I am evaluating a normalization of financial data. Two colleagues feel certain of a certain approach, and I am befuddled. The approach shapes the results how they want (i.e., brings possibly extreme results toward a notional center line) but I'm not convinced it is mathematically cogent.

For background, we are analyzing the performance of loans we provide. We define $purchase\ price$ as the money we lend, $purchase\ amount$ is the total we expect to be paid back, $revenue$ to be the difference of $purchase\ price$ and $purchase\ amount$, and $margin$ to be $revenue - commissions$. For the sake of simplification, let us assume that there are no commissions, such that $revenue \equiv margin$.

We say that the $default\ rate$ as a percentage of margin is as follows, with $defaults$ being $anticipated\ margin - actual\ margin$:

$$default\ rate = 1 - \frac{actual\ margin}{anticipated\ margin} = \frac{defaults}{anticipated\ margin}$$

During the underwriting and quoting phase, we collect all the expected information and generate a couple of risk scores, and then a human makes qualitative judgements of the risk of the customer and of the loan. This risk is captured by the margin we charge and the time length of the repayment schedule. Riskier loans will skew higher in margin or (not XOR) shorter term length, less risky loans lower in margin or longer in term length.

We want to evaluate the performance of various classes of historical loans to decide if we properly priced against the resultant behavior of that business. That is, we want to decide if we adequately priced for risk. We do this by calculating the default rate for one class of business (e.g., from a particular industry or from certain partner sales offices) and comparing it to some benchmark default rate.

We want to normalize the default rates to equivalent units. It may be that one segment of the business hit a 30% default rate and another 20%, but unless we can reflect our anticipated risk on these segments and adjust the rates for comparability, we're concerned our analysis will be flawed. This is where my colleagues' notion arrives.

They feel it appropriate to "normalize" by calculating margin per month as

$$margin\ per\ month = \frac{margin}{term\ length}$$

and taking the mean across all the business as average margin per month

$$average\ margin\ per\ month = \frac{\sum{margins}}{\sum{term\ lengths}}$$

to be a benchmark. With this, they will calculate a (purportedly) normalized default rate as

$$normalized\ default\ rate = default\ rate \times \frac{average\ margin\ per\ month}{margin\ per\ month}$$

They seemed so sure that this gave them the "normalization" they wanted that I assumed my befuddlement was due to my being slow. I algebraically expanded their target formula and arrived at something that didn't make much sense:

$$\frac{defaults}{margin^2} \times \frac{(\sum{margins}) \times term}{\sum{term\ lengths}}$$

There a couple of ways to rewrite that but they seem equivalently unuseful.

My fundamental point of contention is not with the effort of normalization, but the method. Here, we pit the default rate for a class of business against the mean margin/month across all business divided by the margin/month for that class of business. They are right that this corrals the values some, e.g., a lower-than-average risk loan will have lower-than-average anticipated margin, and so its default rate in this analysis will actually be increased. Similarly, the actual default rate for high-risk loans will be decreased. If we expect a class of customer to be inherently riskier, we will price with a proportionately higher margin/month than a class of customer with "average" risk, so achieving a higher-than-average actual default rate for that class of customer doesn't itself worry us. Only if the actual default rate is higher than the target for that class of business/customer will we be worried.

Does this make sense as a means of comparability, and if so, what am I missing?


I could see if the various categories are distributed approximately normally and calculate a z score, but even then I'd like to determine if I'm missing something with this particular approach.


Wow...detailed question!! But very interesting :) Having some background in economics and finance, I will offer some ideas on this, but note that I am not a professional banker or underwriter, so please confer with your colleagues and the literature before using anything from this site (which I am sure you were going to do anyway).

I agree with you that your colleagues' approach seems confused, especially in the sense that its normalization on term length assumes that your risk/reward function is perfectly linear (i.e., twice the margin for twice the term length), which is probably not true since there is a qualitative, human element in setting term length and margin.

From your description, your problem is actually multi-dimensional, and hence cannot be so conveniently normalized to guage performance. Below, I will offer some ideas that may help you better understand where you are mispricing risk.

First, some assumptions and definitions (please correct if I am wrong):

  1. Your firm is risk-averse, and so requires increasing profits from riskier investments. Note that there is no objective way to price risk, as there is always this subjective element to it. However, once you know your risk posture, you can make progress.
  2. Lets change the definition of margin ($m$) to be the profit margin, i.e., $m=\frac{anticipated\space revenue}{purchase\space price}$. This removes the effect of the loan amount from our evaluation.
  3. The expected margin ($p$) is the margin ($m$) adjusted for the default rate ($d$): $p=(1-d)m$
  4. The subjective risk, $r$, is a function of the specific client information $C$. For a given risk level $r$, the possible loan terms can be described by a function of the margin $m$ and term length $t$, hence: $r(C) = f(m,t)$. This is an implict relationship, as both the left and right hand side are the results of human evaluations, but formally this is what you are doing.
  5. The default rate is a function of the risk and the loan terms: $d=g(m,t,r)$.
  6. I am ignoring any internal discount factors you may be using to get the NPV of your loans. You will need to adjust your expected margins for the time value of money if you think this will be relevant.

OK, now lets see what we can do given the above:

The key to evaluating your performance will be to verify that you are acting risk-averse (i.e., consistent with your risk posture). To do this, you will need to do two things, one difficult, one relatively easy:

  1. Hard part: You will need to know what "risk category" or "risk level" your analysts assigned to each loan at the time of application (not ex post facto). If you already have such a system in place, then use those risk categories, if not, you will need to use the assigned margins and payback periods to infer the risk. A simple function that will do this is $r(m,t)=\frac{m}{t}$. This function assumes that if the loan periods are the same, then the one with the higher margin is assumed to have been preceived as riskier. Likewise, if both have the same margin but one has a longer period than another, then it is assumed that the one with the longer period is less risky. The exact risk may be some power of this ratio or some multiple of it, but at least you will be correctly ordering your loans by preceived risk.
  2. Easy Part: Calculate the actual margin, $\hat m = \frac{actual\space revenue}{purchase\space price}$ for each loan.

To get a measure of performance, you will want to perform a regression using your observed triples $x_i \equiv (r_i,t_i,\hat m_i)$, with $r_i$ and $t_i$ being the predictors and $\hat m_i$ being the response. Specifically, we will model $\hat m_i$ as follows:

$\hat m_i = \varepsilon (r_i t_i)^k$, where $k$ is an unknown parameter and $\varepsilon$ is a lognormal random variable on $[0,\infty)$ with logmean $\mu$ and logvariance $\sigma^2$ (both unknown).

I chose the lognormal for computational convenience. A full, albeing more complex, treatment would require generalized linear models, which I think may be too much for this application.

Taking the natural logarithm of both sides, we get the usual linear regression equation with normally distributed errors:

$\ln(\hat m_i) = \ln(\varepsilon)+k\ln(r_it_i) = \ln(\varepsilon)+ks_i = $, where $s_i = \ln(r_it_i)$

You can now estimate $k$ by performing a simple linear regression with $\ln(\hat m_i)$ as the response and $s_i$ as the predictor.

The regression output (from excel, Minitab, or whatever you use) should give you a confidence interval or standard error and degrees of freedom for the slope parameter $k$. You will want to test that $k>1$ vs $k\leq 1$. If $k>1$, it means that you are acting in a risk averse manner and are hence properly pricing your risk.

For a more detailed view, you can make a 3-D plot of your original triples to see if there are certain subsets where the $\hat m$ surface "slopes downward" significantly. You may be good at identifying very low and very high risks but are inconsistent in the middle risk ranges.

  • $\begingroup$ Thank you for such a full response. I had already started the "hard part" conversation and don't quite have the answer yet, but I feel vindicated as having started down that road. +) I'll admit that this is both consistent with the kind of treatment I think this analysis deserves and requires me to knock the rust off my undergraduate math skills (lamentably not very old). I'll review and likely accept this as the answer soon (I'm new-ish to SE, in practice). $\endgroup$ – Daniel Black Mar 7 '14 at 16:13

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