A RAROC Valuation Scheme for Loans and its Application in Loan Origination

In this article, a RAROC (risk-adjusted return on capital) valuation scheme for loans is derived. The critical assumption throughout the article is that no market information on a borrower’s credit quality like bond or credit default swap spreads is available. Therefore, market-based approaches are not applicable, and an alternative combining market and statistical information is needed. The valuation scheme aims to derive the individual cost components of a loan which facilitates the allocation to bank?s operational units. After its introduction, a theoretical analysis of the scheme linking the level of interest rates and borrower default probabilities shows that a bank should only originate a loan when the interest rate the borrower is willing to accept is inside the profitability range for this client. This range depends on a bank’s internal profitability target and is always a finite interval only or could even be empty if a borrower’s credit quality is too low. Aside from analyzing the theoretical properties of the scheme, we show how it can be directly applied in the daily loan origination process of a bank.

A loan is probably the most traditional banking product. However, when different people in different countries or even different people in the same country working in different customer segments speak about a loan, they probably do not speak about the same product. The only common feature is that a lender gives money to a borrower and hopes to get back more than he has lent. Besides that, differences can be substantial. Critical drivers of product structure are the availability of funding and collateral. In some countries, only short-term funding is available to a bank. For this reason, interest rates of loans are rarely fixed over a long-term horizon but can be adjusted by a bank on short notice. In other countries, long-term funding is available, and loans are often fixed-rate or floating-rate loans where the floating rate follows an objective rule like 6M Ibor plus a spread. 3 The most prominent type of loan that is linked to a particular collateral type is the mortgage. Here, often long maturities up to 30 years are observed. However, still, there are some differences between countries. For instance, in the US a borrower can pass the key of a house to a bank when the house loses in value while in Germany defaulting on a mortgage is not that easy and the borrower still is responsible for the residual amount between the loan balance and house value. There are a lot more differences. In some countries, borrowers have prepayment rights on their loans; in other countries, prepayment rights are less popular, but floating-rate loans are embedded with caps and floors. A lot more covenants can be included like interest rates that increase with rating downgrades or minimum requirements on collateral value.
In this article, we develop a loan pricing scheme based on RAROC (risk-adjusted return on capital) as a performance measure and analyze its theoretical properties. Although we provide a theoretical analysis, our primary focus is on applicability, and we aim to capture the most critical features of loan origination in the scheme to make it directly applicable for a bank. A key assumption in this article is that there is no market information on a borrower available. This means that the stream of literature using risk-neutral probabilities for asset pricing, e.g. as in Jarrow, Lando, and Turnbull (1997), which are based on trading strategies in arbitragefree markets are not applicable in this context. There exists a stream of literature on mortgages in the US where a borrower can strategically default when the house price drops. In this setup, elements of this asset pricing framework can be transferred to this particular problem as in Kau and Keenan (1995) and Ciochetti, Deng, Gao, and Yao (2002). However, important practical aspects like economic capital are entirely missing in these approaches, and they are not easily generalizable to other loan types.
Broadly, the literature on loans can be categorized into equilibrium-based pricing models, empirical studies and articles on practical aspects of loan origination. Examples for the first category are Greenbaum, Kanatas, and Venezia (1989) and Repullo and Suarez (2004). The latter analyzed the impact of different regulatory regimes on loan prices. Although providing valuable insights, the framework used in these equilibrium models is too simplistic to be useful in practice. The empirical literature analyses the determinants of loan prices and how loan prices react to certain events like financial crises or changes in regulatory regimes, e.g. Santos (2011) and Martins and Schechtman (2014). This literature provides evidence on pricing schemes as they are applied in bank?s credit risk management, but their usefulness for building these schemes is limited. Some ideas on a designing loan pricing frameworks based on RAROC are presented in Aguais and Forest (2000), Aguais, Forest, Krishnamoorthy, and Mueller (1998), and Aguais and Santomero (1998). However, the description is very sketchy and loan pricing aspects that became important after the financial crisis are, of course, not included since these articles where written well before the crisis.
In this article, a generic framework for loan pricing based on RAROC, which is directly applicable in practice is developed, and its theoretical properties are analyzed. It is assumed that no market information on a borrower like a bond spread or a CDS (credit default swap) spread is available which prohibits no-arbitrage pricing approaches using risk-neutral probabilities. Furthermore, it is assumed that a bank is able to raise long-term funding and that a loan either has a fixed rate of interest or a floating interest rate of the type Ibor plus spread. Besides that, it is assumed that a liquid market in fixed-to-floating swaps exists in the loan?s currency and that basis swaps are available for all tenors of loan interest rate periods. Typically loans pay interest monthly, quarterly, semi-annually, or annually. For all these tenors basis swaps should be available like exchanging a stream of 1M Ibor payments for 6M Ibor. These assumptions are fulfilled mainly for mature capital markets like in the US, UK, and the EUR zone. In addition to the assumptions on interest rate markets, it is assumed that a bank applies statistical rating systems in the spirit of the Basel II Internal Ratings-Based Approach which leads to statistically estimated default probabilities and recovery rates. An overview of these estimation methods can be found in Engelmann and Rauhmeier (2006). This is the core data needed for the RAROC pricing approach of this article.
Another critical component of a loan pricing framework is economic capital. It measures the contribution of a loan to the total credit risk of a loan portfolio. It can be viewed as a capital buffer that is needed to absorb unexpected losses of a loan portfolio. It is assumed that a bank has implemented a framework for economic capital calculation. Depending on its sophistication this calculation can be based on the simple credit portfolio model underlying Basel II (Gordy 2003) or a more advanced approach along the lines of Gupton, Finger, and Bhatia (1997), CSFB (1997), or Wilson (1997a and Wilson (1997b). Finally, some information on a bank?s internal operating costs and its funding costs must be available and included in the pricing framework.
Economically, the default probability of a borrower increases with the interest that he has to pay on loan. We will discuss how this property can be included in the loan valuation scheme and consider the consequences for loan origination. It will be shown that depending on the specific risk characteristics of a borrower, a bank should only approve a loan if the interest rate a borrower is willing to pay is inside the loan's profitability range. This range is defined as the set of interest rates at which a loan is profitable for a bank. In some cases, the profitability range might be empty, which means that the bank should reject the loan application of this particular client.
The general pricing framework presented in this article is not linked to a particular loan segment and is applicable for corporate and retail lending. In the next section, a loan pricing formula and its parameterization will be explained. In Section 3 the RAROC pricing formula will be developed, and the calculation of all its cost components will be derived. After that, in Section 4 the theoretical properties of this scheme will be analyzed by linking the level of interest rates to the default rates of a borrower, and it will be shown how meaningful loan acceptance rules can be derived. In Section 5 some numerical examples are presented for illustration. The final section concludes.
In this section, the loan pricing formula and its parameterization are outlined. This formula builds the basis of the RAROC pricing scheme, which is explained in the next section. Some aspects of the parameterization might not be apparent immediately but will be justified in the next section. The value of a loan will be defined as the expected present value of all future cash flows. These are the interest rate payments, the amortization payments of the loan?s notional, and the liquidation proceeds of collateral in the case of a borrower default. The general expression for a loan's value V at time t is given as where T i is the interest rate payment time in period i, T * i−1 = max(T i−1 ,t), τ i is the year fraction of interest period i, N i the outstanding notional in each period, A i the amortization payments, R i the recovery rate in case of a default in period i, δ(T i ) is the discount factor corresponding to time T i , and v(T i ) the survival probability of the borrower up to time T i . It is assumed that default is recognized in payment times only and that the recovery rate summarizes the liquidation proceeds in case of default discounted back to default time.
The quantities N i , A i , and z i are defined by the loan terms. The amortization payments depend on the loan structure, i.e whether the loan is a bullet loan, an installment loan or an annuity loan. The interest rate z i could be fixed or floating. In the case of a fixed-rate loan, the interest rate in each period is y, where we assume that y is fixed and period-independent. In the case of a floating-rate loan, the interest rate is (L i + s), where L i is an Ibor rate which is fixed at the beginning of each interest rate period i and s is a spread that is assumed constant throughout the loan's lifetime. We will use the notation z i for the interest rate in period i with where f i is the forward rate corresponding to the floating rate L i . To compute forward rates a second discount curve is needed which will be denoted with δ M (t). Forward rates are computed as It remains to explain how the parameters discount factors, survival probabilities, and recovery rates are determined. This is done in a separate subsection for each parameter.

Discount Factors
The pricing formula (1) requires two discount curves, the discount curve for discounting cash flows and the forward curve for computing forward rates for floating-rate loans. The forward curve is computed from the money market and the swap market. Usually the forward curve up to one year is computed from deposit rates and forward rate agreements. Swap rates exist for maturities from one year up to 30 years in some currencies. A swap rate s f ix is the fixed-rate of a swap which periodically exchanges the fixed-rate with a Ibor rate L s with tenor Λ s . If the loan is linked to the same Ibor rate the discount factors δ M can be computed from the relation where U 0 is the start date of the swap, U j are the payment times of the fixed leg and η j are the day count fractions of the fixed leg. Usually a bootstrap algorithm is applied in computing δ M starting from the swap rate with the lowest maturity and moving forward in swap maturities iteratively using the results of the previous calculation to compute the discount factors corresponding to higher maturities.
If the loan is linked to a different Ibor rate L l with tenor Λ l , the above curve cannot be used for computing forward rates. The spread of a basis swap exchanging periodically Ibor payments with tenor Λ s for Ibor payments with tenor Λ l has to be added to s f ix before the bootstrapping starts. We assume the basis swap exchanges L s + s B for L l , where s B is the basis swap spread which depends on the maturity of the basis swap and can be negative. 4 This changes equation (4) to The discount curve δ has to reflect the funding conditions of a bank. It is computed from the fund transfer prices that are provided by a bank's treasury. Typically fund transfer prices are given for a grid of standardized tenors like, 1Y, 2Y, . . ., 10Y and are provided as Ibor + spread. This means, that internally the credit department buys a bond from the treasury department with a notional equal to the amount they intend to lend to a borrower. The coupon of this funding instrument is linked to a Ibor rate L f with tenor Λ f plus a spread s f depending on a loan's maturity. The discount curve δ f can be computed from the relation where W i are interest rate payment times of the funding bond and ξ i are the year fractions of the interest rate periods. The forward rates f j,Λ f are computed by (3) using the swap curve linked to L f . For the calculation of discount factors the notional is normalized to 1. Similar to the bootstrapping of the swap curve, a bootstrapping of the funding curve can be performed starting from the lowest maturity and working iteratively up to the highest. If a loan has a fixed rate of interest or a floating rate linked to the Ibor rate L f we get the discount factors in (1) as δ = δ f .
If the loan's interest rate is floating and its Ibor's tenor Λ l = Λ f again an adjustment by basis swap spreads is needed. Assume that the basis swap for L l and l f exchanges L l + sB for L f where again sB might be negative. Equation (6) has to be adjusted to where the payment times and year fractions in (7) are the same as for the loan. Forward rates f j,Λ l are computed from the swap curve corresponding to the Ibor rate L l . Bootstrapping this relation results in the discount curve needed for discounting a loan's cash flows. Why this is a sensible discount curve for cash flow discounting will become clear in Section 1.

Survival Probabilities
Equivalent to the calculation of a survival probability v(t) up to time t is the calculation of a default probability p(t) = 1 − v(t). Default probabilities with a time horizon of one year are typically one outcome of a bank's rating system. We assume that a bank's rating system has n grades where the n-th grade is the default grade. Again, remember that one key assumption of this article was the absence of market information like bond or CDS spread. Therefore, a bank has to rely on statistical information which is derived using balance sheet information for corporate clients, personal information of retail clients, and expert judgment as inputs. There are typically two ways how banks could extract statistical information about defaults from their rating systems to estimate multi-year default probabilities.
In the first approach, a one-year transition matrix is estimated from the rating transitions that are observed in a bank's rating system. The resulting matrix is denoted with P(1). The entries of the matrix are denoted with p kl , k, l = 1, . . . , n where p kl is the probability that a borrower in rating grade k moves to grade l within one year. The matrix P(1) has the following properties: 1. The entries of P(1) are nonnegative, i.e. p kl ≥ 0, k, l = 1, . . . , n.
If we assume that rating transitions are Markovian, i.e. they depend on the debtor's current rating grade only, and that transition probabilities are time-homogeneous, i.e. the probability of a rating transition between two time points depends on the length of the time interval only, then it is possible to apply the theory of Markov chains to construct transition matrices P(h) for an arbitrary full year h just by multiplying P(1) with itself: Once P(h) is computed, the default probability p k (h) can be read from the last column in the k-th row. Interpolating the values p k (h) gives the term-structure of default probabilities for rating grade k.
In the second approach, banks directly estimate a term-structure of default probabilities, i.e. for each rating grade k a function p k (t) is estimated where p k (t) is the probability that a borrower in rating grade k will default within the next t years. From the term structure of default probabilities given today, conditional default probabilities at future times u can be computed easily. The probability p k (t|u) that a borrower in rating grade k will default up to time t conditional that he is alive at time u is given by One way of estimating p k (t) is by using techniques from survival analysis, where the Cox proportional hazard model has been successfully applied in a credit risk context by numerous authors. Examples are Banasik, Crook, and Thomas (1999) and Malik and Thomas (2010). In this model, p(t) is parameterized as where β i are model coefficients, K i borrower-dependent risk factors like balance sheet ratios for companies or personal data for retails clients and h(t) a borrower-independent baseline hazard function. Borrowers with similar p(1) can be summarized into a rating category k and are then represented by the curve p k (t).
Throughout this article, we will assume that p k (t) is estimated by a Cox proportional hazard model as in (10). However, this is by no means the only way to estimate a PD termstructure. A good overview of available methodologies is provided in Crook and Bellotti (2010).

Recovery Rates
Recovery rates reflect the degree of collateralization of a loan. They can be period-dependent because if a loan is amortizing and the collateral value stays the same over a loan's lifetime, a loan becomes less risky over the years. This should be reflected in an increasing recovery rate. One pragmatic way to include collateral in a loan pricing framework is to provide the collateral value as input. This collateral value should not be the current market value of collateral but include the LGD (loss given default) of the collateral, i.e. the expected loss in value in the case of a borrower default. This loss can stem from price reductions in a distressed sales or reflect the costs of a liquidation process, e.g. for lawyers. Overall, the collateral valuation and LGD estimation process is complex and beyond the scope of this article. Some ideas can be found in articles on LGD estimation in Engelmann and Rauhmeier (2006).
For the purpose of loan pricing, we assume that such a process exists and that the outcome is a collateral cash value C. For the unsecured part of a loan, a bank estimates a recovery rate R u . From this data, the recovery rate R i in each period is computed as In (11) a cap of 100% was introduced. It depends on the specific legal environment of a country's loan market whether recovery rates of more than 100% are possible or not. In case recovery rates can be larger than 100% this assumption can be relaxed.
Note, that when applying this approach consistency is an important issue. In (11) the recovery rate is related to the outstanding notional. In internal risk parameter estimation processes, recovery rates (or, equivalently, LGD values) are often estimated with respect to outstanding notional plus one interest payment. Since loan pricing and risk parameter estimation is usually done in different departments of a bank, one has to take some care to ensure that consistent definitions and assumptions are used throughout a bank. Where this is not the case, appropriate transformations have to be defined.

RAROC Scheme
The main purpose of this section is the derivation of a RAROC scheme for calculating the interest rate of a loan which covers all costs and adequately compensates for the risks associated with a loan. For bank internal purposes, it is important to split the interest rate into its components, i.e. which part of the interest rate reflects funding costs, which part expected losses, or which part basis swap hedging costs. For this reason, a RAROC scheme is derived step-bystep using the general valuation formula (1). Before we start, we have to make an assumption on the disbursement of a loan's notional. This is not reflected in the valuation equation (1).
The assumption in this article is that a loan's notional is disbursed on disbursement dates D j and that on the date D j the amount N D j is paid out to the borrower. The total notional N D is the sum over all disbursements N D = ∑ D j N D j . The first component of the proposed RAROC scheme is the base swap rate. It is only relevant for a fixed-rate loan. For a floating-rate loan, this component is zero. The base swap rate is the fixed-rate that has to be charged by a bank that leads to an identical present value as the stream of Ibor payments. This rate is needed as a reference point to make floating-rate and fixed-rate loans comparable. The base swap rate y s is computed from the condition Solving this equation for y s gives Alternatively, y s can be interpreted as the interest rate that has to be charged for fixed-rate loans to make assets equal to liabilities in a bank's balance sheet under the assumption that all other costs and risks can be ignored. Using this as a starting point we will add all other relevant cost components of a loan to y s in the following steps. To simplify the notation, we will use the abbreviation This is the sum of all parts of total loan balance that are already paid out at time t and the present value of the loan parts that still have to be disbursed.
In the next step, funding costs are computed. This is a bit awkward because funding might be linked to a Ibor tenor that is different from the payment frequency of the loan. To separate funding costs from basis swap hedging costs, we have to use the discount curve δ f and, if the loan is a floating rate loan, compute forward rates from the swap curve corresponding to the Ibor rate L f . For a floating-rate loan, this leads to the condition where W i are the payment times of the funding bonds in (6),N is the average notional in an interest rate period and s f is the spread over Ibor that has to be paid by a borrower to cover funding costs. To give an example for clarification, suppose Λ f = 6M and Λ l = 1M. Since a loan might be amortizing, in each 6M period the notional might change from month to month. Assuming that a repayment of the notional immediately leads to a reduction in the outstanding bonds for funding, the interest paid on the funding bonds has to be reduced with the amortizations. The mismatch in interest tenors is reflected in the averaging of the loan's outstanding notional. If the mismatch is the other way round, i.e. Λ f < Λ l , this problem does not exist. Solving (15) for s f gives where we used the abbreviations A past = ∑ T i <t A i and A PV = ∑ t<T i A i δ f (T i ). For a fixed-rate loan, the solution can be derived from (16) by setting all forward rates f i,Λ f to zero and replacing s f by the fixed rate y f . After solving for y f the funding cost margin s f can be computed as s f = y f − y s .
When the payment frequencies of funding bonds Λ f and the loan Λ l are different, a basis swap is needed for hedging the mismatch in Ibor payments. These hedging costs can be computed from (15) by replacing the discount curve δ f with the loan's discount curve δ and going back to the loan's payment frequency. This leads to where s b, f is the interest margin covering both funding and swap costs. Solving (17) leads to from which the margin for hedging costs s b can be computed as s b = s b, f − s f . Again, the case of a fixed-rate loan is covered by setting f i,Λ l = 0 and replacing s b, f by the fixed rate y b, f . The margin s b associated with basis swap costs is computed as s b = y b, f − y f . So far, we have considered cost components that are independent of a loan's default risk. The next step is taking default risk into account. To derive a margin s EL reflecting expected loss risk, the condition expected assets equals liabilities is applied. We use the abbreviations and In (20), the survival probabilities reflect the fact that a bank will only pay out future tranches of a loan when the borrower is still alive. Using these abbreviations and (1), the interest rate spread s EL,b, f containing expected loss risk and the already computed funding and hedging costs is computed from the condition where again the special case of a fixed-rate loan is included by setting f i,Λ l = 0 and replacing s EL,b, f by a fixed rate y EL,b, f . Solving (21) for s EL,b, f gives The expected loss margin s EL is computed as s EL,b, f − s b, f for the floating-rate loan and as y EL,b, f − y b, f for the fixed-rate loan.
When calculating s f , s b , and s EL the interest margins were motivated from balance sheet considerations. In all the calculation steps, the assets of a bank and its liabilities were matched exactly or in expectation depending on the assumptions in each step. If default risks were independent, the calculations would be finished at this step because by the law of large numbers the variance in a loan portfolio's losses will become arbitrarily small if the number of loans is sufficiently large and without any volume concentration. This would result in deterministically matched assets and liabilities. Credit risks, however, are not independent since all borrowers are affected by macroeconomic risk resulting in dependent defaults. In a bad macroeconomic environment, credit losses are higher than expected while in a benign scenario they are lower. To avoid bankruptcy in recession years, banks have to hold an equity capital buffer than could absorb losses beyond expectation.
There are minimum requirements on the size of the capital buffer from regulators in Basel Committee on Banking Supervision (2006) and Basel Committee on Banking Supervision (2011). For less sophisticated banks, a simple approach using fixed weights like 8% of outstanding loan balance are applied in the Standardized Approach. Here, the calculation of the capital buffer E is simple and E is independent of credit risk parameters like PD and LGD.
Most of the bigger banks, however, apply the Internal Ratings Based Approach which allows banks to compute minimum capital buffers from internal estimates of PD and LGD using the formula where PD = 1−v(1) is the one-year default probability of the borrower, LGD can be computed from the collateralization at the loan's start, and ρ is the asset correlation which is defined in Basel Committee on Banking Supervision (2006) depending on borrower segment. 5 A lot of banks use the regulatory rules for determining capital that is allocated to a loan. The regulatory rules, however, do not reflect concentration risk and, therefore, miss important economic properties of a credit portfolio. For this reason, especially globally active banks have set up internal credit risk models to compute capital buffers against unexpected losses for managing their loan portfolios. These models are used to compute the loss distribution of a bank's credit portfolio better reflecting economic reality. The most popular approaches are based on Gupton, Finger, and Bhatia (1997) and CSFB (1997). Once the loss distribution is computed a risk measure is derived. Usually expected shortfall is used to compute portfolio risk and the risk is distributed among the single credits in a process called capital allocation. For details see Kalkbrener, Lotter, and Overbeck (2004) and Kalkbrener (2005). Since allocation and calculation of economic capital is not the focus of this article, it will not be discussed any further. For the purpose of this article, it is assumed that a bank has set up a process for economic capital computation which results for a particular loan in a contribution of economic capital E to total portfolio risk either by an internal model or following the regulatory rules.
The capital E is allocated to the loan and cannot be used for other investments. A bank defines a target return w t on its equity capital. The level of w t is a political decision by a bank depending on the market environment it is operating in. The equity capital E is not lying in a safe but invested in assets like government bonds where it generates a return w r . The difference to w t has to be generated by interest income of the loan. This leads to an additional interest rate margin s UL , the unexpected loss margin, which is computed as Finally, the operating costs of a bank, like staff salaries or office costs, have to be earned by a loan's interest. These costs are summarized in an additional cost margin c. The calculation of c depends on the institutional details of a bank and there is no general rule that is applicable to any bank. To include these costs into RAROC an adjustment is required reflecting the fact that only surviving borrowers can cover the costs. This leads to a cost margin s c which is computed as Note, that this assumption is not required for economic capital. The reason is that by construction the expected loss margin s EL should be sufficient to cover expected loss and economic capital is only a buffer against unexpected events. Once a borrower defaults and a loss provision is built, the capital is freed and can be used for other investments of the bank.
Putting all cost components together gives the hurdle rate z h of a loan, i.e. the interest rate that covers all costs and profitability targets of a bank. It is computed as Note, that this calculation is true only if the PD of a borrower does not depend on the interest rate z. If this is not the case (26) has to be replaced by a numerical algorithm as discussed in the following two sections.
If the interest rate z is given, the return on equity capital, or, equivalently, a loan's RAROC can be computed as This equation allows a bank to measure the impact of interest rates different from the hurdle rate z h on the return on economic capital. Furthermore, (27) can be used to measure the performance of already existing loans.

Properties of RAROC
When looking at (27) it seems that if z becomes arbitrarily large, so does RAROC. This means that this performance measure suggests that banks should charge as high as possible interest rates to maximize profitability. Obviously, this reasoning is flawed since at a certain interest rate level a borrower is unable to service his debt and will default. In order to make RAROC realistic, we have to link default probabilities and interest rates.
When building internal models for the default risk of a borrower, banks often include a variable known as the debt service ratio (DSR) into the list of explanatory risk factors. DSR computes the ratio of annual interest and amortization payments on all loans of a borrower and the available funds to pay interest. In the case of a company these funds are net profit before interest and taxes, in the case of a retail client it is net annual income. The interest rate z of a loan enters DSR linearly as DSR = β 0 + z · β 1 . The coefficient β 0 contains payments on other existing credit products a borrower might have while β 1 is the ratio of the loan's balance divided by available funds. If β 1 is small then PD can be approximately considered as independent of z. The larger β 1 , however, the more this approximation leads to wrong conclusions.
To analyze the properties of RAROC when default risk and interest rates are coupled, we use a simplified setup to maintain analytical tractability. We assume a bullet loan with a balance of one that pays interest annually at times T i = 1, . . . , m and t = 0. Furthermore, we assume funding costs, hedging costs and operational costs of a bank are zero. In addition, we assume the interbank curve is flat and all zero rates are zero, i.e. all discount factors are equal to one. Finally, we assume w r equals zero and R i is a constant R. Concerning economic capital we assume that a bank follows the Basel Standardized Approach, i.e. that economic capital E is independent of PD. This leads to a simplified RAROC formula with a simplified s EL as For the latter equation, note that in (22) A i is zero for all i < m for a bullet loan and one for i = m. The default part V D simplifies because R i is constant and discount factors are one which results in a telescope sum that can be simplified to 1 − v(m).
To model survival probabilities, we assume DSR is part of the risk factors and we condense all other risk factors into the constant β 0 . Furthermore, we assume a constant h. This leads to the survival probability v(i) at time i of The properties of RAROC under these assumptions are summarized in Theorem 1.
Theorem 1 Under the assumptions of (28), (29) and (30) where the constants β 1 , h and R are required to fulfill β 1 > 0, h > 0, and 0 < R < 1, RAROC(z) as a function of the loan interest rate z has the properties: There exists a unique interest rate z max with RAROC(z) ≤ RAROC(z max ), ∀z ∈ R The proof of Theorem 1 is provided in the appendix. The economic interpretation of Theorem 1 is quite intuitive. The first relation means that if a bank pays huge interest until the verge of bankruptcy, RAROC becomes arbitrarily small. If, on the other hand, the borrower pays huge interest which brings him close to bankruptcy, RAROC becomes arbitrarily small, too. Therefore, if a loan brings either the borrower or the bank into trouble, this is adequately reflected by RAROC. Somewhere in between these extreme cases, there is an optimum from the perspective of the bank which allows the bank to generate high income while keeping default risk manageable if the client is creditworthy.
Theorem 1 has direct consequences on the loan origination process of a bank. There exists only a finite range of interest rates that should be considered as acceptable from a bank's perspective. Given the profitability target of a bank w t only loans should be accepted with a RAROC greater or equal w t . This translates directly into a set of acceptable interest rates.
Theorem 2 Define the acceptance range A for a loan as the set of interest rates leading to a RAROC greater or equal w t A = {z : RAROC(z) ≥ w t }.
Then exactly one of the three cases is true: 1. A is empty 2. A consists of one point z max

A consist of an interval
The proof of Theorem 2 follows directly from Theorem 1. In the case of RAROC(z max ) < w t , A is empty and if RAROC(z max ) = w t then A = {z max }. Finally, if RAROC(z max ) > w t , there exists an interval [z l , z u ] where RAROC(z) ≥ w t . The smallest interest rate of this interval is the hurdle rate z h which is the minimum interest rate that covers all costs and risks associated with the loan. Note, when PD is a function of z, the hurdle rate can no longer be determined by (26) but has to be computed by a numerical algorithm finding min z RAROC(z) ≥ w t . Although there exist interest rates z with z > z max and RAROC(z) > w t it does not make sense for a bank to charge them. It can achieve the same profitability at a lower interest rate making it more likely that the client will take the loan from the bank and not from a competitor.
To conclude this section, we remark that we suppose that Theorem 1 holds in a more general setup. In numerical examples when using (23) for calculating economic capital E we still get a unique maximum RAROC in numerical examples. While it is quite easy to show that Parts 1 and 2 of Theorem 1 still hold, the third part becomes rather complex since the most general Basel formula includes PD as a function of z and the asset correlation ρ as a function of PD and, therefore, as a function of z which makes the analytical treatment of RAROC in this case rather difficult. Yet from our numerical experiments, we suppose that Theorem 1 holds in the more general setup.

Numerical Example
To illustrate the RAROC pricing scheme, we consider fixed-rate loans with and without amortization and with and without collateralization. We consider a ten-year fixed-rate loan paying an interest rate of 4% with quarterly interest payments. The loan's notional is N = 1, 000, 000 which is paid in one tranche at the loan's start date. We consider a bullet loan, i.e. a loan without amortization payments and an installment loan with an amortization rate of 5% annually. This means that in addition to the interest payment, the installment loan pays back 1.25% of the initial notional, i.e. A i = 12, 500 every quarter. Furthermore, the impact of collateral is illustrated. We assume that in this case, collateral with a cash equivalent value of C = 600, 000 is available. For the unsecured parts of the loan, we assume a recovery rate R u = 20%. This leads to a total of four different loans. For these loans RAROC is computed in the first part and hurdle rates and maximum RAROC in the second part.
To carry out these calculations, information about interest rate markets and institutional details of the bank is required. In the first step, the information on funding and interest rate markets is collected. We assume that the funding of a bank is expressed as a spread over 12M Ibor rates, i.e. the bank funds itself by issuing bonds paying annual interest linked to a 12M Ibor rate. Furthermore, swap rates of fixed-to-floating swaps and basis swaps have to be included to account for the tenor mismatch in funding and lending. Assuming the European conventions, we have quotes for swaps exchanging a fixed-rate against a 6M Ibor rate. Furthermore, we need the spreads of basis swaps exchanging a 6M Ibor rate against a 12M Ibor rate because of the funding tenor Λ f = 12M, and we need the spreads of basis swaps exchanging a 3M Ibor rate against a 6M Ibor rate because of the loan's tenor Λ l = 3M. The data is summarized in Table 1.
The front part of the discount curves that is bootstrapped from fixed-to-floating swaps is build from deposit rates. In the example of Table 1 the 3M and 6M deposit rate are used for computing the front part of δ M,6M . The data in Table 1 are not real market quotes but serves for illustration only.
For the evaluation of default risk, we assume that a bank has established a rating system with six grades and uses a Cox proportional hazard model (10) to estimate term-structures of default probabilities. We assume that the loan's interest rate is part of one risk factor, all other risk factors are summarized in the coefficient β 0 and h is a constant as in (30). The parameters for each rating grade are summarized in Table 2 while the default probabilities for each rating grade are illustrated in Figure 1 using the interest rate of the example, 4%.  It remains to define economic capital and the operating costs of the bank. We assume an annual operating cost margin c = 0.50%. Economic capital is computed following the regulatory rules for corporate clients with an annual turnover above 50 million EUR where we use both the Standardized and the Internal Ratings Based Approach in our examples. In case of the Standardized Approach we assume that the company does not have an external rating. Finally, we assume a target RAROC w t of 10%.
Cost components and RAROC are computed for the collateralized bullet loan (Loan I), the unsecured bullet loan (Loan II), the collateralized installment loan (Loan III), and the unsecured installment loan (Loan IV). The borrower rating is "3", i,e, we assume a borrower with a one-year default probability of roughly 1%. The results are summarized in Table 3 when E is computed as 0.08 · N D and in Table 4 when E is computed by (23).  The quantities y s and s f show the effect of the amortization rate. Both the swap curve and the funding spreads curve are steep. Since an amortization rate reduces the effective maturity of a loan both quantities are lower for amortizing loans. This effect is not seen in s b because both basis swap spread curves are flat. The expected loss margin s EL is considerably higher for the unsecured loan. For the amortizing collateralized loan the expected loss margin is lowest because this loan becomes less risky when the outstanding balance is reduced due to the amortizations. This effect is not seen in E in Table 4 because economic capital is based on a one-year horizon in the Basel II setup. We see that in both tables, only the collateralized loans pass the RAROC target of 10%. The unsecured loans show a RAROC below 10% and should be rejected if a bank strictly sticks to its profitability target.
In the seconds example, we compute z h , z max and maximum RAROC for Loan IV. Again we present the results for both regulatory regimes. The outcome for the Standardized Approach is displayed in Table 5 while the numbers for the Internal Ratings Based Approach are shown in Table 6. We see that for the high risk clients no hurdle rate z h exists. This means that it is not possible for a bank to set an interest rate that makes the loan profitable. Therefore, a loan application of these clients should be rejected. We see that for Rating "3" in both cases the hurdle rate is above 4%. This is consistent with the results in Tables 3 and 4 where RAROC was below the profitability target of 10% for Loan IV when an interest rate of 4% was used. Consistent with intuition, in both cases z h is increasing with borrower default risk while z max and RAROC max are decreasing.

Conclusion
In this article, a loan pricing scheme was developed using the performance measure RAROC. Motivated by balance sheet considerations, i.e. the desire to match assets and liabilities, a calculation scheme is developed which explicitly returns all relevant cost components, funding costs, swap hedging costs, expected loss costs, target return on economic capital, and internal bank costs. For fixed-rate loans, a formula for the base swap rate was given in addition. These cost components are essential for internal fund transfer pricing processes between separate functions in a bank.
The proposed pricing scheme is applicable for loans with the deterministic interest rate, i.e. fixed-rate loans and floating-rate loans linked to Ibor rates. We have analyzed the scheme mainly for the case where term-structures of default probabilities are estimated using a Cox proportional hazard model. This was mainly motivated by the analytical tractability of this model. However, the scheme does not depend on this modeling assumption and could work with any term-structure of default probabilities regardless of its determination.
In a theoretical analysis, it was shown in a slightly simplified setup that if the borrower default probability increases with a loan's interest rate then RAROC become −∞ in the limiting cases of arbitrarily large negative and positive interest rates which means that both the cases of bank and borrower bankruptcy are treated within economic intuition by RAROC. It was further shown that RAROC has a unique maximum and that at most a finite interval of interest rates exists at which a bank should accept a loan application. In cases where interest rates a borrower is willing to accept are outside this interval or when the acceptance range is empty, a bank should reject a loan application.
Numerical examples illustrated the application of this loan pricing framework. The examples suggest that the main results of the article hold in a more general setup than we were able to formally prove. The main challenge of applying this framework in practice is finding empirically a link between a loan's interest rate and borrower default rates. In real data sets important information for determining this relationship like the total interest a borrower is paying on all his existing loan products or timely income information is often missing in retail data sets which makes the parameters β 0 and β 1 of our examples very hard to estimate.
This results allows us to compute (we use a bit sloppy notation in the end) For the proof of Part 3, we will show that the first derivative of RAROC with respect to z is between 1 and −∞ and it is monotonically decreasing which implies that there exists exactly one root of dRAROC(z)/dz which proofs the theorem. We start with dRAROC(z)/dz where we use the abbreviations L := 1 − R and D := z − z EL : We have lim which can be seen after applying the rule of L'Hospital 2m − 1 times. Note that the highest exponent of q in the numerator is 2m − 1 because the coefficient of q 2m is zero and the highest exponent of the denominator is 2m. This leaves one q in the denominator after 2m − 1 times applying L'Hospital's rule while there is none in the numerator.
Since d(z−z EL ) dz is continuous it must have at least one root. To show that this root is unique, we show that d(z−z EL ) dz is decreasing monotonically by proving that d 2 D dz 2 = d 2 (z−z EL ) dz 2 is negative for all z.
∑ m i=1 hi · q hi + h(m − i) · q h(i+m) ∑ m i=1 q hi 2 + L log(q) 2 β 2 1 ∑ m i, j=1 q h j h 2 i 2 · q hi + h 2 (m 2 − i 2 ) · q h(i+m) ∑ m i=1 q hi 3 − L log(q) 2 β 2 1 2 · ∑ m i, j=1 h j · q h j · hi · q hi + h(m − i) · q h(i+m) Since log(q) < 0.0) the first term of this expression is negative. To prove that the full expression is negative, it is sufficient to prove that each coefficient of q hi in the numerator of the second term is non-negative and at least one is strictly positive. In total there are 3m − 1 terms q hk with k = 2, . . . , 3m. For k = 2, . . . , m + 1 only the first part of the double sum is