book value of leverage

A simple analog is the leverage of a property bought on an (unsecured) commercial loan

Suppose the house was bought for $600k with $480k loan. After a few years, loan stays at $480 (to be paid off at maturity), but house doubles to $1.2m.

Book value of EQ is still 600-480 = $120k, but current EQ would be 1.2m - 480k = 720k.

The book value of leverage was and is still 600/120 = 5.0

The current value of leverage would be (1200k)/720k, which is lower and safer.

Now the bleak picture -- suppose AS value drops from 600k to 500k. Book leverage remains 600/120 = 5.0

Current value of leverage is 500/(500 - 480) = 25.0. Dangerously high leverage. Further decline in asset valuation would wipe out equity and the entire account is under water. Some say the property is under-water but i feel really we are talking about the borrower and owner of the property -- i call it the account.

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(Book value of) Leverage in "literature" is defined as

   (book value of) ASset / EQuity (book value)

Equivalently,

   (LIability + EQ) / EQ .... (all book values)

The denominator is much lower as book value than the current value. For a listed company, Current value of total equity is total market cap == current share price * total shares issued so far. In contrast, Book value is the initial capital of the founder + actual dollars raised through the IPO, ignoring the increase in value of each share. Why is this book value less useful? We need to understand the term "shareholder equity".  This term logically means the "value" of the shares held by the shareholders (say a private club of .... 500 teachers). Like the value of your house, this "value" increases over time.