Money and Banking

Balance sheets, leverage and loss absorption

Let $A$ denote total assets, $L$ liabilities and $E$ equity, measured on a consistent accounting or valuation basis. The balance-sheet identity and two simple capital measures are:

If an asset loss $X$ is recognised immediately and liabilities remain unchanged, $A\prime =A-X$ and $E\prime =E-X$. The percentage loss of equity is $X/E$, while the new equity ratio is $(E-X)/(A-X)$. Using the original asset denominator would produce a different measure. An institution with no additional loss absorbers exhausts its equity when $X=E$; it is therefore wiped out by an asset loss fraction $E/A$. The regulatory leverage ratio uses a specific exposure measure and Tier 1 capital, so it is not simply the reciprocal of accounting leverage.

Profitability, margins and operating efficiency

Let $\Pi$ be net income for a period and let bars indicate average balance-sheet stocks over that same period. Return on assets and return on equity satisfy:

The last equality requires the same net-income numerator and compatible average stocks. Higher leverage increases the sensitivity of shareholders' returns to profits and losses; it does not by itself improve the profitability of the underlying assets.

For interest income $II$ and interest expense $IE$, net interest income is $NII=II-IE$. A conventional net interest margin is $NIM=NII/\overline{IEA}$, where $\overline{IEA}$ denotes average interest-earning assets. Some datasets instead divide by average total assets. The denominator must always be stated. A loan–deposit spread, $I{I}_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{n}\mathrm{s}}/\bar{Q}-I{E}_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{s}}/\bar{D}$, is a difference between two yields with different denominators; it is not automatically equal to NIM. Here $\bar{Q}$ is average lending and $\bar{D}$ average deposits. The cost-to-income ratio is operating expenses divided by operating income under the stated accounting definition.

Present value, duration and interest-rate shocks

For deterministic cash flows $C{F}_t$ discounted at a constant per-period yield $y$, the price is $P={\sum }_{t=1}^{T}C{F}_t/{(1+y)}^t$. Macaulay duration measures the present-value-weighted payment date, while modified duration converts it into first-order price sensitivity:

This convention assumes one compounding period per unit of $t$. Use a duration measure consistent with the yield and compounding convention. Enter a 150-basis-point yield change as $0.015$, not $1.5$. The approximation ignores convexity, changes in cash flows and nonparallel yield-curve movements. For material shocks, reprice the actual cash flows or add an appropriate convexity adjustment.

Expected credit loss and loan pricing

For one exposure and a specified horizon, probability of default $PD$, loss given default $LGD$ and exposure at default $EAD$ give the basic expected-loss calculation:

The recovery rate must use the same claim basis as EAD and should reflect the assumed timing and costs of recovery. This elementary product is not a complete accounting expected-credit-loss model: discounted cash-flow shortfalls, changes in exposure and multiple economic scenarios can matter.

A simple pricing decomposition adds funding cost, operating cost, expected loss and a target margin. An exact one-period break-even coupon requires explicit payoff assumptions. If a loan of principal $Q$ pays $(1+r)Q$ when it performs and recovers $\rho Q$ in default, with default probability $p$, funding rate $f$ and end-of-period costs $cQ$, then:

The denominator $1-p$ matters because the coupon is not collected in default. Do not substitute this formula into an exercise that assumes a different recovery basis or simply defines an additive pricing convention. Expected profit, accounting provisions and unexpected-loss capital also measure different things.

Borrower participation and adverse selection

In a simple limited-liability model, a borrower receives principal $Q$ and its project pays $Y$ with probability $q$, otherwise zero. If successful-project repayment is $(1+r)Q$, the borrower's expected payoff is:

With a zero outside option and strict positive participation, $r<Y/Q-1$. At equality the borrower is indifferent; whether it participates must be specified. If the types that remain in the market change when the rate passes a threshold, expected repayment is piecewise. A higher promised interest rate can then lower expected bank revenue by driving safer borrowers out of the pool.

Securitisation and tranche losses

Let total pool loss $x$, tranche attachment point $a$ and detachment point $d$ use the same units, with $0\le a<d$. The amount of loss allocated to the tranche and the fraction of its principal lost are:

For a tranche covering pool losses from 5% to 15%, a 9% pool loss consumes four percentage points of pool principal, equal to 40% of that tranche. It is not a 4% loss of the tranche itself. This static waterfall ignores the timing of interest and principal, fees, triggers and recoveries; a transaction model must follow its actual contractual rules.

Liquidity insurance and expected utility

Suppose a fraction $\lambda$ of savers needs early consumption $c_1$ and the remaining fraction consumes $c_2$ later. If one unit retained until the later date yields $R$ and early liquidation yields one, the basic resource constraint per initial saver is:

where $W$ is initial wealth. When $u_1\left(c\right)=\sqrt{c}$ and $u_2\left(c\right)=\alpha \sqrt{c}$, the interior optimum satisfies $\sqrt{c_2}=\alpha R\sqrt{c_1}$, or $c_2={\alpha }^2{R}^2{c}_1$, together with the binding resource constraint. Boundary and incentive constraints must also be checked. An allocation can improve on autarky without being the exact optimum.

For a risky withdrawal payoff $X$, expected utility is $E\left[u\left(X\right)\right]={\sum }_s{p}_{s}u\left(X_s\right)$. With concave utility it generally differs from $u\left(E\left[X\right]\right)$. In particular, the late-consumer utility from a run lottery is $\alpha {\sum }_s{p}_s\sqrt{X_s}$, not $\alpha \sqrt{{\sum }_s{p}_s{X}_s}$. This distinction is essential when comparing withdrawal with waiting.

Run capacity, insurance and liquidation

If liquid resources total $H$ and each early claimant is promised $c_1$, the number paid in full is $\lfloor H/c_1\rfloor$, with residual resources $H-c_1\lfloor H/c_1\rfloor$. The residual claimant's payment must be included when calculating expected utility under a random queue. A guarantee of principal $W$ does not necessarily eliminate an incentive to withdraw when the immediate contractual payment is greater than $W$.

If a redemption needs cash $C$, immediately available liquid assets are $H$, and securities sell at haircut $h$ relative to book value, the book value that must be sold is:

These expressions assume enough saleable assets exist and that the haircut is constant. The loss is larger than simply multiplying the cash shortfall by $h$, because the sale must raise the full shortfall after the discount.

Capital and liquidity regulation

Under a simplified standardised credit-risk calculation, risk-weighted assets are $RWA={\sum }_i{w}_{i}EA{D}_i$, where $w_i$ is the applicable risk weight. A capital ratio is the specified eligible capital divided by RWA, for example $CET1/RWA$. Actual regulatory RWA can also include market risk, operational risk and adjustments. The unweighted leverage ratio divides eligible Tier 1 capital by the regulatory exposure measure, which can include off-balance-sheet items.

Two complementary liquidity measures are:

Eligibility, valuation haircuts, inflow caps and funding weights are part of the definitions. The ratios should not be calculated by inserting arbitrary balance-sheet categories. Capital absorbs losses; liquidity resources meet payments. Passing one ratio does not prove that all solvency, funding or concentration risks are controlled. The regulation chapter explains the applicable standards and historical examples.

Money stocks, payments and the money multiplier

In a stylised system, let currency held by the public be $C$, deposits be $D$ and bank reserves be $R$. Broad money in this simplified definition is $M=C+D$; the monetary base is $B=C+R$. Writing $c=C/D$ and $r=R/D$ yields the accounting relationship:

This identity is not a mechanical causal rule by which central-bank reserves determine bank lending. Currency preferences, reserve holdings, loan demand, funding costs and monetary-policy implementation all matter. Nor does this stylised $M$ reproduce every official monetary aggregate. Loan origination creates a loan asset and a matching deposit liability; repayment of principal destroys those two claims. Payment to another bank changes the distribution of deposits and reserves across banks rather than necessarily changing their system-wide totals.

Stablecoin backing and payment business economics

A simple stablecoin backing ratio is market value of reserve assets divided by redeemable token liabilities, measured at the same date and on the same valuation basis. A ratio of at least one does not by itself guarantee immediate redemption: custody, legal priority, settlement, liquidity and operational arrangements remain essential. A discount to the reference unit is $(P-P^{*})/P^{*}$, where $P$ is the observed token price and $P^{*}$ the promised reference value.

For a payment business processing $N$ transactions, earning average revenue $v$ and incurring variable cost $c$ per transaction, contribution before fixed costs is $N(v-c)$. With fixed cost $F$ and positive unit contribution, operating break-even volume is $N^{*}=F/(v-c)$. Transaction counts, payment values and fees as a percentage of value are distinct units; a model that mixes them will not reconcile. Customer balances held in a closed-loop system are also not automatically accounting revenue or profit.