Interest expense is the largest item on most banks' income statement, but this expense frequently receives significantly less attention than expense items that are far less important to the profitability of the institution. The discussion that follows will show you how to do some simple calculations to help you set prices for deposit products in ways that can significantly improve the profitability. The discussion will be focused on how to determine the marginal cost of funds for depository products.

"What is our marginal cost of funds?" is a question that is infrequently asked and less frequently answered at many institutions. For some products, it can be relatively simple, while for others it can be fiendishly complex and require statistical analysis. The discussion that follows will help to calculate the simple straightforward costs and to know what is involved for the difficult calculations. To calculate marginal costs, you must first gather the necessary data. For checking, savings, and money market (MM) products, this is fairly simple--you just need the aggregate balance for the product, and the weighted average interest rate over the period--probably a month. If your products have different interest rates depending upon the account balance, you should treat each tier separately in this analysis. You should go back as far possible; three years of history is probably the minimum to do a good analysis.

For certificate of deposit (CD) products, the data collection is somewhat more difficult, as you will need the total balance, the amount that matured during the period, the amount that rolled over, the maturing deposits that were withdrawn, the non-maturing deposits that were withdrawn, and the new deposits added during the period. This may not be available in an master customer information file (MCIF) extract and may require a special extract from Information Systems.

Getting data on a monthly basis is probably the easiest, as this is frequently captured for an MCIF extract. Unfortunately, monthly isn't necessarily the best choice as the number of days varies; for some types of analysis, this will introduce unwanted variability, but if monthly data is the only choice available, you can generally use it without too much difficulty.

Once you have the data from your internal systems, you will need to enhance it with some publicly available data on market interest rates. Many institutions use Treasury rates for transfer pricing, but for marginal cost analysis, swap rates are generally a better choice, as swap rates closely follow the top of the market for bank CD offerings and provide a very good proxy for the competitive market. Federal Home Loan Bank (FHLB) and other outside borrowing rates are perhaps the best proxy for the competitive market CD market, but using FHLB rates requires more code to download the rates and requires that you do your own averaging over the period. The best data would be competitor's actual rates, but gathering history for competitors' rates would be very difficult. Swap rates are readily available from the St. Louis Federal Reserve's FRED2 system for several different time intervals, making swap rates the most practical choice for marginal cost analysis.

Once you have both in-house data and swap rate data, merge them so that you have a table with period (date), balance, nominal interest rate, and swap (benchmark) interest rate. For checking, savings and money market accounts, use the one year swap rate. For CD products, use the swap rate term that most closely matches the CD term. For a money market product, the table would have the following structure:

When you plot the balance against the nominal interest rate, you will probably get a plot that shows no discernible pattern, similar to the plot in Figure 1, but if you plot the balance against the *delta to benchmark* as shown in Figure 2, you will probably get a reasonably clear pattern where the the balance increases as the delta to benchmark increases. For many institutions, the pricing decisions based upon Figure 2 can be worth hundreds of thousands or millions of dollars.

To calculate the marginal cost, we want to get the slope of the line that best fits the data in Figure 2. An eyeball calculation would be to take a point at each end and calculate the slope--(0%,$1.05e8) and (-0.2%,$9.75e8) for a slope of

(5e+05 - (-2500000))/(0 - (-0.2))

## [1] 1.5e+07

1.5e7 or $15 million per month balance increase per 1% increase relative to the benchmark. A more precise way to calculate the slope would be to do a linear regression

## ## Call: ## lm(formula = amt_balance_change.MM ~ delta_to_benchmark.MM, data = lbtDf2) ## ## Coefficients: ## (Intercept) delta_to_benchmark.MM ## 41253 11038442

## Analysis of Variance Table ## ## Response: amt_balance_change.MM ## Df Sum Sq Mean Sq F value Pr(>F) ## delta_to_benchmark.MM 1 3.82e+13 3.82e+13 152

which gives a slope of 11038442 or $11 million per month balance increase per 1% increase relative to the benchmark. To convert this to a marginal cost we have to calculate the cost of adding $11 million to an existing $99 million MM product by increasing the interest rate by 1% (0.083% per month) assuming a benchmark rate of 0.5% (0.0416% per month):

\[ \begin{aligned} \text{marginal cost}&=&\frac{0.00083*99e6+(0.000416+0.00083)*11e6}{11e6} \\ &=&0.008718 \text{ per month}\\ &=&10.5\% \text{ per year} \end{aligned} \]

Why is the marginal cost so high? In order to get customers to add $11 million, the bank must pay a premium (the 0.00083*99e6 in the formula) to existing depositors who really don't care about getting the additional interest. If product balances are high, it may make sense to open a new product and allow rate-sensitive depositors to switch into the new high-rate product until the marginal cost of the original product is reasonable.

The marginal cost of funds for savings accounts can be calculated similarly. For interest bearing checking products, the marginal cost can be calculated if there are no fees for going below a minimum balance. If there are minimum balance fees, the interest rate would need to be adjusted to include the fee as an implied negative interest rate. Checking deposits tend to be very seasonal, with declines in December through February for Christmas spending and a spike in April for tax refunds. There is also a lot of noise introduced by weekly, bi-weekly payroll cycles being superimposed over a monthly data extract cycle. These two characteristics make getting a good regression model difficult for checking account data.

Doing the modelling in the format described is quite straightforward, but it can't really be used for large changes in the balance, especially in a declining environment. $11 million leaving a $100 million balance is 11% of the balance--but would be about 5.5% of a $200 million balance. A future article will address how to improve the model to account for this difference.

Calculating the marginal cost of Certificate of Deposit (CD) products can be significantly more difficult than the similar calculation for money market, savings and checking accounts. The example that follows will illustrate some of these difficulties. First we need to address a terminology problem. For CDs, we will look at the amount maturing, and call that the product balance to make things work similarly to the way that they did for Money Market products. For these plots, we will assume that the amount maturing each month is the same--a completely invalid assumption in real life, but it makes the illustration much easier to follow. If you plot the balance against the nominal rate as in Figure 3, you will probably get a plot with no discernible pattern. If you plot the balance against the delta to benchmark as in Figure 4, you still probably won't get much of a pattern; this is very unlike the MM product.

Why is there no pattern? Many customers looking at CDs will compare prices with MM funds, with CDs of slightly shorter terms, and with CDs of slightly longer terms; if the other products are more competitive with the customer's perception of the yield curve, the customer will put money into the MM or the shorter/longer CD. They will *substitute* a more attractive CD if the originally desired CD isn't priced to their liking. To illustrate this in a data plot, do a contour plot of the CD Balance Change vs. the CD delta to benchmark AND the MM delta to benchmark. In this plot you will probably see a clear pattern as shown in Figure 5. CD products typically have a great deal of product substitution, so that the price of the product relative to its benchmark is important, but so is the price relative to other CD products and MM products. It is impossible to estimate this graphically; a multi-variate regression model is the only approach that will work for this situation.

To calculate the marginal cost of the CD product, you have to assume the prices of the substitute CD terms--but if there a large amount maturing in one substitute CD term, that would assume a high marginal cost for that CD term. Since the bank doesn't want to set prices so that multiple products have a high marginal cost, **both** prices have to be calculated simultaneously. For this reason, it doesn't make much sense to try to calculate the marginal cost for CD products. Setting rates is the ultimate goal, and the marginal cost is just input into that goal.

The best way to set CD prices is to model demand for each CD product including substitution effects to get a formula that represents the demand for each CD as a function of its price and the price of other CDs. Then all of the formulas can be solved simultaneously to obtain a price sheet that will reduce the cost of funds for the deposit portfolio. Although Excel has the statistical tools to do this, automating a large number of regression models in Excel would be difficult.

Fortunately, there are inexpensive or free statistical packages that can easily handle the task of developing the demand models. Solving the system is straightforward with commercial mathematical modeling languages like AMPL or GAMS.

The data in this article was synthesized to illustrate common characteristics of money market and certificate of deposit demand data. Data manipulation and chart preparation was done using the R statistical language. Historical swap rates from the St. Louis Federal Reserve FRED2 system were downloaded and incorporated using the fImport package. The charts were prepared using ggplot2. The display of mathematical formulas in the HTML web page uses MathJax, which allows formulas to be written using LaTeX syntax.

The discussion in this article follows portions of the *Estimating Deposit Volume Price Sensitivity* presentation by the author at the 2005 Institute for Operations Research and Management Science (INFORMS) Annual Meeting. The presentation received an Honorable Mention in the Financial Services Section contest for Best Presentation. The abstract is available on the INFORMS web site.

Many banks and check cashing services are swamped with paycheck cashing transactions on Friday afternoons and Saturday mornings. It can be prohibitively expensive to hire additional staff to handle these peak periods, since even part-time staff would be idle for the rest of a 20 to 30 hour work week, and people who are qualified to be tellers can generally find full-time work. It can be difficult to determine policies that handle this peak period volume efficiently.

In some circumstances, it can make sense to charge a higher check-cashing fee during these peak times to encourage check-cashers to move their check-cashing transactions to other times when a branch is underutilized or to use less labor-intensive deposit services. The article that follows is a brief (very) summary of the findings of my Doctor of Engineering Praxis Optimal Time-variable Pricing for Check-cashing Transitions on when variable pricing works in time-shifting demand, and when it would back-fire horribly. The article is divided into the following sections:

- Product Substitution and Check Cashing
- Check-cashing Volume vs. Pricing
- Estimating Per-visit Labor Cost
- Creating an Optimization Model to Maximize Income
- When Time-variable Pricing Works, and When it Does Not
- Conclusions

In designing a pricing policy, understanding product substitution is critical. When we go to the grocery store to buy milk, if the price on your favorite brand of fat-free milk is higher than you want to pay, you will probably buy or “substitute” a lower-priced brand for your favorite. The same behavior holds true in the check-cashing market: if the fee is too high, the check casher will go somewhere else or substitute a different transaction sequence for their favorite sequence as a way to avoid the check-cashing fee. Depending upon check hold policies and account rules, there are several possible substitution alternatives:

- Pay the check-cashing fee
- Deposit a portion of the check and get the remainder in cash
- Deposit the check and withdraw all or part of the check in a teller transaction on the same or a subsequent day
- Deposit the check and withdraw all or part of the amount in an ATM transaction on the same or a subsequent day
- Argue with the teller until the teller gives in and waives the fee as a way to deal with a growing line of disgruntled customers.

Because most of the substitution patterns generate an additional teller visit or require a longer teller visit, any fee increase that encourages substitution can actually increase teller workload.

Understanding how check-cashing demand changes as the check-cashing fee increases is the cornerstone of implementing a time-variable pricing policy. This requires historical data on fee changes and transaction volumes. At a typical bank this can become quite complex, as customers typically do multiple general ledger (GL) transactions during one visit. To get accurate demand information for check-cashing visits, the institution must first categorize all common groups of transaction into visits, and then do regression models on demand for each visit type. Categorizing GL transactions into visit types and then converting GL transaction history into visit type demand data is time-consuming and computationally intensive. Academic computer scientists would describe it as “non-trivial”, though with the advent of solid-state disk drives and inexpensive memory, this is today a much easier problem then it was in 2004.

The basic steps in this process are

- Collect GL transaction data covering the period(s) when there were check-cashing fee changes
- Collect information on dates for other policy changes and prices for other products
- Identify common groups of GL transactions that form common customer visit types. This will typically require the use of Apriori associations or some other type of associations modeling tools. This step is computationally fairly straightforward.
- Group GL transactions into common customer visit types. This step is computationally intensive, and is by far the most difficult step in the process.
- Run regression models to estimate the changes in demand as function of check-cashing fee prices. Controlling for other fee and interest rate changes can be challenging.

Before attempting to do this analysis, estimate the resources required for Step 4 and make sure that you will have the necessary computer resources before proceeding.

To choose optimal check-cashing fee prices, an institution must estimate the labor cost for each visit type identified in the previous step. To do this, the institution must collect payroll data at the shift level in order to calculate the number of tellers working at a given time in each branch. When combined with the visit type data from the previous step, it is possible to use linear regression to estimate the number of teller minutes required for each visit type. This information can be interesting all by itself and may lead to changes in policy that are unrelated to check-cashing fees.

Once the demand sensitivity data and labor cost data have been estimated, it is time to generate an optimization model that chooses a fee timetable and staffing schedule that maximizes income while meeting various constraints on minimum and maximum work-week and other work-schedule quality metrics. Even if time-variable pricing is not used, the work-schedule optimization can be beneficial to the organization.

The most natural formulation for this model is a non-linear (fee price) integer (shift scheduling) model. Because there are few non-linear integer solvers available and because this is a computationally difficult problem, it is generally better to formulate the problem as a strict integer problem. In this case, the fee price is allowed one of several discreet values, probably in 0.25% increments.

In general, if a bank allows deposit of a check and relatively quick withdrawal of funds, check-cashing fees can backfire due to the substitution of a deposit visit and a withdrawal visit for the single check-cashing visit. For a dedicated check-cashing service where there are no deposit accounts and this no substitution alternatives, time-variable pricing could be an efficient way to handle peak demand.

Time-variable pricing probably would not work well at most banks and credit unions due to the ability of check-cashing customers to avoid the fee by substituting other groups of transactions that avoid the fee and which are more labor-intensive for the institution. For a dedicated check-cashing service, this could be a way to handle peak-volume problems efficiently. To get the full benefit of a time-variable pricing policy, an institution should do a staff-scheduling optimization model as well so that tellers are as steadily utilized as possible and so that wait times are reasonable throughout the business week.

Analyzing the possible substitution patterns prior to undertaking the analysis for estimating optimal prices is time well spent; if there are any reasonable substitution patterns for paying check-cashing fee, there is little point in looking at time-variable pricing.