# Cashing a blank check

January 26, 2021. Suppose you find a blank check on the ground, and unscrupulously decide to cash it in. If overdrawing gets you nothing, how much should you cash it in for? Assuming wealth follows the 80-20 rule, the answer is: almost nothing!

#### Introduction

In the film “Blank Check” (1994), 11-year old Preston Waters is handed a blank check, and cashes it in for a million dollars. Luckily, this is precisely the amount of money that the check’s signer, a convict attempting to launder his ill-gotten gains, has left with the bank’s president. But what if Preston overdrew, asking for, say, $10$ billion? This would probably have raised the suspicions of the complicit bank president and the check would have bounced altogether. When I was a kid, I thought it was incredibly lucky for Preston to find the check in the first place. I now think drawing the precise amount of money held in trust is infinitely luckier. But this raises the question: if you find a blank check, and you don’t want it to bounce, how much should cash it in for?

#### Expected return

I’ll assume we know nothing about the identity of the signee, and that if they have a balance of $b$, and we make out the value of the check to be $v$, then the check will bounce if $v > b$. Our strategy will be to calculate the expected return for $v$ and then maximise it. If $f(b)$ is the probability distribution for bank balances, then the expected return for $v$ is simply $v$ multiplied by the probability $b> v$:

$E(v) = v \int_v^\infty f(b) \, db = v[1 - F(v)] = v \bar{F}(v),$

where $F$ is the cumulative distribution function, and the $\bar{F} = 1 -F$ the tail. To maximise this, we assume the curve is smooth, differentiate and set to $0$, using $\bar{F}’ = -f$:

$E'(v) = \bar{F} - vf(v) = 0 \quad \Longrightarrow \quad v = \frac{\bar{F}(v)}{f(v)}.$

Any $v$ which satisfies this equation is an extremum.

#### Long and short tails

Now the question is how to model the distribution of bank balances. This is the sort of thing expected to follow a power-law curve like the Pareto distribution, the proverbial “80-20” curve. This is simply defined by its power-law tails:

$\bar{F}(v) = \left(\frac{L}{v}\right)^\alpha,$

where $L$ is the minimum amount to keep a bank balance open (say a monthly fee), and $\alpha > 0$ is a shape parameter we will “leave blank” for the moment. This is well-defined since it heads to zero. The probability density for $v \geq L$ is

$f(v) = -\bar{F}'(v) = \frac{\alpha L^\alpha}{v^{\alpha + 1}}.$

The optimal draw then obeys

$v = \frac{\bar{F}(v)}{f(v)} = \left(\frac{L}{v}\right)^\alpha \cdot \frac{v^{\alpha + 1}}{\alpha L} = \alpha v.$

For $\alpha \neq 1$, the only solutions are $v = 0$ and $v = \infty$! For $\alpha > 1$, we can plot the expected return $E(v)\propto v^{1-\alpha}$, and see that it monotonically decreases, with the maximum at $v = L$. Preston should only have asked for a few bucks! But perhaps this is an artefact of the infinite power-law tail. A more realistic choice is the truncated Pareto distribution, where the power law is confined to $L \leq v \leq H$ for an upper limit $H$, say the personal wealth of Jeff Bezos or Elon Musk. The density for the truncated Pareto distribution is simply a conditional probability, conditioned on being in the interval $[L, H]$:

$f(v) = \frac{\alpha L^{\alpha}v^{-(\alpha+1)}}{1 - (L/H)^\alpha},$

and the tail is

$\bar{F}(v) = \int_v^H \frac{\alpha L^{\alpha}v^{-(\alpha+1)}}{1 - (L/H)^\alpha} dv = \frac{(L/v)^\alpha - (L/H)^\alpha}{1 - (L/H)^\alpha}.$

Thus, we now have to solve

$v = \frac{\bar{F}(v)}{f(v)} = \frac{(L/v)^\alpha - (L/H)^\alpha}{\alpha L^{\alpha}v^{-(\alpha+1)}} \quad \Longrightarrow \quad v = (1-\alpha)^{1/\alpha} H.$

If $\alpha < 1$, then we do get a finite answer, proportional to the upper bound, so for instance if $\alpha = 0.5$, and we take the upper limit to be around 100 billion dollars, then Preston should ask for

$v \sim \sqrt{1-0.5} \times 10^{11} \approx 70 \text{ billion dollars},$

or $0.7$ of some other reasonable guess for $H$. But if $\alpha \geq 1$, the prefactor is not real, and as for the full Pareto distribution, the maximum expected return occurs at $L$. And indeed, wealth typically does obey an approximate Pareto distribution with $\alpha > 1$. For instance, the proverbial “80-20” rule corresponds to $\alpha \approx 1.16$, and this analysis of the Forbes 400 richest people in the world finds a shape parameter of $\alpha = 1.49$. So once again, a perfectly rational Preston Waters would ask only for the monthly fee! But this would make for a far less entertaining movie.

Written on January 26, 2021