Why Leverage Can Become More Expensive at the Margin

The specific question here is whether a credit surface merely rises with leverage or becomes progressively steeper as leverage grows. That is the question of convexity: whether the next increase in loan-to-value costs more than the previous one. Understanding it gives readers a concrete way to judge why high-leverage borrowers can face accelerating pressure, while also showing why the answer depends on the distribution of collateral outcomes and on whether leverage is measured by the loan or by the promised repayment.

Convexity is a marginal claim

A rising credit surface says that more leverage costs more. A convex credit surface says something stronger: the cost of the next increment of leverage is larger than the cost of the preceding increment. The first statement concerns the direction of the spread. The second concerns the way that direction changes.

Imagine moving along the loan-to-value axis in equal steps. On a gently rising surface, each step might add roughly the same amount to the spread. On a convex surface, the first step could be manageable, the second more expensive, and a later step substantially more expensive. The borrower is not just paying for a larger loan. They are moving into a part of the collateral distribution where the lender’s chance of receiving the promised amount is deteriorating more rapidly.

That interpretation follows from the pricing mechanism. The loan value associated with a promise is built from state-by-state collateral-limited payoffs. As the promise rises, the marginal increase in loan value is the probability that the promise remains solvent. If the distribution of collateral values causes that solvency probability to decline in a sufficiently regular way, the spread can rise at an increasing rate when the loan value grows.

Convexity therefore describes an acceleration in marginal credit pressure. It does not mean that all borrowers are charged the same premium, nor does it mean that a market has stopped lending at a particular threshold. It says that moving farther along the leverage dimension changes the price relationship more sharply. The distinction is important for credit supply: a market can continue to offer loans while making the high-leverage end increasingly unattractive.

The Federal Reserve paper gives a hierarchy of sufficient conditions for convexity of the loan credit surface, the surface indexed by loan value or LTV. The strongest and most familiar condition is log-concavity of the collateral-value density. A density is log-concave when its logarithm bends downward; economically, this rules out some forms of excessively irregular tail behavior. Log-concavity implies an increasing promise hazard rate, meaning that the rate at which promises become insolvent does not fall as the promise grows.

A weaker condition is increasing promise hazard rate, abbreviated IPHR. A still weaker condition is increasing loan hazard rate, or ILHR, which measures how the default exposure changes when the amount of loan value raised increases. The paper also gives direct restrictions on how quickly the density may decline. These conditions form a hierarchy: the later conditions impose less structure while still guaranteeing the convexity needed for the loan surface.

The point of the hierarchy is not to give readers a checklist of technical tests.