This article is supposed to illustrate using Cramer-Lundberg model from actuarial science and applying to personal finance. This model was invented in 1930s for insurance companies. Insurance companies never seem to go bankrupt yet ordinary people are often get financial struggles. While it's easy to blame insurance companies for cheating at the game and ripping us off (and there is certainly a lot of truth to that), perhaps we can also learn the game they're playing and apply their models to our personal finances.

Financial ruin is when your bank account goes to zero. At that point you're unable to pay for anything. The goal of ruin theory is to understand when ruin happens (and with what probability) and adopt behavior that reduces that probability as close to zero as possible.

This is the model everybody is familiar with. Every day you're earning money (e.g. from a job or your online shop) and every day you're spending money (e.g. food, groceries, rent, etc.). The money you earn is your income, while the money you spend are your expenses. Let's label them $W_+$ and $W_-$ respectively. Then your net profit $\Delta W$ is:

$$\Delta W = W_+ - W_-$$

Your total wealth over time $W(t)$ is then your initial wealth $W_0$ (i.e. the money you initially had in the bank) plus your net profit through time $\Delta W \cdot t$:

$$ W(t) = W_0 + \Delta W\cdot t$$

Your bank account can be visualized like this, where your income grows your account while your expenses take away from it:

If we let $t\rightarrow\infty$, the condition to avoid ruin is that on average you have to save more money than you spend it (i.e. $\mathbb E[\Delta W] > 0$). In this case, probability of ruin is $\Psi = 0$ i.e. you will never go bankrupt.

If on average you spend more money than you save (i.e. $\mathbb E [\Delta W] < 0$) then the probability of ruin $\Psi = 1$ i.e. bankrupcy is certain.

Despite the fact everybody internalizes this model on some level, the issue with this model is glaringly obvious. It doesn't account for unplanned expenses. People don't usually go bankrupt because they're irrational, but because something happens that.

Building on the previous model, your still have your income ($W_+$), expenses ($W_-$), net profit ($\Delta W$), initial capital ($W_0$) and your net worth ($W(t)$). We now also introduce unplanned expenses $X_k$. How do we characterize unplanned expenses if they're, well… unplanned? We'll have to make some assumptions still. We're going to assume they form Poisson point process distributed in time. This means:

1. Two unplanned expenses cannot happen at the same time (although they can get arbitrarily close)
2. Unplanned expenses are independent of each other.
3. They happen at a certain frequency $\lambda$, and it's less exponentially likely that two unplanned expenses happen closer or farther from each other than this frequency.

Your total wealth equation over time now looks like this: $$W(t) = W_0 + \Delta W \cdot t - \sum _{k=1}^{N(t)}X_k$$ This is the classic Cramer-Lundberg model of wealth. For simplicity sake, in the following sections we'll only look at single risk $X$ happening at frequency $\lambda$ e.g. \$1000 every 6 months, with deviations from that frequency being exponentially unlikely.

The wealth in your bank now looks like this where your income $W_+$ grows your account, while your planned expenses $W_-$ and unplanned expenses $X$ take away from it:

Both of these look like “expenses”, but they key is in their interpretation and behavior.

• Planned expenses ($W_-$) is something tied to your income. For example, if you're selling
• Unplanned expenses ($X$) are independent and happen at a frequency $\lambda$ and intensity $X$. They don't depend on your income, nor your current wealth, nor your other planner or unplanned expenses.

In order to calculate probability of ruin, it's useful to look at exponential function of wealth: $$M(t) = e^{RW(t)}$$

Setting the adjustment coefficient $R$ such that $M(t)$ is a martingale with zero drift, we get the equation: $$R\Delta W = \lambda(\mathbb e^{RX} - 1)$$

This equation has a solution if and only if the following inequality is satisfied: $$\Delta W > \lambda X$$ The trouble is that the equation is non-linear but we can approximate $R$ by using the expansion $e^x = 1 + x + \frac{x^2}{2} + o(x^2)$, in which case we get (eq. 1): $$R \approx\frac{2(\Delta W - \lambda X)}{\lambda X^2} $$

Note that adjustment coefficient $R$ depends on the following parameters: - Your income vs. expenses - i.e. your net profit $\Delta W$ - Your risk tolerance - i.e. frequency $\lambda$ and intensity $X$

If we take $t\rightarrow\infty$, the probability of ruin $\Psi$ is given by Lundberg inequality: $$\Psi < e^{RW_0}$$

Solving for $W_0$ and plugging in the approximation for the adjustment coefficient $R$ from the previous section:

$$W_0 > \frac{1}{R}\ln\frac{1}{\Psi} \approx \frac{\lambda X^2}{2(\Delta W - \lambda X^2)}\ln\frac{1}{\Psi}$$

This inequality includes four variables ($W_0, \lambda, X, \Psi$):

• Your income vs. expenses - i.e. your net profit $\Delta W$
• Your risk tolerance - i.e. frequency $\lambda$ and intensity $X$
• How much money you need to avoid ruin - i.e. your the size of your bank account $W_0$

The inequality answers the question: - “How much money $W_0$ do I need to have in the bank, assuming I'm saving $\Delta W$ every month and want to be able to absorb a risk of an unplanned expense happening at frequency $\lambda$ and size $X$, such that my probability of ruin is less than $\Psi$? (e.g. $\Psi = 10^{-6}$ or one in a million)?”

What we can see from this that probability of ruin exponentially decreases the more money you have in the bank. For example, to protect yourself from an unplanned expense \$1000 that might happen every 6 months ($\lambda = 1/6\text{mo}, X = \$1000$), having \$6000 in the bank is going to make you several orders of magnitutde safer than \$3000 in the bank.

But your bank account does not tell the full story. Your income vs. expense also matters. We saw the inequality $\Delta W> \lambda X$ which says you must always save at least to be able to cover unplanned expenses and violating this inequality makes your probability of ruin certain.

We also saw that if we want to keep the probability of ruin constant, the amount of money you need in the bank is inversely proportional to your income ($W_0 \sim\frac{1}{\Delta W}$) . Therefore, there is a balance to be had between $W_0$ and $\Delta W $ in order to keep the probabilty of ruin low.

This formalizes the common wisdom: - “Keep 3-6 months of living expenses” (condition on $W_0$) - “Live below your means” (condition on $\Delta W$)

The model assumes infinte time horizon $t\rightarrow\infty$ to calculate ruin. But common sense says it's unlikely you will find yourself bankrupt if you lived above your means only for a short period of time.

The main limitation is that it assumes unplanned expenses are exponentially distributed and independent. In real life we often encounter cascade risks or events that are not Poisson.