The Laws of Motion of the Broker Call Rate in the United States

In this paper, which is the third installment of the author’s trilogy on margin loan pricing, we analyze 1 , 367 monthly observations of the U.S. broker call money rate, which is the interest rate at which stock brokers can borrow to fund their margin loans to retail clients. We describe the basic features and mean-reverting behavior of this series and juxtapose the empirically-derived laws of motion with the author’s prior theories of margin loan pricing (Gari-valtis 2019a-b). This allows us to derive stochastic diﬀerential equations that govern the evolution of the margin loan interest rate and the leverage ratios of sophisticated brokerage clients (namely, continuous time Kelly gamblers). Finally, we apply Merton’s (1974) arbitrage theory of corporate liability pricing to study theoretical constraints on the risk premia that could be generated in the market for call money. Apparently, if there is no arbitrage in the U.S. ﬁnancial markets, the implication is that the total volume of call loans must constitute north of 70% of the value of all leveraged portfolios.

"Since those who rule in the city do so because they own a lot, I suppose they're unwilling to enact laws to prevent young people who've had no discipline from spending and wasting their wealth, so that by making loans to them, secured by the young people's property, and then calling those loans in, they themselves become even richer and more honored." -Plato, The Republic, 380 B.C.
"Neither a borrower nor a lender be: For loan oft loses both itself and friend."

Introduction
This paper is inspired chiefly by two of the author's theoretical formulas for margin loan pricing by stock brokers (Garivaltis 2019a-b): 1. The instantaneous monopoly price of margin loans to Kelly (1956) gamblers: 2. The negotiated interest rate under instantaneous Nash (1950) Bargaining 1 with Kelly gamblers: In these formulas, r * L denotes the (continuously-compounded) margin loan interest rate charged by the broker over the differential time step [t, t+dt], where ν := µ−σ 2 /2 is the asymptotic (or logarithmic) growth rate of the stock market index, σ is the annual volatility, and µ is the annual (arithmetic) drift rate. r B denotes the broker's cost of funding ("broker call money rate") for the duration [t, t + dt]. These formulas are of great interest on account of their simplicity and their practicality; naturally, the broker charges more if the underlying growth opportunity dS t := µ dt + σ dW t (5) d(log S t ) = (µ − σ 2 /2) dt + σ dW t A. Garivaltis is more favorable (higher ν, lower σ). Because all the action (the broker posts a monopoly price, or the principals Nash bargain over both the price and quantity of margin loans) happens over the differential time step [t, t + dt], the formulas apply equally well to a general situation whereby the stock market index S t is governed by time-and state-dependent parameters µ(S t , t) and σ(S t , t). The affine relationships (3) and (4) imply that the net interest margin r * L − r B must shrink whenever the broker call rate r B increases; ceteris paribus, for a 100 basis point fluctuation in the broker call rate, only 50 bps will pass through to the consumer (or 75 bps under Nash Bargaining).
The purpose of this article, then, is to use empirical data to divine the general laws of motion of the U.S. broker call rate r B (t), and to study the logical consequences for the random behavior of margin loan interest rates, risk premia, and the leverage ratios of continuous time Kelly gamblers. The U.S. broker call money rate, which is published daily in periodicals like The Wall Street Journal and Investor's Business Daily, is so-named because stock brokers must be prepared to repay these funds immediately upon "call" from the lending institution.
The paper is organized as follows. Section 2 describes our data set, which consists and use it to construct a classical method-of-moments estimator of the analogous Section 3 juxtaposes the empirical specifications (7) and (8)  In order to find agreement with the author's prior work on margin loan pricing (Garivaltis 2019a-b), we must deal with the continuously-compounded annual interest rate, as follows: Figure 1 gives a plot of the time series (y t ) T t=1 ; the grey bars on the figure indicate NBER recessions, during which rates have usually fallen precipitously. For the sake of smoothing out the choppy appearance of (y t ) T t=1 , Figure 2 plots the 12-month simple moving average which is plotted in Figure 4. To help visualize the internal correlation structure of the call money rate, Figure 5 gives a plot of the sample autocorrelation function where j ∈ {0, ..., 12} denotes the number of lags, in months. The sample correlation coefficient for successive monthly observations isρ 1 = 59.7%. In order to control for any confounding effects that the interim observations (y t−j+1 , y t−j+2 , ..., y t−1 ) could possibly have on the observed relationship between y t−j and y t , Figure 6 supplements the sample correlogram with a 24-month plot of the sample partial autocorrelation function. As illustrated by the figure, the partial autocorrelations start to lose their statistical significance for lags in excess of 12 months.

Reversion to the Mean
Drawing some inspiration from the sample autocorrelation function as depicted in Figure 5, we proceed to estimate a stationary first-order autoregressive model of (y t ) T t=1 . This amounts to the linear stochastic difference equation or equivalently, where L denotes the lag operator. The deep parameters are α, ρ, and σ, and the stochastic shocks ( t ) T t=1 are assumed to be unit white noise, e.g. they are serially uncorrelated, E[ t ] ≡ 0, and Var[ t ] ≡ 1. The contemporaneous disturbance t is assumed to be uncorrelated with Call Rate t .
Under this terminology, the long-run mean of the (continuously-compounded) interest rate is given by and Of course, the (aptly named) parameter ρ in this AR(1) model is equal to the Pearson correlation coefficient of successive monthly interest rates: cess (y t ) ∞ t=1 is given by If we let θ := 1 − ρ and re-arrange the empirical specification (13), we obtain the following equivalent representations: where θ represents the rate of monthly mean reversion per 100 basis points of deviation from the equilibrium level. The coefficients α, ρ can be recovered from the new parameters µ, θ via the relations α = θµ and ρ = 1 − θ. Table 2 gives the parameter estimates that obtain when fitting the empirical rela- tionship y t+1 = α + ρy t + σ t via ordinary least squares (OLS). The linear regression is illustrated in Figure 7, which plots the broker call rate versus its lagged values.
Thus, our empirical law of motion for the call money rate is This means that for every 100 basis points of deviation from its long-run average of 3.94%, the broker call rate is expected to close the gap at a rate of 40 basis points per month. However, this mean-reverting behavior is corrupted by random disturbances whose average (root-mean-squared) magnitude is 2.36% per month. Solving the first-order difference equation (13) for Call Rate t in terms of Call Rate 0 , one gets the expression (cf. Hamilton 1994) Thus, our general forecast for the broker call rate t months hence (normalizing today's ) .
Example 1 (Out-of-Sample Predictions). As of this writing, Bankrate.com reports the following information about the U.S. call money rate: Figure 6: 24-month plot of the sample partial autocorrelation function of the U.S. broker call rate.
• The current U.S. call money rate (as of this writing) is also y 12 = 4.25%.
Thus, from the standpoint of a month ago, today's call money rate would have been forecasted to be 4.13%, for a prediction error of 0.12%. From the standpoint of a year ago, today's call money rate would have been forecasted to be 3.94%, for a prediction error of 0.31%. These errors compare favorably with the root-mean-squared errors plotted in Figure 8.

AR(2) Model
Taking our cue from the fact that Bankrate.com only reports the two most recent monthly observations of the broker call rate, we proceed to estimate a (stationary)  second-order autoregressive model for the sake of lowering our root-mean-squared prediction error. Thus, we have the empirical specification or equivalently, The long-run mean is and the unconditional variance (cf. Fuller 1976) is When expressed in mean-deviation form, our empirical specification amounts to or equivalently, A particular solution is of course given by y p t ≡ µ. In order to solve the associated homogeneous equation we require the roots of the characteristic equation which are Thus, the general solution of the difference equation (34) is (38) Figure 10 compares the 12-month forecasts of our estimated AR(1) and AR (2) {0.764, -0.308} AR(2) forecast exhibits a significantly slower rate of mean-reversion than its AR (1) counterpart. On that score, Figure 11 plots the two models' responses to an exogenous 100 basis point impulse in the broker call rate. After 6 months, the persistent effect on the broker call rate amounts to 14 basis points under the AR(2) model; at the 12-month mark, the marginal effect dissipates to just 3 basis points.

Vasicek Model
To better understand the short-term (intra-month) fluctuations of the broker call rate, we use our monthly AR(1) parameter estimates to help fit an Ornstein-Uhlenbeck model of interest rate evolution in continuous time (cf. Mikosch 1998). Vasicek (1977) was the first researcher who used Ornstein-Uhlenbeck processes to model the meanreverting behavior of interest rates. In our context, we have the following stochastic Equivalently, we have the integrated form (cf. Mikosch 1998) where W t is a standard Brownian motion and dW t := √ dt is its instantaneous change in position over the differential time step [t, t + dt]. The parameter µ := E[Call Rate t ] represents the stationary mean, or long-run equilibrium level, of the broker call money rate. The parameter denotes the instantaneous rate of mean-reversion, e.g. the expected rate of change in the interest rate as a percentage of its current deviation from the long-run average.
Finally, the parameter represents the local variance of interest rate changes per unit time.
The solution of the Ornstein-Uhlenbeck equation (cf. Mikosch 1998) is A. Garivaltis and the stationary (long-term) standard deviation is The corresponding t-month ahead forecast is and the root-mean-squared forecast error is In order to reconcile the conditional forecast function (45) with the AR (1) forecast In order to reconcile the long-run standard deviation (44) with its AR(1) counterpart Thus, the following three equations summarize our estimated law of (continuous) motion for the U.S. broker call rate.
Integral Form:

Implications for Margin Loan Pricing
For the sake of this section, so as to avoid any confusion, all interest rates, standard deviations, drifts, etc. will now be reported as numbers belonging to the unit interval [0, 1] (rather than as percentages between 0 and 100).
In the author's prior work on margin loan pricing in continuous time (Garivaltis 2019a-b), he derived the simple theoretical relationship where C is a constant that is independent of the broker call rate and independent of the time t. This was done by assuming that the broker's sole (representative) client is a continuous time Kelly gambler (cf. Luenberger 1998) who borrows cash over each differential time step [t, t + dt] for the sake of leveraged betting on a single risk asset (say, the market index) whose price S t follows the geometric Brownian motion Here, we have used the symbol dW S (t) to denote the standard Brownian motion that drives the asset price; the drift and volatility are µ S and σ S , respectively. The Equivalently, the broker faces the inverse (instantaneous) demand curve where the parameter ν S := µ S −σ 2 S /2 represents the expected compound (logarithmic) growth rate of the market index (say, the S&P 500). Thus, our constant C is given by Given the backdrop of our mean-reverting empirical model of the broker call rate, The theoretical pricing formula (53) implies that the margin loan interest rates charged Bearing in mind that Call Rate t = 2(Margin Rate t − C), we get the law of motion A. Garivaltis where the time t is measured in months and W t is the standard Brownian motion that drives the broker call rate. Thus, the leverage ratio of the (representative) Kelly gambler reverts to its long-term mean of (µ S − µ)/(2σ 2 S ) at the same rate θ as the broker call rate and the margin loan interest rate. Given our empirical findings, we have the concrete (monthly) law of motion:

Arbitrage Pricing of Call Loans
In this section, we use Merton's (1974Merton's ( , 1992 no-arbitrage approach to corporate liability pricing to derive theoretical formulas for the broker call rate and the net interest margin that banks should earn on such loans. On that score, we let r denote the risk-free rate of interest, and we let R denote the broker call rate, where ρ := R − r > 0 is the corresponding risk premium. The broker himself charges his retail customers a margin loan interest rate of R > R. We assume that the (representative) brokerage client borrows D dollars to finance the purchase of a single share of a risky stock or index, whose initial price at time 0 is S 0 . The client's initial equity is As usual, we assume that the asset price follows the geometric Brownian motion where µ is the annual drift rate, σ is the annual volatility, and W t is a standard Brownian motion 2 . Interest is assumed to compound continuously over the loan term [0, T ], so that the client's accumulated margin loan (debit) balance at time t is De Rt .
Thus, his equity fluctuates according to the random process E t := S t − De Rt .
If the broker was willing or able to continuously monitor the client's account for solvency, then there would be no credit risk, for, on account of the continuous sample path of (E t ) t∈[0,T ] , the broker could liquidate the account the instant that E t = 0 (or some other threshold E). Thus, under continuous monitoring, there is certainly no risk to the bank that funded part of the margin loan; in this case, the no-arbitrage axiom dictates that R = r. In order to have R > r in equilibrium, we must start with a situation whereby it is possible for the retail client to default on his margin loan.
Thus, as in Fortune (2000) and Garivaltis (2019a), we assume that the broker does not monitor the client's account for solvency until some given maturity date, T .
However, if the broker is willing to maintain a dynamically precise short position in the risk asset (cf. Fortune 2000 and Garivaltis 2019a), then it is possible, in the sense of Black and Scholes (1973), to completely "eliminate risk" through continuous trading in the underlying. In this happenstance, the no-arbitrage principle implies a unique margin loan interest rate R > r, but it fails to give us a characterization of A. Garivaltis the call money rate, since there is no actual risk to the bank that funded the margin loan. Thus, in order to generate risk premia in the call money market, we must make the twin assumptions: 1. The broker does not check the client's portfolio for solvency until the maturity date, T .
2. The broker is not willing or able to hedge his own default risk.
In this environment, we now have the possibility of a "default cascade" whereby the client defaults on his margin loan at T , and this in turn causes the broker to default on his debt to the money market. Accordingly, we will assume that the broker borrows Equivalently, this means that for 0 ≤ t < T , we have Following Fortune (2000) and Garivaltis (2019a), we assume that the retail client will abandon his account at T if E T ≤ 0, leaving the broker with collateral worth S T .  Thus, the broker's assets at the end of the loan term amount to and the broker's final equity is equal to If the broker's final equity is ≤ 0, then he himself will default on his debt to the money market, leaving his creditors with collateral in the amount of min(S T , De RT ).
Thus, the final payoff that accrues at T to the bank that made the call loan is where we have made use of the fact that d < D and R < R. Table 4 summarizes the three possible credit events faced by the call lender.
Assuming that the bank's call money was itself borrowed at the risk-free rate r, the bank's final profit (loss) is Making use of the identity min(x, y) = x + y − max(x, y), we have where K := de RT . That is to say, the bank's (random) profit π T amounts to the final payoff of the following portfolio: • Long one share of the stock • Short d dollars at the risk-free rate of interest • Short one European-style call option at a strike price of K := de RT .
Naturally, the bank can hedge its (net long) exposure to the underlying (e.g. the bank has de facto written a covered call) by shorting a dynamically precise amount of the retail client's portfolio. In order to prevent riskless arbitrage opportunities, the time-0 expected present value of the bank's profit with respect to the equivalent martingale measure (Q) must be zero: Recalling the Black and Scholes (1973) formula where and and simplifying, we get the following equation characterizing the broker call rate: where ρ := R − r is the risk premium for call money, and the ratio d/S 0 represents the percentage of the portfolio that has been financed by call money. As usual, N (•) denotes the cumulative normal distribution function.
Note that the broker call rate R does not depend on the drift µ or on the margin loan interest rate R that the broker charges its clients. The characterization (79) of R is not particular to the numerical levels of d and S 0 ; it only depends on their ratio d/S 0 . Similarly, the numbers r and R only matter to (79) in so far as their difference ρ := R−r is featured prominently. That is to say (cf. Merton 1974 andMerton 1992), the risk premium for call money depends only on the following credit characteristics: • T (the loan term); • d/S 0 (the loan-to-value ratio); • σ (the volatility of the collateral).
The bank's net exposure to the underlying in state (S t , t) is equal to (cf. Wilmott  This means that the sum total of broker and client equity must amount to only 27.7% of the value of all leveraged portfolios. These figures contradict the well-known legal constraint (e.g. U.S. Regulation-T) on retail margin debt: To avoid this logical contradiction, we must admit the possibility that the banks and financial institutions that lend call money to stock brokers in the United States may be earning substantial arbitrage profits on the spread over the risk-free rate.
Note well that varying the term of the call loan is of no great help in resolving the puzzle; indeed, Figure 15 plots the implied maturities T that would rationalize different values R of the broker call rate, assuming the parameters r := 2.088%, σ := 40%, and d/S 0 := 50%. For the currently observed call rate of 4.25%, we get an implied loan term of 1.75 years and an implied delta in the amount of 6.6% of the retail client's portfolio. broker call rate will revert to the mean at an expected rate of 40.3 basis points per month, but this reversion is disturbed by monthly innovations whose root-meansquared magnitude is 2.362%. Buoyed by the fact that Bankrate.com reports the two most recent observations of the broker call money rate (4.25% as of this writing), we constructed an AR(2) model that reduced the monthly root-mean-squared prediction error (in-sample) by 6.5 basis points, to 2.297%.

Summary and Conclusions
We proceeded to reconcile this empirical law of motion with following theoretical relationship (Garivaltis 2019a), based on instantaneous monopoly pricing of margin loans to Kelly gamblers: Margin Loan Interest Rate t = 1 2 (Broker Call Rate t ) + 1 2 (ν S − σ 2 S /2), where ν S denotes the long-run compound annual (logarithmic) growth rate of the stock market, and σ S is its annual volatility. Under this arrangement, only half of the random movements in the broker call rate get passed on to retail consumers. Assuming Thus, the margin loan interest rate will display the same rate of (continuous) meanreversion as does the broker call rate; the unanticipated instantaneous changes in the margin rate (= 1.495 dW t ) will be half the size of the corresponding movements in the broker call rate. We then derived a stochastic differential equation that governs the (monthly) leverage ratios (b t ) of continuous time Kelly gamblers: Hence, our empirical finding is that the long-term average interest rate on margin loans should be 5.9%, and that the leverage ratios of sophisticated brokerage clients should oscillate randomly about an equilibrium level of 2.03 : 1.
Finally, we used Merton's (1974) no-arbitrage method to uniquely characterize the correct risk premium ρ := R − r that commercial banks should earn on their loans to stock brokers. We assumed that brokers loan money to retail clients at a marked-up rate of R > R; to generate risk premia in the market for call money, we had to assume that stock brokers are not willing or able to short their customers' portfolios for the sake of hedging the default risk.
Thus, we modeled a situation whereby commercial banks are exposed to the risk of a cascaded default, meaning that the retail client defaults on his margin loan and the brokerage in turn defaults on its debt to the money market. The commercial bank can hedge this risk by shorting the dynamically precise fraction of the retail client's portfolio at time t, where T is the maturity date of the call loan, ρ is the risk premium for call money, d/S t is the percentage of the client's portfolio that is financed with call money (as opposed to broker equity and client equity), and σ is the annual volatility of the collateral.
Under very conservative assumptions (40% annual volatility and a 90-day loan term), we concluded that call lenders' current level of exposure to the stock market amounts to ∆ = 4.4% of the value of all leveraged portfolios in the United States.
Comparing the current broker call rate of 4.25% with the prevailing U.S. Treasury yields, we found that the implied loan-to-value ratio is north of 70%. This is absurd on account of U.S. Regulation-T, which caps the loan-to-value ratios of retail margin borrowers at 50%. In order to alleviate this apparent contradiction, we must live with the possibility that U.S. banks who deal in the market for call money could in fact be earning substantial arbitrage profits on the spread of the broker call rate over the risk-free rate.
Northern Illinois University