Abstract
Loan pricing is a surprisingly multifaceted undertaking touching on asset-pricing theory, accounting principles, corporate finance, and the fundamentals of risk management. Taking a tour of these modern financial concepts, this chapter builds on previously discussed economic-capital ideas to construct a risk-adjusted return on capital. Each loan decision is treated as a separate investment project with these risk-adjusted returns acting as a (fair) enterprise-wide decision criterion; this is accomplished by incorporating cost recovery, firm capital structure, portfolio concentrations, and the risk-return characteristics of the individual loan. The following discussion, quite naturally given our focus, touches solely on the quantitative aspects of loan pricing. Market forces and an institution’s mandate or strategy play an equally important role, but are beyond the scope of this work.
Look beneath the surface; let not the several quality of a thing nor its worth escape thee.
(Marcus Aurelius)
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Notes
- 1.
The reader is directed to EBA [16] for an interesting (and timely) dip into these conceptual waters.
- 2.
- 3.
This may be hard to believe for a reader having struggled to get a mortgage loan, but this point is quite true in commercial lending.
- 4.
It is, as we’ll see, broader than just the lending margin. Tenor, for example, is also quite important. Lending margin, however, is a useful one-dimensional description of the principal lever in the lending decision.
- 5.
It’s hard to be entirely definitive, but modern asset pricing probably started with Williams [41].
- 6.
See Mas-Colell et al. [27, Part IV] for more on this important, but complex question.
- 7.
During the grace period, where the loan outstanding does not change, the associated N values can naturally be zero.
- 8.
Strictly speaking, the term coupon should be used exclusively with bonds, not loans. Old habits, and expressions, nevertheless die hard. The reader should feel free to replace the term “coupon” with interest-rate payment as desired.
- 9.
This is the concrete representation of the number of days between payment dates used to determine the magnitude of the cash-flow.
- 10.
There is the possibility of providing fixed-rate loans to lending clients, but these would be swapped back to floating anyway.
- 11.
Moreover, by the publication of this book, it will certainly have been replaced.
- 12.
We further assume that this (transformed) measure is induced with the money-market account as the choice of numeraire asset. This is typically generically referred to as the risk-neutral measure.
- 13.
- 14.
The measure \(\mathbb {Q}^{T_{i}}\) is induced with the zero-coupon bond with maturity T i as the choice of numeraire asset. This is typically referred to as the forward measure; see, for example, Brigo and Mercurio [12, Section 2.5] or Björk [6, Chapter 19]. Implicitly, there is a change-of-measure from \(\mathbb {Q}\) to \(\mathbb {Q}^{T_{i}}\); this is accomplished with none other than the Radon-Nikodym derivative, \(\frac {d\mathbb {Q}^{T_{i}}}{d\mathbb {Q}}\). This clever choice of numeraire allows us to simplify dramatically our integrands by separating out the dependent instantaneous short rate, r, and the LIBOR rate, L i.
- 15.
Brigo and Mercurio [12] provide much more background and detail on this point.
- 16.
An Arrow-Debreu security is, of course, rather more general; this would be a special case. See Arrow and Debreu [3] for the gory details.
- 17.
OIS discounting remains a relatively recent practice and is (probably) poised to change somewhat with the advent of reference-rate reform. See Hull and White [21] for a discussion of its origins.
- 18.
The link between short-term interest rates, the business cycle, and default probabilities probably argues against this decision. For simplicity of pricing, however, it is probably best to avoid this potential rabbit hole of complex economic relationships.
- 19.
F τ(⋅) denotes the cumulative distribution function of τ. The default probability is, by its very form, equivalent to this (as-yet-unspecified) function of τ.
- 20.
Indeed, one can think of the difference between the risk-free and survival rates as a kind of representation of the individual firm’s credit spread.
- 21.
It is common, at this point in most default-risk pricing discussions, to place some additional structure onto the survival probabilities. This involves the (very useful) notion of the hazard function. For our purposes, however, this quantity will not be required. The interested reader is referred to Taylor and Karlin [38, Chapter 1] or Stuart and Ord [37, Section 5.34] for more background on hazard functions.
- 22.
This can get quite involved. See, for example, Ametrano and Bianchetti [2] for a discussion of some of the finer points involved in this exercise.
- 23.
More details can be found in Bolder [10, Chapter 9].
- 24.
Meucci [29] provides an insightful description of these two different tracks in quantitative financial analysis.
- 25.
Nor are these loans typically fair-valued, beyond informational purposes, in financial reporting. The clear exception is loan securitization, of course, but that is not the current topic of discussion.
- 26.
- 27.
This viewpoint would, with a few small label changes and brushing over some firm-specific elements, equally apply to virtually any financial institution.
- 28.
Typically, derivative valuations on both sides of the balance sheet roughly cancel one another out. While centrally important for the operations of almost any financial institution, they do not figure importantly in this discussion. Chapter 10 touches on how derivative contracts enter into the economic-capital picture.
- 29.
For many financial institutions, funding is typically augmented by (or even predominately comprised of) client deposits.
- 30.
An old paper—which still nicely frames the key issues—about the interplay between growth, dividends, and stocks prices is found in Gordon [17].
- 31.
A randomly selected Swedish firm’s long to medium-term growth target, for example, should probably not be determined independently of the expected evolution of Swedish output.
- 32.
If a firm owns equity assets, this would also include dividend income. Marketable securities’ revenue is also a bit more complex than simply coupons, but here we seek to keep it conceptually simple.
- 33.
Because loans are typically held at amortized cost, changes in their fair value do not flow through profit-and-loss. To account for expected lending losses, a separate loan-impairment computation is performed. While structurally slightly different, it is conceptually analogous to the final net expected loss term in Eq. 6.16. This element, in all its glory, will be considered more formally in Chap. 9.
- 34.
It is naturally part of the analysis, but it should not be the sole decision criterion.
- 35.
It also must be consistent with the firm’s overall risk appetite and capital position.
- 36.
James [22] is an interesting look into the genesis and rationale behind this measure.
- 37.
As usual, it is always a good idea to understand the specific recipe used to cook the dish.
- 38.
In practice, Eq. 6.24 allows us to compute the RAROC of any possible sub-portfolio. It is only necessary to keep track of the marginal contribution and consumption to economic capital for each individual asset.
- 39.
Most retail commercial banks, for example, fund themselves with short-term deposits, but make long-term mortgages loans. Swapping both sides back into a common reference rate enormously helps manage net margins.
- 40.
This is defensible, because the hedging decision can be thought of separately and independently from the lending choice.
- 41.
Exchange-rate risks are also hedged, but even if they were not, this element would show up in the economic-capital calculation.
- 42.
Following this reasoning, an m value is also readily calculable for fixed-rate treasury asset investments. This would permit extension of these ideas across all firm assets.
- 43.
We need not even make use of the change-of-measure trick to evaluate our expectation taken with respect to the equivalent martingale measure (induced by a collection of forward measures).
- 44.
Recall that the (expected) basis risk associated with the different LIBOR tenors is embedded in f. The worst-case aspect of this risk should also find its way into the market-risk component of economic capital.
- 45.
This consequently forces us to make use of the forward-measure numeraires introduced in our initial loan-pricing development.
- 46.
Funding financed lending activity also, of course, earns LIBOR. Since it simultaneously costs LIBOR, this effect cancels out.
- 47.
These asset-specific values are also, from a logistical perspective, more readily available over time for inclusion in the calculation.
- 48.
Indeed, in many cases, we may have u ≡ 0.
- 49.
- 50.
This necessitates the scaling by the appropriate day count.
- 51.
The mathematical structure of this approximation—treated here in a very abstract manner—is discussed in detail in Chap. 5.
- 52.
A non-exhaustive list would include the regional and sector identity of the obligor, the amount outstanding, the credit class, the loss-given-default, the and size of firm.
- 53.
Albeit subject to the reasonableness of the variety of assumptions made in Chap. 5.
- 54.
Equation 6.40 would hold with equality if Δi ≡ 1 for all i = 1, …, β or the payments are made annually. When the payment frequency is not annual, we will almost invariably annualize the RAROC estimate anyway.
- 55.
It is admittedly a kind of risk-weighted average, but this effect is relatively small and we’ll ignore it for our purposes.
- 56.
Presumably, although it is not easily verified, the origin of the term stems from the need for a project’s expected return to jump over this rate to be accepted.
- 57.
Other entities can, and certainly do, have analogous strategic objectives to be attained.
- 58.
This piece is, in fact, already incorporated into our net lending income.
- 59.
Whatever the time horizon, we have an annualized rate in mind.
- 60.
In reality, due to term premia and credit risk, the LIBOR rate should exceed the risk-free rate.
- 61.
Such questions, often referred to as risk budgeting, can become quite involved. See Scherer [32] for a useful entry point into this world.
- 62.
Countercyclical buffers are (typically) excluded to avoid introduction of economic cyclicality into the pricing decision. This point, depending on one’s perspective and objectives, is open to debate.
- 63.
Refer to Chap. 1 for a review of these economic-capital buffers.
- 64.
We will relax this assumption in a moment and explore the implications for the RAROC calculation.
- 65.
These values will be addressed in the subsequent table.
- 66.
If we wish to compute a RAROC for a treasury investment, the denominator would include slightly different ingredients. See Fig. 6.4 for more detail.
- 67.
To be really precise, we should write the set of approximation parameters as θ 0 ≡ θ to reflect their link to the starting time point—that is, i = 0—associated with the calculation.
- 68.
This depends on the level of the loan’s associated risk-weighted assets; this quantity is extensively discussed in Chap. 11.
- 69.
For small to medium firms, the process of price discovery is complicated by a relatively small number of potential lenders. Moreover, the flow of information regarding these lender’s pricing intentions is not entirely transparent.
- 70.
In this example, the existing loans are assumed to have an average maturity of four years.
- 71.
Practically, the calculation of this effect is conceptually straightforward. If N and O denote the new and existing (i.e., old) loan exposures, respectively, then
$$\displaystyle \begin{aligned} \hat{\mathcal{E}}_{\theta}(N) = \hat{\mathcal{E}}_{\theta}(O+N)- \hat{\mathcal{E}}_{\theta}(O), \end{aligned} $$(6.47)where, \(\hat {\mathcal {E}}_{\theta }(\cdot )\) denotes the economic-capital approximation. This computation thus requires three separate evaluations of the approximation approach from Chap. 5.
- 72.
A lag is necessary, since these values are used for cash-flows and economic-capital calculations. At time i, we need to know the commitment as of time i − 1 to properly compute these quantities.
- 73.
As one would expect, regulators also put limits and provide guidance on the appropriate choice of μ. Chapter 4 discusses this point in more detail.
- 74.
For a bullet-style loan, the grace period extends to the final maturity, when the full notional amount is repaid.
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Bolder, D. (2022). Loan Pricing. In: Modelling Economic Capital. Contributions to Finance and Accounting. Springer, Cham. https://doi.org/10.1007/978-3-030-95096-5_6
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