit or loss at time 0 for the bank: P D ZC 0 + ( P D ZC ) exp(s ZC T ) = exp(s T ) (2.) The probability of default of the counterparty, P D ZC, should be interpreted as market implied probability of default and can be related to the (implied) ZC spread, s implied ZC : exp(s implied ZC T ) = The fair rate for the ZC bond is thus given by P D ZC (2.2) s ZC = s + s implied ZC (2.3) The fair rate for the zero coupon bond includes the funding rate. 2.2 Elastic case In the elastic case, the derivation is based on the elastic funding assumption: Elastic unding assumption: The funding costs of the bank are adjusted immediately after each new transaction and fully reflect the new asset composition Under this assumption the funding rate is not given like in the inelastic case, but needs to be determined from an economic principle. To this end, the assumption is used that it should be equally attractive for investors to invest in the equity of the bank as in any other financial asset. This means that under the risk-neutral measure (and zero interest rates): E[E T ] = E 0, (2.4) where E 0 denotes the equity of the bank at time 0 and E T the equity of the bank at time T. Consider a simple balance sheet with a single asset A and funded completely with equity, for which the equity investor receives the funding rate s. The asset is a zero coupon bond with a maturity, T, and a probability of default, P D A. Then E 0 = A and E T is a random number that takes values A exp(s T ) if A is not in default and 0 if A is in default, where it is assumed that recovery is zero. The fair funding rate, s, is the rate that ensures (2.4) ( P D A )A exp(s T ) + P D A 0 = A (2.5) This results in exp(s T ) = P D A (2.6) 3

4 urthermore the fair rate on the asset, s A, can be determined by the requirement that at time 0 the equity investor should not make a profit or loss, which requires the value of the pay-off of the assets to equal the value of the equity (at time 0): E[A T ] = E[E T ]. (2.7) This requires that the return on the asset compensates the funding cost. Indeed in the simple case of a single asset, this implies that the fair rate on the asset equals the funding rate s A = s, (2.8) which gives in the familiar result exp(s A T ) = P D A. (2.9) Now the bank adds a ZC bond, ZC, with the same maturity T and a default probability P D ZC. Because of the elastic funding assumption the funding rate of the complete equity will re-adjust to the new fair funding rate. To calculate the new fair funding rate, also the joint default probability, the probability that A and ZC both default before time T, denoted by P D A&ZC, is required. As stated the assumption is that, with the new ZC bond, the funding rate is immediately adjusted to the new asset composition. or the equity investor there are now four cases: A and ZC do not default, A defaults and ZC survives, ZC defaults and A survives, and A and ZC both default. The new funding rate, s new, satisfies again (2.4), which gives ( P D A P D ZC + P D A&ZC )(A + ZC) exp(s new T ) +(P D A P D A&ZC )ZC exp(s new T ) +(P D ZC P D A&ZC )A exp(s new T ) +P D A&ZC 0 = A + ZC (2.0) The new funding rate is given by exp(s new T ) = A + ZC ( P D A )A + ( P D ZC )ZC (2.) This may be compared to the old funding rate in (2.6), which will be denoted by s old from now on. Note that depending on the difference between the PD of the zero coupon bond and the PD of the assets, the new funding rate will be higher or lower P D ZC = P D A s new P D ZC < P D A s new P D ZC > P D A s new = s old < s old > s old 4

5 If the PD of the zero coupon bond equals the PD of the asset A, then the funding cost will remain the same. If the PD of the ZC bond is smaller than the PD of the asset A, then the funding cost will go down, and if the PD of the ZC bond is larger than the PD of the asset A the funding cost will go up. This is consistent with the intuition that if the quality of the assets deteriorate the funding costs should go up and vice versa. urthermore if the zero coupon bond is much smaller than the asset A, the funding cost will change only by a small amount. Nevertheless this small change in funding cost will compensate for the funding cost for the ZC bond, as is shown next. Now that the new funding rate is such that the value of the pay-off of the equity at time T is A + ZC, the fair rate on the ZC-bond can be calculated from (2.7). The fair rate is determined by ( P D A P D ZC + P D A&ZC )(A exp(s A T ) + ZC exp(s ZC T )) +(P D A P D A&ZC )ZC exp(s ZC T ) +(P D ZC P D A&ZC )A exp(s A T ) +P D A&ZC 0 = A + ZC. (2.2) Since s A is known from (2.9), the ZC rate can be easily calculated exp(s ZC T ) = P D ZC. (2.3) This is the main result of this section. It shows that under the assumption of immediate adjustment of funding costs of the bank after a transaction, the funding costs do not affect the fair rate. This may be contrasted with the fair rate in the inelastic case, (2.3), which includes the funding cost. The above derivation was based on an extremely simple initial balance sheet with only a single asset. In the appendix, it is shown that the result holds also for a general asset mix, and a non-zero correlation (or dependence-structure) with the ZC-bond. 3 Inelastic vs Elastic Which of the two assumptions is more realistic? The assumption that a transaction does not affect the funding cost or that funding cost are immediately adjusted to take the new asset composition including the new transaction into account? It seems that the inelastic case is more realistic, since investors will not immediately after each transaction of the bank adjust the compensation they require for funding. Although this statement may be difficult to test in practice, since the adjustment required in the example of the ZC-bond is of the order ZC/A and may be too small to be observable. 5

6 Also most funding will not be overnight but (much) longer term funding (amongst others to mitigate liquidity risk). Therefore even if investors would be aware of all the changes in the asset composition of the bank, only the cost of a small part of the total funding could be adjusted. Nevertheless it is clear that if the asset composition changes over time the funding costs will change as well. Therefore the inelastic assumption seems not to be fully realistic either. Although it might be argued that on small time scales (e.g. M) the funding costs can be assumed to be fixed. Another argument in favor of the inelastic assumption is that this better represents the way banks are managed. A derivatives trading desk is simply confronted with a certain funding curve at which it can borrow/lend (perhaps separate curves are defined for borrowing and lending), but the impact of the transaction of the desk on funding cost of the bank is not taken into account. However an important argument in favor of the elastic assumption is that it allows for consistent valuation of defaultable bonds and other derivatives. Under the inelastic assumption the valuation of credit bonds would include the funding cost of the bank, as in (2.3), however since these are traded and the market value can be observed, they should be valued without funding cost. Therefore this would lead to an ad hoc distinction of trades valued including funding cost (under the inelastic assumption) and without. However the question at the start of this section Which of the two assumptions is more realistic? is in my opinion not the right question, since it basically asks how the treasury of the bank is organised. That a typical bank uses fixed funding costs does not automatically imply that using fixed funding costs in the valuation leads to the right value. In particular, it is imporant in my opinion to have a consistent framework. The assumption of elastic funding rates seems more appropriate then, since it leads to a consistent valuation of e.g. traded credit bonds. Also, if the ZC-bond, in the example in section 2, is large enough it can be put in a separate entity (an SPV) and the funding of of this SPV will solely be determined by the quality of the ZC-bond and not by the funding cost of the bank. A similar argument is made by Hull and White in [7]. They argue that funding costs should not be accounted for in the derivatives value. Allthough both assumptions reflect some parts of reality, I believe the elastic funding assumption is more appropriate for the valuation of derivatives. 4 Black-Scholes under Inelastic and Elastic unding Assumption This section considers the derivation of the Black-Scholes model under the inelastic and elastic funding assumption. The derivation under the inelastic assumption has been done by Piterbarg in [] and his derivation is followed here. The derivation 6

7 under the elastic assumption is new to my best knowledge. In the following three rates are used: r rf the risk free rate, for which often the OIS rate is used. r R the repo rate. r the funding rate for the bank. 4. Black-Scholes under the Inelastic assumption or simplicity, assume the underlying asset is a non-dividend paying stock and that the transaction is uncollateralized. The stock, S(t), follows the following dynamics: ds(t) = µs(t)dt + σs(t)dw (t), (4.) where µ denotes the drift, σ the volatility and W (t) a Brownian motion under the real world measure. Consider a derivative on the stock, V (t, S). To replicate the derivative a portfolio consisting of (t) units of stock and an amount γ(t) of cash is used. Where as usual V (t, S) (t) =. (4.2) S Piterbarg [] distinguishes the different contributions to the cash amount γ(t). an amount V (t, S) that is borrowed/lent unsecured at a funding rate r 2. an amount (t)s(t) that is secured by the stock and therefore earns/costs the repo rate r R The growth of the cash amount is thus given by dγ(t) = [r V (t, S) r R (t)s(t)]dt (4.3) Using Ito s lemma a PDE can be derived [], similar to the usual Black-Scholes PDE V (t, S) V (t, S) + r R S + σ2 t S 2 S2 2 V (t, S) S 2 = r V (t, S) (4.4) Note that the value of the derivative depends on both the repo rate and the funding rate. The conclusion is that under the inelastic funding assumption the value of a derivative (in general) depends on the funding rate. 4.2 Black-Scholes under the Elastic assumption Under the elastic funding assumption the derivation is different. In that case, the derivative and Delta hedge V (t, S) (t)s(t) (4.5) are added to the existing assets. As in section 2.2 I assume the portfolio needs to be funded by equity. Since for a small time dt the portfolio is risk free, the reasoning 7

8 leading to (2.3) (or its generalisation (A.)) applies here as well. Assuming no counterparty risk, the result is that the additional funding costs are determined by the risk free rate, r rf, without additional spread: dγ(t) = r rf [V (t, S) (t)s(t)]dt (4.6) Note that this is different from the argument in [5], where it is argued that the additional funding costs for a derivative are (in our notation) r rf V (t, S)dt, which is not consistent with the reasoning here, since the derivative is risky. At least, in the model developed here, only the combination of derivative and hedge will have an additional funding costs proportional to the risk free rate. In any case, from (4.6) the resulting PDE is V (t, S) t V (t, S) + r rf S + σ2 S 2 S2 2 V (t, S) S 2 = r rf V (t, S) (4.7) Under the elastic funding assumption the value of the derivative in the Black- Scholes model is independent of the funding rate. 4.3 A simple derivative To illustrate the difference of the two PDE s derived under the two assumptions, consider a simple derivative that at maturity, T, pays the stock, S(T ) V (T, S) = S(T ) (4.8) The Black-Scholes values according to the two assumptions are inelastic assumption: V (0, S) = exp( (r r R )T )S(0) (4.9) elastic assumption: V (0, S) = S(0) (4.0) The result for the inelastic assumption can be understood from noting that the option needs to be funded and the rate r is paid for this funding, whereas the hedge, a short position in the underlying stock, yields the repo rate, r R. The result from the elastic assumption is consistent with the simple hedge strategy to short the stock and from the cash received buy the derivative. 5 Conclusion In this paper the valuation of derivatives including funding costs is considered. Two opposite assumptions for the funding are considered: the inelastic and elastic funding assumption. It is shown that under the elastic funding assumption the funding costs do not affect the valuation of a zero-coupon bond or the valuation of a derivative in the Black-Scholes model. 8

9 One may observe that although this paper focuses on derivatives, the conclusions apply more generally to mark-to-market valuation. In particular, to the valuation of banking book items. Acknowledgements I would like to thank Drona Kandhai, Jaroslav Krystul, Tim Mexner, and Nikolai Zaitsev for useful discussions. References [] Piterbarg, V., unding beyond discounting: collateral agreements and derivatives pricing, Risk magazine, feb 200. [2] Pallavicini, A., D. Perini, and D. Brigo, unding Valuation Adjustment: a consistent framework including CVA, DVA, collateral, netting rules and rehypothecation arxiv:2.52v2 [q-fin.pr], dec 20. [3] Morini, M., and A. Prampolini, Risky funding: a unified framework for counterparty and liquidity charges, 200 [4] ries, C., Discounting Revisited. Valuations under unding Costs, Counterparty Risk and Collateralization, 200 [5] Burgard C. and M. Kjaer, In the Balance, Risk magazine, nov. 20. [6] Dermine J.,und Transfer Pricing (TP), Beyond the Global Banking Crisis in B. Swarup ed. Asset- Liability Management for inancial Institutions: Balancing inancial Stability with Strategic Objectives, Bloomsbury, London, 202. [7] Hull J., and A. White, The VA debate, Risk magazine, aug A Derivation for general loss distribution As in section 2 consider a balance sheet where the assets A are funded completely by equity. At time zero a zero-coupon bond is added, ZC, again funded by equity. The purpose of this appendix is to re-derive equation (2.) in case the assets A do not consist of a single asset with a single PD. The following notation is used: L T the loss at time T on the assets A. ρ A (L T ): the loss density for the assets A at time T (under the risk neutral/pricing measure). P D ZC (L T ): the probability of default of the zero coupon bond given a loss L T (on the assets A) at time T. 9

10 P D ZC : the unconditional probability of default of the zero coupon bond. LGD ZC : the loss given default of the zero coupon bond (e.g 60%). Given a loss L T, the total equity at time T is in expectaton: E[E T L T ] = A L T + ZC( P D ZC (L T )LGD ZC ) (A.) To calculate the time 0 value, the expectation value with respect to the risk neutral measure is taken: E[E[E T L T ]] = dlρ A (l) [A l + ZC( P D ZC (l)lgd ZC ] since = A EL A + ZC( P D ZC LGD ZC ), (A.2) dlρ A (l)p D ZC (l) = P D ZC, (A.3) and the expected loss is defined by EL A = dlρ A (l)l. (A.4) Requiring that the return on equity compensates the loss, (2.4), yields which results in [A EL A + ZC( P D ZC LGD ZC )] exp(s new T ) = A + ZC, (A.5) exp(s new T ) = A + ZC A EL A + ( P D ZC LGD ZC )ZC If there is only a single asset A, the expected loss can be expressed as and the result reduces to exp(s new T ) = EL A = P D A LGD A A A + ZC ( P D A LGD A )A + ( P D ZC LGD ZC )ZC (A.6) (A.7) (A.8) which is the same as (2.), except that the LGD can differ from 00%. Similar as in section 2.2, the fair rate for the ZC bond, s ZC is determined by (2.7). This gives dlρ A (L) [(A l) exp(s A T ) + ZC( P D ZC (l)lgd ZC ) exp(s ZC T ))] = (A EL A ) exp(s A T ) + ZC( P D ZC LGD ZC ) exp(s ZC T ) = A + ZC (A.9) Since the fair rate on the assets satisfies A exp(s A T ) = (A.0) A EL it follows that exp(s ZC T ) = (A.) P D ZC LGD ZC This extends the result (2.3) to a general asset mix and LGD. 0