First-Principles Derivation of A Bank

What is a Bank?

A bank borrows money from people with money but low-risk appetite, and lends money to people who need money to take potentially rewarding risks.

Say we live in a toy society with $10 in circulation and 2 people (a farmer and a cook). In this society, the farmer grows vegetables and the cook processes vegetables for food – both activities essential for survival. Assume that both parties start with $5 each. The farmer sells vegetables for $1 per serving and can produce 2 servings per day. The cook prepares food for $2 per serving and also has the capacity to produce 2 servings per day. Every day, the farmer sells 2 servings of vegetables to the cook for a total of $2. Then, the cook prepares 2 servings of food and sells 1 serving to the farmer for $2. In this perfectly balanced society, everybody eats, and money never needs to “rest.”

Let's break the above equilibrium by adding a time component. In this society, vegetables take three days to fully grow. The farmer purchases meals every day for $2, but he can only sell his 6 servings of vegetables on the third day. The first day goes by fine: the farmer spends $2 on food, leaving him with $3 in savings. The second day, he again pays $2 and is left with just $1. On the third day, he falls $1 short of the money needed to purchase food. He might ask the cook to lend him $1 so he can buy food. If the cook agrees, the farmer will eat, and the next day he will sell his 6 servings of vegetables at $1 per serving – earning a profit of $6, repaying the cook’s $1, and ending up back at $5 in his bank account so that the cycle can continue. However, if the cook—having a low-risk appetite—refuses to lend, the farmer (and eventually the cook) would starve. This is where a bank comes in: all parties deposit their money in exchange for an IOU. On the third day, the farmer can borrow $1 from the bank to eat and then repay it the next day. This is the primary function of a bank—taking funds from those who prefer safety and lending to those willing to take a risk.

Let's Make a Bank

Now that we know what a bank is, let’s build one. You file the paperwork, set up a branch, and open for business. Soon, numerous depositors arrive and deposit a total of $100. Excellent! A few moments later, your first borrower arrives, asking to borrow money collateralized by some assets he intends to buy. How much can you lend him?

Naively, you think: “I have $100 in deposits, so I can lend him $100. I’ll charge 5% simple interest so that within a year I’ll have a $5 profit.” Awesome. However, after lending him the money, a depositor shows up the next day asking to withdraw $1. You suddenly find yourself with $0 in reserves – not enough in cash to pay back depositors! So, you reverse time and try again.

This time, you consider that with $100 in deposits, your depositors are estimated to withdraw 5% of their funds within the year. To be safe, you keep a reserve ratio of 10%, meaning you have $90 available to lend. You charge 5% simple interest, so within a year you expect to make a $4.50 profit. Again, you lend the money and you actually require him to leave collateral worth 105% of the loan (to meet your 5% haircut requirement). Unfortunately, the market price of his collateral drops by 6%. Suddenly, after liquidating the collateral you’re left with $89 plus $10 in reserves, meaning you owe $100 to your depositors but have only $99 in assets. The depositors make a mad dash to withdraw their money, so you reverse time and try once more.

This time, you start with $100 in deposits and still keep a 10% reserve. Additionally, you raise $1 from each of your four friends who want to invest in the bank—and you add $1 yourself. These $5 become your tier 1 capital; funds available to satisfy depositor withdrawals in worst-case scenarios. If the bank turns a profit, these earnings are split five ways. If the bank takes a loss, however, you and your friends lose your $5 to make the depositors whole. This is essentially what happens when a bank issues stock. Feeling safe with an 8% tier 1 capital buffer, you decide to loan out $62.50 (since $5 buffer / $62.50 loan = 8%). But then, the collateral drops by 30% because it was Dogecoin! You liquidate the collateral as quickly as you can, recovering only $46.50, and even after using $5 of your tier 1 capital to repay depositors you’re left owing $100 while having just $89 in assets. You reverse time and try one last time.

Now, you have $100 in deposits, keep 10% in reserve, and have raised $5 in tier 1 capital. This time, you calculate a multiplier based on the riskiness of the loan—known as the risk weight. For a crypto-collateralized loan (where the borrower deposits crypto as collateral because they might not pay back), the risk weight is 5, reflecting crypto’s higher risk. For a corporate bond-collateralized loan (where corporate bonds serve as the backup), the risk weight is 1, because corporate bonds are less risky since the corporation might step in. For a US Treasury collateralized loan, the risk weight is 0, because US Treasuries are deemed very safe (the US government would cover defaults). You want to maintain an 8% tier 1 capital buffer. This means:

• For crypto-backed loans, you can hold up to $12.50 because: $5 tier 1 capital / (5 * $12.5 loans) = 8%.
• For corporate bonds, you can hold up to $62.50 because: $5 tier 1 capital / (1 * $62.5 loans) = 8%.
• For US Treasuries, you can hold virtually an infinite amount, since $5 tier 1 capital / (0 * $X loans) is not constraining.

Say you charge a 5% interest rate for loans collateralized by US Treasuries. You can lend out all $90 (after reserves), and make a profit of $4.50 in a year if you do not use any of your tier 1 capital. Now assume someone is willing to pay 9% for a crypto-backed loan and 6% for a corporate treasury-backed loan. You’d want to optimize the following equation:

maximize crypto_loans * 0.09 + corporate_loans * 0.06 +
        (deposits * (1 - reserve_ratio) - crypto_loans - corporate_loans) * 0.05

given:
  deposits         = $100
  reserve_ratio    = 0.1
  t1_capital       = $5
  crypto_loans     > 0
  corporate_loans  > 0
  crypto_loans + corporate_loans <= deposits * (1 - reserve_ratio)
  t1_capital / (crypto_loans * 5 + corporate_loans * 1) >= 0.08
  

As you add more loan types to this optimization problem, it becomes increasingly complex. Yet, this structure can reveal useful characteristics, such as deriving minimum interest rates for a product given a particular risk weight. In solving this simplified version of the maximization problem, you find that profits are maximized by lending $62.50 collateralized by corporate bonds, and the remaining $27.50 collateralized by US Treasuries. In fact, you discover that lending collateralized against crypto doesn’t become optimal until the crypto interest rate reaches 10%.

Risk Weights Feel Arbitrary and Inefficient. How Can I Get Paid to Correct Them?

Let’s suppose there is a person who believes that the 10% interest rate for crypto-backed assets is absurd. He would be willing to make loans backed by crypto for 9% interest. He holds $100 in 10-year US Treasuries that pay 6%. How can he earn extra?

The most direct solution is to create his own bank and set risk weights that he deems fair. However, risk weights are often determined by government regulators—and even banks may think the weights are too high, but they cannot lower them legally lest regulators become nervous about maintaining enough tier 1 capital for depositor protection.

Another solution might be to sell his $100 treasury portfolio to make crypto-backed loans at a 9% interest rate. Yet, that is not the most efficient use of capital.

A more capital-efficient approach is to borrow against his $100 treasury portfolio, using it as collateral. The bank then provides him with $95 in cash (incorporating a 5% haircut) at a 5% interest rate. (The bank is content with this arrangement because, with the haircut and shorter duration, the loan is safer than the 10-year US Treasury collateral backing it. Fully collateralized loans must be safer than the collateral itself.) He can then make crypto-backed loans at 9% interest and pocket the 3% difference. On a default-free, “happy path,” he makes an additional 2.7%—boosting his overall return on the treasury portfolio from 6% to 9.7%. In the event of a default, however, the bank might seize his collateral. Essentially, his service is to assume the risk differential between crypto-backed and treasury-backed loans.

As you can imagine, the bluntness of one-size-fits-all banking regulation creates a lot of holes for inefficiencies to arise. I'll be thinking about these inefficiencies, and will follow up with more blogs about it in the future!