Discounting Damage: Non-Linear Discounting and Default Compensation. Valuation of Non-Replicable Value and Damage
DDiscounting Damage:Non-Linear Discounting and DefaultCompensation
Valuation of Non-Replicable Value and Damage
Version 0.3.6(work in progress)
Christian Fries ∗†‡
June 12th, 2020 ∗ † Department of Mathematics, University of Munich, Theresienstraße 39, 80333 München, Germany ‡ DZ Bank AG Deutsche Zentral-Genossenschaftsbank, Platz der Republik, 60325 Frankfurt am Main,Germany a r X i v : . [ q -f i n . M F ] A ug bstract In this short note we develop a model for discounting.A focus of the model is the discounting, when discount factors cannot be derived from marketproducts. That is, a risk-neutralizing trading strategy cannot be performed.This is the case, when one is in need of a risk-free (default-free) discounting, but default pro-tection on funding providers is not traded. For this case, we introduce a default compensationfactor ( exp(+˜ λT ) ) that describes the present value of a strategy to compensate for default (likebuying default protection would do).In a second part, we introduce a model, where the survival probability depends on the requirednotional. This model is different from the classical modelling of a time-dependent survivalprobability ( exp( − λT ) ). The model especially allows that large liquidity requirements areinstantly more likely do default than small ones.Combined the two approaches build a framework in which discounting (valuation) is non-linear.The non-linear discounting presented here has several effects, which are relevant in variousapplications: • If we consider the question of default-free valuation, i.e., factoring in the cost of defaultprotection, the framework can will lead to over-proportional higher values (or cost) forlarge projects (or damages). The framework can lead to the effect that discount-factorsfor very large liquidity requirements or projects are an increasing function of time. It mayeven lead to discount factors larger than one. This may have relevance in the assessmentof event like climate change. • For the valuation of defaultable products, e.g., like a defaultable swap, the frameworkleads to the generation of a continuum of (defaultable) par rate curves (interest rate curve)and the valuation of a payer and a receiver swap differs by more than just a sign.
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HRISTIAN P. Contents . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . 9 . . . . . . . . . . . . . . . . . . . . 10 . . . . . . . . . . . . . . . . . 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . 14 . . . . . . . . 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 . . . . . . . . . . . . . . . . . . . . . 24 . . . . . . . . . . . . . . . . . . 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . 29 c (cid:13) HRISTIAN F RIES V ERSION
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HRISTIAN P. The concept of valuation tries to determine an equivalent present value for a future value. Here,the value can be positive (a claim) or negative (a liability, or a damage). Apart from the factthat future values may be uncertain, which requires some notion of an expectation, a conceptfor risk and a price (or value) assigned to risk, it is required to define the dependency on time.A time-value. This is usually called discounting.In mathematical finance, one possible approach to derive a value of a financial product is todefine it through its current market price. If a financial product does not have a market price,one may try to associate the value of this product with a function of other market observedproducts by establishing a relation among these products, e.g. a replication strategy .Under suitable (and fairly strong) assumptions, a mathematical theory is applicable that repre-sents the present value as an expectation of (the distribution of) future values under a (stochas-tic) model, parametrized solely by market observables. This approach constitutes a market-implied valuation with a model using market-implied parameters.Market implied valuation is a reasonable approach in many situations, but maybe not in all. Themost critical assumption in this approach is the ability to perform a replication (hedging). Withthat regard, it should be stressed that even if replication could be performed in theory, a market-implied valuation is not admissible if such a risk-neutralizing replication is not performed inpractice.In the following, we will often use the word value, regardless if it is a cost or a benefit, becauseits just a matter of the sign. Furthermore, the sign of the value, depends on the observer, in abilateral contract one counterparty’s claim is the other counterparty’s liability.
The market-implied valuation of a liability gives rise to a possibly counter-intuitive dependencyon market-implied default probability. Consider a loan, where a counterparty borrows a unit in time t , to be paid back in T . To compensate for interest and the risk to default, the amountpaid back · exp( r f ( T − t )) with some rate r f . If the counterparty has a larger probability tofail on paying back, the rate r f will be larger.Consider a liability where a counterparty is liable to pay unit at a future time T > t . Thisamount is just a fraction, namely exp( − r f ( T − t ) , of the payment in the previously mentionedloan. Hence, the value of this liability in time t is that fraction of the corresponding loan takenin t , namely · exp( − r f ( T − t ) .Now, if the creditworthiness of the counterparty decreases, the rate r f will increase (the com-pensation contracted on loan will increase) and hence, the value of any existing liability willdecrease. This effect is reasonable from the lenders perspective, since the probability that the c (cid:13) HRISTIAN F RIES V ERSION
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P.borrower defaults on the payment increased. However, the effect appears awkward from theborrowers perspective. Since the value of a liability is negative for him, he profits from anincrease in the probability of default. Yet, this is reasonable, since the default is an option forhim, in which he does not need to pay.
While the valuation of a liability in the previously discussed form is well-grounded, it cannotbe applied to access the present (time t ) value a future damage that occurs in time T . Considersome environmental damage that needs to be repaired or compensated by all means. Assumesome model predicts that the time T value (i.e., cost) of this damage is V ( T ) . It seems temptingto consider the time t value as discounted V ( t ) = V ( T ) exp( − r f ( T − t )) . This may appearreasonable since it is the value that has to be contracted in t to achieve a corresponding paymentin T .Note, however, that such a discounting includes the possibility to default on the liability. How-ever, for the damage, there is no option to default on it. For that reason, one may concludethat the right way of discounting in this case, would be to use some (idealized) risk-free rate r (lower than r f ) such that V ( t ) = V ( T ) exp( − r ( T − t )) . But then, this approach depends onthe ability to perform a risk-free replication, which is - if at all - possible only for liquid marketassets.Furthermore. the existence of a risk-free interest rate is a illusion, or at best, an approximationonly valid for very short maturities.In this note, we like to take a look at discounting for values (cost) of events, that cannot bereplicated but have to be compensated by all means. Examples are damages evaluated in climatemodels or economic damages by pandemics. We will derive two frameworks: • a discounting based on a diversification of default risk, which may lead to discountfactors larger than , exhibiting the impact of a possible mismatch of market-implied andrealized default probabilities, and, • a valuation, where the (realized) default probability depends on the requested notional,i.e., is non-linear in the notional.We will combine the two aspects. Since the market implied default intensity is a market-expectation for common cash-flows and the realized default intensity is state-dependent, large(or huge) cash-flows may receive discount factors larger than . With the possible exception to factor in the option of our own extinction. c (cid:13) HRISTIAN F RIES V ERSION
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HRISTIAN P. For a discussion on the role of discounting to determine present values of future events, e.g. re-lated to damages related to climate change we refer the reader to [8, 1] and references therein.In [3, 2] the long maturity limit of interest rates is discussed, linking to the problem of valuationof long-term projects. We refer the reader to [3] and references therein.As noted in [3] (and other related publications), a major issue with discounting is that undercertain assumptions the discount factor is an exponential function of maturity, exp( − r ( T − t )) and this exponential weight for future cash-flow, which results in an underweight of futureevents.Let P ( T ; t ) denote the time t value of a zero-coupon bond with maturity T and P ( S, T ; t ) the time- t value of a forward bond, that is the time- t value of the price to be paid in S toreceive in T . The exponential discounting follows from the assumption of time-consistence( P ( S, T ; t ) = P (0 , T − S ; t ) = P ( T − S ; t ) ) and the absence of arbitrage. via P ( T ; t ) = P ( S ; t ) P ( T − S ; t ) . (1)However, the relation (1) assumes a re-investment strategy, that is, a trading strategy and ne-glects the possibility that the bonds used in the strategy defaults. In Section 2 we will shortly review discounting as it arises in the context of risk-neutralvaluation. This makes the assumptions that claims can be replicated by trading in a market.Valuations are hence market implied.In Section 3 we ask how to provide funding for a future cash-flow, when all market tradedinstruments are defaultable. Instead of bonds, we consider funding providers, that is, counter-parties that can provided funding (that is, a zero-bond), but which are subject to default. Theneed to compensate for default by diversification induces a discount factor that can be largerthan 1.In Section 4 we assume that the default probability of the funding provider depends on therequired fund. This means, than the discount factor depends on the notional. That is, discountingis non-linear in the notional.We conclude in Section 7 by numerically investigating the properties of the model. c (cid:13) HRISTIAN F RIES V ERSION
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HRISTIAN P. Discounting as a time-value-of-money can be derived from a replication strategy, e.g., mappingfuture liabilities to current market prices, [7, 6].Consider a counterparty borrowing (unsecured) the amount M from the market. The marketrequests an interest rate from the counterparty. This rate is called the funding rate . Expressed asa continuously compounded rate, if the repayment of the borrowed amount M and all accruedinterest occurs in T , then in T the counterparty has to pay back the amount M · exp T (cid:90) r f ( τ )d τ .Assuming a positive funding rate r f , the amount paid back is larger than the amount M origi-nally borrowed.The funding rate r f is often decomposed into two parts, r f = r + λ . The rate r is considered therisk-free rate, while λ is a counterparty specific component reflecting the counterparty specificdefault risk.Hence, r f is considered to be higher than an idealized risk-free rate r , due to the perceived riskthat the borrower can default, i.e., he can fail to pay back at the future time T .If the counterparty is a net borrower, i.e., at any future point in time it borrowed money fromits investors, then any inflow of cash can be considered to earn the rate r f by reducing therequirement to borrow money, hence reducing the funding costs. Under this situation, N ( t ) =exp (cid:16)(cid:82) t r f ( τ )d τ (cid:17) constitutes a numéraire for the counterparty (similar to a Bank account).Given that future values are stochastic the (risk-neutral) valuation of future cash-flows becomesthe discounted expectation V ( t ) = E Q (cid:18) V ( T ) N ( t ) N ( T ) |F t (cid:19) = E V ( T ) exp( − T (cid:90) t r f ( τ )d τ ) |F t , (2)where N ( t ) is the funding numéraire.This funding discounting can be understood from the assumption that the counterparty bor-rowed money from its investors and guaranteed the return r f to them.In general the funding r f is a stochastic process and future cash flows V ( T ) are randomvariables. c (cid:13) HRISTIAN F RIES V ERSION
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HRISTIAN P. In (2) the probability measure Q is induced by the assumption of risk-neutral replication.That is, a risk-neutral valuation relies on the fact that contracts can secure future payments.Parameters derived from this context are market-implied parameters . They reflect the marketperceived (or market-implied) probabilities associated with the events. These parameters are afunction of the time t at which the contracts are traded.In contrast to the risk-neutral measure, the objective probability measure P of future eventsmay differ from Q and parameters related to the real probability of events may differ frommarket-implied parameters.In the following, a parameter with a tilde denotes a parameter related to the objective probabilitymeasure, whereas the same symbol without the tilde denotes the corresponding market-impliedparameter. The popular fundamental object for building interest rate curves is the zero coupon bond: P ( T ; t ) is the time t value of receiving in T . We may distinguish a default-free zero couponbond (denoted here by P ◦ ) and a defaultable zero-coupon paying (denoted here by P d ).Interest rates are an alternative (equivalent) form of expressing the system of zero bonds. Theircompounding can be understood as a convention in their definition and is not necessarily relatedto a possible trading strategy. For example, we can express P ◦ by a continuously compoundingyield r ( T ; t ) or as a forward rate L ( t, T ; t ) , r ( T ; t ) = − log( P ◦ ( T ; t )) / ( T − t ) , L ( t, T ; t ) = ( 1 P ◦ ( T ; t ) − / ( T − t ) .Similarly, a defaultable zero bond can be used to define an (implied) survival probability, whichis just λ ( T ; t ) = − log( P d ( T ; t )) / ( T − t ) − r .In the following we will often use the notation of continuously compounded rates, exp( − r ( T ; t )( T − t )) and exp( − λ ( T ; t )( T − t )) , but this is only used because these expressions appear maybemore familiar. With respect to interest rates, we have to distinguish rates used to accrue a collateral contracts(EONIA, e -STR, SOFR) and rates of unsecured lending. With respect to the latter, there aremany. Any counterparty issuing bond to receive funding by that creates a counterparty specificinterest rate curve - the funding curve. c (cid:13) HRISTIAN F RIES V ERSION
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P.With respect to the curve used to accrue collateral, these rates do not include the cost ofproviding funding, since the daily settlement of collateralized contracts effectively removescounterparty risk.Hence, collateralization rates can be merely used to derive the information of the basis r - anidealized risk-free rate. The rate is hypothetical, because there is no long-term risk-free fundingat this rate.In the following, we consider that fund is provided at the funding rate r f and use the risk freerate r as a basis to decompose the funding rate. The difference between the market-implied risk-neutral measure P and the objective measure Q becomes apparent if we consider default events.Assume that V ( T ) is a deterministic time- T value and τ HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. The market-implied valuation of a liability is associated with the possibility that the liablecounterparty may default on its liabilities. An increase of the probability to default on theliability, reduces the present value of that liability in the markets perspective.In this Section, we like to consider the valuation of cash flows that have to be paid in allcircumstances. That is, default is not an option.To ease notation, we will consider t = 0 and use the idealization from (3), that is considerthe three parts exp( − rT ) (risk-neutral valuation), exp( − λT ) (implied survival probability)and exp( − ˜ λT ) (realized or objective survival probability). The discussion straightforwardlygeneralizes to the case of stochastic rates. In the risk-neutral valuation, we could try to cover the case of a non-performing counterpart.We seek protection for the default case having (market-implied) probability − exp( − λT ) , thatis, the market price of this protection is − exp( − λT ) . Hence, buying protection the valuationbecomes the risk-neutral valuation V ( T ) exp( − rT ) exp( − λT ) (cid:124) (cid:123)(cid:122) (cid:125) value of defaultable cash-flow + V ( T ) exp( − rT )(1 − exp( − λT )) (cid:124) (cid:123)(cid:122) (cid:125) cost of defaultable protection = V ( T ) exp( − rT ) . The approach to value the liability by incorporating the market price of default protectiondepends strongly on the ability to buy that protection, on the business model that protection isactually bought, and on the reliability of the protection seller.The approach is possibly valid for liquid market products, but unlikely feasible for catastrophicor systemic damages,An alternative approach to ensure the payment is to diversify the default risk. To start, consideran idealized setup and assume that we can contract payments with an objective survival prob-ability exp( − ˜ λT ) from different counterparties and that their default events are independent.In that case, we can split a payment X into n parts across these counterparties and receive inexpectation n (cid:88) i =1 n X exp( − ˜ λT ) = X exp( − ˜ λT ) . c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P.Assuming that the default events are independent, the variance of these repayments is givenby n (cid:88) i =1 n X exp( − ˜ λT )(1 − exp( − ˜ λT )) = 1 n X exp( − ˜ λT )(1 − exp( − ˜ λT )) .Choosing X = V ( T ) exp(+˜ λT ) we see that we receive V ( T ) in expectation with a variance(risk) given by n V ( T ) (1 − exp( − ˜ λT )) .The risk can be reduced by increasing n .To summarize, we contract (distributed among multiple parties) the payment X = V ( T ) exp(+˜ λT ) .A risk-neutral valuation of the future payment X = V ( T ) exp(+˜ λT ) would give us V ( T ) exp( − rT ) exp((˜ λ − λ ) T ) and for ˜ λ = λ we see the same value as for a discounting with the risk-free rate.The discount factor exp( − rT ) exp((˜ λ − λ ) T ) contains three parts: • exp( − rT ) is a factor representing the (risk-neutral) time-value of money. • exp( − λT ) is a discount we receive from a funding provider, due to its ability to default.This survival probability is fixed at trade time t and is market-implied. • exp(+˜ λT ) is the inverse of the true (objective) survival probability and acts as a com-pensation of the (diversified) objective default risk. Note that it is observed at time T ifthe funding is performed or not.Reducing the risk by diversification, the default probability is observed at the future pointin time T under the objective probability measure, whereas the risk-neutral expectation isperformed under the market-implied risk-neutral measure observed at the valuation time. In astochastic model, we will be exposed to the risk of future changes in ˜ λ − λ . In the previous Section, diversification produces the required funding in expectation. This isunsatisfactory and should be elaborated further.Assume that we distribute the total payment of X ∗ among n counterparties with i.i.d. survivalprobabilities ˜ p , such that each entity pays n X ∗ . Let Z denote the random variable representingthe sum of the defaultable payments (a sum of independent Bernoulli distributed randomvariables). Then we receive in expectation . µ = E ( Z ) = X ∗ ˜ p . (4) c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P.The variance of the payment is σ = V ( Z ) = 1 n ( X ∗ ) ˜ p (1 − ˜ p ) . (5)For n large, the random variable Z can be approximated by a normal distribution and we canestimate the probability that the payment stays above a given threshold µ − cσ as P ( Z ≥ µ − cσ ) = 1 − α = Φ( − c ) .For α = 1% we find c ≈ . . We now require that the amount X = V ( T ) is paid with a given probability (confidence level) − α . Thus we require µ − cσ = X .Plugging in (4) and (5) which express µ and σ in terms of the amount X ∗ that has to becontracted, this gives X ∗ ˜ p − c √ n X ∗ (cid:112) ˜ p (1 − ˜ p ) ! = X ,From which we determine the amount that has to be contracted to ensure the payment of X with a given probability − α : X ∗ = X p − c √ n (cid:112) ˜ p (1 − ˜ p ) = X ˜ p − − c √ n (cid:112) ˜ p − − . (6)With ˜ p = exp( − λT ) we now see that this gives X ∗ = X exp(+˜ λT ) 11 − c √ n (cid:113) exp(+˜ λT ) − .The market price of these contracts (that is a risk-neutral valuation) would then give X exp( − rT ) exp((˜ λ − λ ) T ) 11 − c √ n (cid:113) exp(+˜ λT ) − .We find that the need to diversify the funding risk modifies the discounting. The discount factornow consists of three parts: Alternatively, one might use the Cantelli inequality to estimate the probability that the payment stays above agiven threshold. It is P ( Z ≥ µ − cσ ) ≥ − / (1 + c ) or α = 1 / (1 + c ) , i.e., c = (cid:112) /α − P ( Z ≤ µ − σ (cid:114) α − ≤ α .For α = 1% we have c = √ 99 = 9 . . This is a much rougher estimate. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. • exp( − rT ) is a factor representing the (risk-neutral) time-value of money. • exp((˜ λ − λ ) T ) , which is due to the fact that each funding supplier has a default risk andthere may be a mismatch between market-implied and realized default risk. • The factor − c √ n √ exp(˜ λT ) − , which is due to the fact that we like to ensure the paymentsat a given confidence level via diversification among n funding suppliers. Note that thisfactor is larger than . For n → ∞ the factor converges to .The assumption of independence of the n independent funding suppliers is a strong. If weconsider only a limited amount of say 10 funding suppliers, we find c/ √ n ≈ / . A rough(first order) estimate for the additional factor is − c √ n (cid:113) exp(˜ λT ) − ≈ c √ n (cid:112) ˜ λT . which indicates, that the factor can become a significant adjustment in the discounting. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. In the previous Section we considered n independent funding providers for the given futurefunding requirement X .The fact that the discount factor depends on the true (objective) default probability motivates afurther generalization: if we take the view of some (defaultable) funding provider, providingthe amount X (or a fixed fraction from it), it is reasonable that there is an upper bound tothe fund that can be provided or - similarly - the objective (realized) default probability ofthe funding provider depends on the amount X . This assumption then introduces a notionaldependency of the discount factor.In addition, it is natural to consider the temporal distribution of funds provided by a fundingprovider. For example, if a funding provider provided the amount X in t and is required toprovide the amount X shortly after in t , then it is more likely that he defaults on X if X was high.Both aspect are becoming relevant if we assume that there is a limited amount of fundingproviders, each having a limited capacity for providing funds at a certain survival probability.In other words, we like to consider two generalizations: • we assume that an individual funding provider has a limited capacity, that is, it canprovide funding (or put differently: pay for a damage) only within a certain limit. • we assume that there is only a limited amount of funding providers.With respect to a funding provider having a limited capacity, we assume some interdependencyof the funds he has to provide. This will make discounting a portfolio problem, similar to aCVA.Concerning the second assumption, one may argue that every individual could act as a fundingprovider, such that n becomes the number of inhabitants, which makes n large. However, inthat case, every funding provider can provide only a very limited amount. A first idea to introduce a notional dependency would be to have a default intensity ˜ λ dependenton the amount X that has to be provided. It is reasonable to assume that the dependency issuch that we default with higher probability only on the additional amount.However, we would like to have that the need to provide fund instantaneously increases thedefault probability of a funding provider on that specific fund, that is, if fund has to be providedin T , then the survival probability up to time T − (infinitesimal before T ) is different form thesurvival probability up to time T + (infinitesimal after T ). This effect cannot be captured by c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P.a classical model with a time-dependent intensity unless the intensity is allowed to become aDirac distribution.Instead, we directly model a discontinuous change in the survival probability.At a future time t the funding provider will default on a payment of X with probability − ˜ p ,i.e., the expected fund provided in t is X · ˜ p . We assume that ˜ p depends on the notional amountto be provided and that the ability to provide fund applies on a marginal basis, that is, for theexpected fund provided ˜ X = X · ˜ p we have d ˜ X = ˜ q ( x )d x with some given monotone function ˜ q . Furthermore, we assume that the marginal survivalprobability ˜ q depends on the past fund provided, that is we assume that at time t i we have d ˜ X i = ˜ q ( a ( t i ) + x )d x .The term a corresponds to the accumulated liabilities and models how the need to provide fundat previous times impacts the ability to provide fund at current times. A possible model for thefunding consumption level a is a ( t i ) = (cid:88) t k HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. The modelling approach differs from that with a bounded default intensity λ ( t ) , where thesurvival probability is continuous in time exp( − (cid:82) t λ ( s )d s ) . However, it is possible to establisha simple link between the two approaches - and this link will also help to understand thedifferences.Assume that the accumulated funding requirement a ( t ) = (cid:82) t d ˜ X ( s ) is normal distributed, say a ( t ) follows the stochastic differential equation d a ( t ) = µ d t + β d W ( t ) , i.e. a ( t ) = µt + βW ( t ) . In this model we have a linear increasing funding requirement andallow for some diffusion. The important aspect here, is that the funding requirement are in-finitesimal.Furthermore, assume that the (marginal) survival probability is an exponential function of therequired fund, i.e., ˜ q ( x ) = exp( − x ) . Then we find for some incremental funding requirement ∆ X ( t ) ∆ ˜ X ( t ) := ∆ X ( t ) · ˜ p ( t, ∆ X ( t )) = ∆ X ( t ) (cid:90) ˜ q ( a ( t ) + x )d x ≈ ∆ X ( t )˜ q ( a ( t )) . (7)For ∆ X ( t ) and ˜ q ( a ( t )) being independent, we find from E (˜ q ( a ( t ))) = E (exp( − βW ( t ))) = exp( − λt ) with λ = µ − β that E (cid:16) ∆ ˜ X ( t ) (cid:17) ≈ E ( X ( t ) exp( − λt )) .This last expression corresponds to a “discounting” with a survival probability exp( − λt ) .While this is a very simple construction, the funding requirements are mostly a linear functionof time and translate state (x) to time (t), the analogy illustrates the difference to our approach:The analogy holds for small funding requirements ∆ X ( t ) , where past funding requirementsare diffusive. Assuming the linearization used in (7) it trivially creates independence of thediscounted notional X ( t ) and the discount factor.Hence, the fundamental difference in our approach is that we consider large notionals and theirimmediate effects on the survival probability. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. In Section 3.3 we derived that to ensure the availability of funding of X in T within someconfidence level c we need to contract an amount X ∗ that is chosen higher to compensate forthe default. Under some simplified model we derived that X ∗ is X ∗ = X p − c √ n (cid:112) ˜ p (1 − ˜ p ) = X ˜ p − − c √ n (cid:112) ˜ p − − . (8)Here, ˜ p denotes the objective survival probability (in contrast to the market-implied survivalprobability).For the case that we are only interested in matching the funding requirement in expectation( c = 0 ) the formula (8) simplified to X ∗ = X p .This has the simple interpretation that we need to contract p -times the original amount tocompensate for a default of a funding-provider.Using this approach now on a marginal basis with the notional dependent survival probabilitythis translates to the requirement X ∗ (cid:90) ˜ q ( a ( t i ) + x )d x ! = X . (9)Put differently, with ˜ p ( t i , X ) := 1 X X (cid:90) ˜ q ( a ( t i ) + x )d x ,we have X ∗ ˜ p ( t i , X ∗ ) = X .Here ˜ p ( t i , X ∗ ) is the effective survival probability for the amount X ∗ . In our applications,we usually know X (the value that needs to be funded) and seek the corresponding factor / ˜ p ( t i , X ∗ ) . Thus, in our implementation, we are rather interested in the function ˜ p ∗ ( t i , X ) = ˜ p ( t i , X ∗ ) where X ∗ as a function of X is given by (9).In the following, we call ˜ p ( t i , X ) the survival probability (the expected percentage amountof X achieve by contracting X ) and / ˜ p ∗ ( t i , X ) the default compensation factor, that is / ˜ p ∗ ( t i , X ) − is the percentage amount of X required in addition to X to ensure X inexpectation. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. Performing valuations under this model results in a “portfolio problem” where the valuationof a product depends on the presence of other products, similar to a CVA or MVA. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. For the implementation of the capacity of a funding provider, we need to implement a (stochas-tic) process that keeps track of the funding provided in the past (to calculate the level a ) andprovides the effective funding ˜ X = X/ ˜ p ∗ ( t, X ) for a funding request X .We consider a piecewise constant x (cid:55)→ ˜ q ( x ) with ˜ q ( x ) = ˜ q j for x j < x < x j +1 . Then thecumulated survival probability of the funding amount X , given a funding consumption b is ˜ p ( t i , b, X ) := b + X (cid:90) b ˜ q ( ξ ) d ξ and y (cid:90) x ˜ q ( ξ ) d ξ = ˜ q l ( y − x ) for k > l , (cid:80) k ≤ j HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. We summarize some properties of the model. We verify them in the numerical experiments inSection 7. Our model introduces a portfolio effect. The value of a portfolio of two products is differentfrom the sum of the values of the two products, valued individually. Such a portfolio effect isalso common in risk-neutral valuation, e.g., when valuing under counterparty risk.Since our model introduces a temporal dependency, where a survival probability for a fundingdepends on the accumulated past funding, we investigate financial products with periodicpayments, see Section 7.2.4. For a classical linear product, like forward (rate) agreements or swaps, the volatility of thestochastic payments does not impact the valuation. This is due to the product valuation beinglinear and due to the existence of a static hedge. This will be no longer the case in a modelwith notional dependent survival probability since the default probabilities are state dependent.In this case, scenarios with larger payments will obtain different weight. This is (similar to) awrong-way risk.To investigate the effect, we consider a forward rate agreement or a swap (having stochasticpayments) and investigate their behaviour under our model. See Section 7.2.2 and 7.2.3. Considering a defaultable interest rate swap with a notional dependent survival probability willintroduce a continuum of par forward rates and par swap rates.Consider a stochastic interest rate term-structure model with stochastic forward rates L ( T i , T i +1 ) = T i +1 − T i P ( T i ) P ( T i +1 ) − , where P ( T ) is the stochastic process modelling the value of a zero-couponbond with maturity T .In this model, consider the valuation of a swap, i.e., the stream of payments M · ( L ( T i , T i +1 ; T i ) − K ) in T i +1 ( i = 1 , . . . , n ) , c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P.where M denotes the notional. Performing a classical valuation with respect to a numéraire N ,the time- t value of this stream of payments can be expressed as V swap ( M, K, T , . . . , T n +1 ; t ) = M n (cid:88) i =1 ( L ( T i , T i +1 ; t ) − K ) · P ( T i +1 ; t ) ,where P ( T i +1 ; t ) = E Q N (cid:18) N ( t ) N ( T i +1 ) | F t (cid:19) , L ( T i , T i +1 ; t ) = 1 P ( T i +1 ; t ) E Q N (cid:18) L ( T i , T i +1 ; t ) N ( t ) N ( T i +1 ) | F t (cid:19) .The swap’s par-rate K ∗ = K ∗ ( M, T , . . . , T n +1 ; t ) is now defined as the rate for which V swap ( M, K ∗ , T , . . . , T n +1 ; t ) = 0 .In a classical valuation theory, the par-rate does not depend on the notional M and we find thatthe rate for a single period is given by the forward rate K ∗ ( M, T i , T i +1 ; t ) = L ( T i , T i +1 ; t ) .We now consider the par-rate of an unsecured swap with a notional dependent survival prob-ability. This will result in the par rate to depend on the notional itself. In fact, the par-ratewill receive a spread that depends on the slope (first derivative) of the survival probability as afunction of the notional and the interest rate volatility of L .To understand this effect, let X = L − K denote the stochastic cash-flow and p = p ( X ) = p (0) + p (cid:48) (0) X a notional dependent survival probability with p (0) , p (cid:48) (0) being deterministic.Then we have Xp ( X ) = Xp (0) + Xp (cid:48) (0) X = Xp (0) + X p (cid:48) (0) = Xp (0) + X ∂ log( p ( X )) ∂X (cid:12)(cid:12) X =0 p (0) Now, if K ◦ is such that for X ◦ = L − K ◦ we have E( X ◦ p (0)) = 0 , then we find for K = K ◦ + ∆ K ! = E ( Xp ( X )) = − ∆ KP ( T i +1 ; t ) p (0) + E (cid:18) X ∂ log( p ( x )) ∂x (cid:12)(cid:12) x =0 p (0) (cid:19) that ∆ K = 1 P ( T i +1 ; t ) E (cid:0) X (cid:1) ∂ log( p ( x )) ∂x (cid:12)(cid:12) x =0 .In this equation, the right hand side has a (weak) dependence on ∆ K , due to X = X ◦ − ∆ K .Instead of solving for ∆ K it is - for obtaining the right intuition - sufficient to approximating X by X ◦ and see that ∆ K ≈ P ( T i +1 ; t ) E (cid:0) ( X ◦ ) (cid:1) ∂ log( p ( X )) ∂X (cid:12)(cid:12) X =0 . (10) c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P.This is the impact of the notional dependent survival probability on the par rate: The term E (( X ◦ ) ) is the variance of the underlying cash-flow, the term ∂ log( p ( x )) ∂x (cid:12)(cid:12) x =0 is the slope of thelog-survival probability.Note that ∂ log( p ( x ) ∂x (cid:12)(cid:12) x =0 is negative if the survival probability decreases for increasing (positive)notional and that in this case, the spread ∆ K is negative. We verify this behaviour in ournumerical experiments in Section 7.2.5. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. We use classical models to simulate the evolution of (market) risk factors, based on Itô pro-cesses, like a Black-Scholes model, Bachelier model or LIBOR market model.We use these models to simulate the “funding requirements” of classical financial derivatives(like a swap or a forward agreement), and the valuation is performed under a risk-neutralmeasure.We chose this setup for the sake of comparison. Since the behaviour of these products is wellknown under these modes, we can investigate the impact of a notional dependent discounting.For λ = ¯ λ , c = 0 , α = ∞ we recover the classical risk-neutral valuation.The parameter ¯ λ is the objective future default intensity, while λ is the market-implied defaultintensity (a discount on defaultable loans). A reasonable approach could be to set λ = 0 andjust consider some excess default intensity ¯ λ .The parameter c defines the quantile level of risk we are willing to allow for funding mismatches.If c = 0 then funding is provided only in expectation.The parameter n specifies the number of independent funding providers and impacts the riskto miss a required funding.The parameter α controls how fast the funding system recovers.In our experiments it is sufficient to specify the funding rate r f = r + λ (or alternatively,consider λ = 0 ). All cash-flows are considered defaultable. In that case ˜ λ can be interpretedas the mismatch of the realized (objective) default intensity and the market implied defaultintensity. If payments are stochastic, risk-neutral valuation values future scenarios by taking their ex-pectation under a risk-neutral measure. This approach is justified by the ability to replicatepayments by trading activities. If replication is not possible, the scenarios should be simulatedunder the objective measure and instead of expectation, risk measures (like expected shortfall)should be considered.That said, we conduct our analysis under the risk-neutral measure, since a change of measurewould not impact the qualitative behaviour. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. We present some numerical results, illustrating the behaviour of the model. The experimentsin this sections are available in [4], using [5] (version 5.0.5). Source code is provided throughthe referenced repositories. As illustrated, a state dependent default probability may translate to a time-dependent sur-vival probability, in expectation, in a simplified model, where the funding requirements areinfinitesimal, and distributed over time.We consider a process d S = µ d t + σ d W . We use an exponential, state dependent, instantaneoussurvival probability q ( x ) = exp( − x ) This implies a stochastic survival probability for the interval from to T being exp( − λ (0 , T ) T ) with λ = µ − . σ .Figure 1 depicts the numerical result for µ = 0 . , σ = 0 . . While this results is fairly trivial, it maturity dependency of survival probability under continous funding requirements survival probability in exponential state dependent model analytic survival probability (classic intensity model) maturity Figure 1: Maturity dependence of the (expectation of) the stochastic survival probability in amodel of state-dependent instantaneous funding requirements (red). The behaviouragrees with a deterministic exponential survival probability.can be seen as a unit test of the implementation of the state dependent model. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. We consider the value X ( T ) to be log-normal distributed following a Black-Scholes model d X = rX d + σX d W ( t ) . We assume that X ( T ) represents a future cash-flow requirement(e.g. the cost to compensate a damage).A risk-neutral valuation of X ( T ) would result in X (0) , independent of the parameter σ . If X ( T ) is considered to be a defaultable cash-flow, where default is considered independent of X , we would arrive at a value X (0) exp( − λT ) . Following our discussion in Section 3 thiswould imply that we need to contract the amount X (0) exp(˜ λ − λT ) to compensate for thedefault - at least, in expectation.Considering a notional-dependent default probability, we consider the default-compensatedamount X ( T ) / ˜ p ∗ ( T, , X ( T )) . This is the amount we have to diversify among defaultablefunding providers to get X ( T ) in general. If ˜ p ∗ is neither a constant nor homogenous, this willintroduce a non-linearity and hence a risk-neutral valuation will dependent on the volatility σ .If we consider a simple model with a piecewise linear function p generated from piecewiseconstant q , the function x (cid:55)→ x ˜ p ∗ ( T, , x ) will be piecewise constant too. We take ˜ p ∗ ( T, , x ) = (cid:40) . for x < La for x > L .Then X ( T ) / ˜ p ∗ ( T, , X ( T )) corresponds to a pay-off of a European option. It is X ( T ) / ˜ p ∗ ( T, , X ( T )) = X ( T ) + (1 − /a ) max( X ( T ) − L, .From this, the volatility dependence of the pay-off becomes obvious. Figure 2 depicts thesituation for T = 5 , a = 0 . .An obvious upper bound to the risk-neutral valuation of the default-compensate payoffs is (1 − /a ) X (0) . In the example of Section 7.2.2 the impact of the non-linear discounting appears as a simple,almost linear interpolation between the two extreme factors and /p . The situation looksdifferent if we consider a future cash-flow requirement of X ( T ) − K , that is, a forward agree-ment with forward value K . We assume a log-normal X ( T ) as above. The amount X ( T ) − K can be positive or negative, where we consider the positive value a damage (liabilities) andthe negative value a gain. We used a non-linear discounting, i.e., default-compensation thatcompensates only the positive amounts. In relative terms the effect then appears much strongerand it is not bound by applying the factor /p to the all paths, see 3. We consider damages to be potentially unbounded, but gains bounded. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. volatiltiy dependency of compensation cost risk free default compensated (constant survival prob.) default compensated (state dependent survival prob.) benchmark (using option) volatiltiy Figure 2: Volatility dependency of the default-compensation cost of X ( T ) compared to no-compensation (red) or the compensation corresponding to a constant instantaneousdefault intensity. We consider a sequence of values (damages) X ( T i ) − K , for i = 1 , . . . , n . (11)If the non-linear discounting model is applied to the cash-flow stream (11), the compensationfactor (and the survival probability) of the value T i depends on the events T j , j < i . In ad-dition, since there is a positive correlation between X ( T i ) and X ( T j ) being high, there is afeedback effect. In case of a non-linear discount factor, we have that the sum of the individualcompensated values is different from the compensated sequence of values.In Figure 4, 5 we depict the valuation of the single amount X ( T i ) − K as a function of T i ,conditional to the prior compensation of the amounts X ( T j ) − K for j < i . Figure 4 showsthe result for K = 0 , Figure 5 for K > .Since in a classical (risk-free) setup the valuation of X ( T i ) − K is independent of prior val-uations of X ( T j ) − K and the risk-free valuation of any of those is X (0) − KP ( T i ) , for therisk-free valuation we will see a horizontal line for K = 0 and an upward sloping curve for K > . Likewise, a constant compensation factor will result in a parallel shift of the risk freeline. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. volatiltiy dependency of compensation cost risk free default compensated (constant) default compensated (state dependent) benchmark (using option) volatiltiy Figure 3: Volatility dependence of the compensation cost of X ( T ) − K . The setup correspondsto that in Figure 2, excepts that the cash-flows allow for negative values (here inter-preted as gains).A state-dependent survival probability and hence a state-dependent compensation leads toa strong maturity dependency. For the case K = 0 the value is just a maturity dependentinterpolation between the two constant cases.For K > the behavior in our test is as follow: for low maturities almost all szenarios resultin negative values (which we interpret as gains) and the compensation factor is (path-wise) ,given the same result as the risk-free case. For higher maturities more szenarios show positivevalues (positive funding requirements), consuming the capacity of the funding provider, de-creasing the marginal survival probability. In that case, the funding compensation can exceedthat of a constant compensation factor, due to the asymmetry between positive and negativevalues. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. maturity dependency of comensation cost (conditional to previous compensations) default compensated (state dependent) risk free default compensated (constant) maturity Figure 4: Maturity dependence of the compensation cost of a sequence of funding requirements X ( T i ) , ( i = 1 , . . . , n ). maturity dependency of comensation cost (conditional to previous compensations) default compensated (state dependent) risk free default compensated (constant) maturity -0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.0500.050.10.150.20.250.3 Figure 5: Maturity dependence of the compensation cost of a sequence of funding requirements X ( T i ) − K , ( i = 1 , . . . , n ). c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. We numerically verify the result that a notional dependent survival probability generates acontinuum of (defaultable) par-rates (swap rates and forward rates).The model used is a standard forward rate model (LIBOR market model) with an exponentiallydecaying forward rate volatility. The initial forward rate curve is flat (at 5%).The notional dependent survival probability was . if the notional stayed below a certainthreshold, this leads to the induced interest rate spread being (almost) zero for small notion-als.Figure 6 shows the dependency of the 20Y par swap rate on the notional.In Figure 7 and 8 the same model is used to calculate the (par) forward rate curve for differentnotionals of the forward rate agreement and different volatilities of the interest rate, verifyingthe intuition derived above. notional dependency of a par-swap rate of a swap risky swap rate risk free swap rate -2000 -1500 -1000 -500 0 500 1000 1500 2000 notional Figure 6: The par swap rate of a 20Y swap with a notional dependent survival probability(red). We model the survival probability as is the value of the cash-flow is below afixed threshold and hence the swap rate states at the risk-free rate (green) for smallnotionals. Large notionals generate a spread. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. notional dependency of the par forward rate -1000.0 -800.0 -600.0 -400.0 -200.0 200.0 400.0 600.0 800.0 1000.0 maturity Figure 7: The forward rate curve, i.e., the curve of the par rate of a forward rate agreementfor different notionals. The shape of the curves is a consequence of an exponentialdecay in the forward rate volatility as a function of time-to-maturity. While this leadsto a mean reversion of the short rate is also leads to a flattening of the spread curve,as higher maturity rates have approximately the same volatility. The slightly non-smooth shape of the curve is due to the fact that they are obtained via a Monte-Carlosimulation of the model. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. notional dependency of the par forward rate -1000.0 -800.0 -600.0 -400.0 -200.0 200.0 400.0 600.0 800.0 1000.0 maturity Figure 8: .The forward rate curve, i.e., the curve of the par rate of a forward rate agreementfor different notionals. The same experiment as in Figure 7 except for the volatilitybeing twice as large. This results in an increase of the spread by (approximately) afactor of four, which is in line with the derivation (10). c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. We considered a default compensation factor ( exp(+˜ λT ) ) that takes the role of the cost to buydefault protection, for the case where default protection is not available as a traded asset. Notaccounting for tail risk, i.e., achieving protection only in expectation, the factor is the inverseof the objective survival probability. Factoring in the risk to fail in providing sufficient defaultprotection, the factor increases, depending on the number of funding providers and their defaultprobabilities.We then established a model where the objective survival probability is not an exponentialfunction of time, but a function of the fund required. This leads to a non-linear discount factorand hence to a non-linear default compensation factor.We provide a prototypical open source implementation of the framework and investigate thebehavior of the model, i.e., the impact of a default compensation on the present value. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. [1] Emmerling et al. “The role of the discount rate for emission pathways and negativeemissions”. In: Environmental Research Letters 14 (2019). DOI : 10 . 1088 / 1748 -9326/ab3cc9 .[2] Francesca Biagini, Alessandro Gnoatto, and Maximilian Härtel. “he Long-Term SwapRate and a General Analysis of Long-Term Interest Rates”. In: SSRN (2015). DOI : .[3] Dorje C. Brody and Hughston Lane P. “Social Discounting and the Long Rate Interest”.In: Mathematical Finance DOI : .[4] finmath.net. finmath-experiments: Experiments related to mathematical finance usingfinmath-lib . URL : http://finmath.net/finmath-experiments .[5] finmath.net. finmath-lib: Mathematical Finance Library: Algorithms and methodologiesrelated to mathematical finance . URL : http://finmath.net/finmath-lib .[6] Christian P. Fries. “Funded Replication. Valuing with Stochastic Funding”. In: SSRN1772503 (2011). DOI : 10 . 2139 / ssrn . 1772503 . URL : http : / / ssrn . com /abstract=1772503 .[7] Christian P. Fries. Mathematical Finance. Theory, Modeling, Implementation . John Wiley& Sons, 2007. DOI : . URL : .[8] Christian Gollier and Martin L. Weitzman. “How should the distant future be discountedwhen discount rates are uncertain?” In: Economics Letters 107 (2010), 350–353. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM // WWW . CHRISTIANFRIES . COM // FINMATH ISCOUNTING D AMAGE : N ON -L INEAR D ISCOUNTING F RIES , C HRISTIAN P. Notes Suggested Citation F RIES , C HRISTIAN P.: Discounting Damage: Non-Linear Discounting and Default Com-pensation (April, 2020). Classification Classification: MSC-class: 91G30JEL-class: C63.Keywords: Valuation, Discounting, Interest Rates Modelling, Funding, Environmental Damage34 pages. 8 figures. 0 tables. c (cid:13) HRISTIAN F RIES V ERSION HTTP :// WWW . CHRISTIANFRIES . COM //