Puzzle #2 asked how a $6 million mortgage pool with a weighted average LTV of 80% could have a total property value of $9 million when the average LTV implies a value of $7.5 million. Chun Lee NG gave a good general answer.

Puzzle #2: Follow-up Questions asked two more questions:

1. Suppose there are just two mortgages in the pool. Can you give a specific example of two loans that fits the description of the puzzle?

2. If weighted average LTV can be inconsistent with the ratio of total loans to total value, does this mean that it is a flawed measure of loan pool quality?

To answer the first follow-up question, you can set up the math and solve a quadratic equation if that is your idea of fun. See ltv-math for the gory details. Note that the weighted average LTV calculation requires the loan amounts to be squared. This explains how the weighted average LTV can be inconsistent with the simple ratio of total loans to total value.

The solutions to Puzzle #2 fall on an oval (I don’t think it’s a true ellipse):

One of the solutions is marked with a large dot. For the first loan, the marker corresponds to a loan amount of $4 million and a property value that is also $4 million (the characteristics of the first loan are indicated by the outer axis labels in red). The marker also shows that for the second loan the amount is $2 million and the value is $5 million (the second loan is described by the inner axis labels in blue). So the first LTV is 100% and the second is 40%. The weighted average LTV is (4 * 1 + 2 * 0.4) / 6 = 0.8.

As for as the second follow-up question, I think that it is important to understand the quirks of the weighted average LTV calculation, but I do not believe these necessarily make it a poor measure of portfolio quality. The total property value of a portfolio is less important than the individual LTVs. Beyond a certain point, a surplus of property value cannot improve the quality of an individual loan. The probability of loss cannot fall below 0%. And that surplus property value does nothing for the other loans in the portfolio, which must stand on their own.

**Other Posts on Puzzle #2:** Puzzle #2, Puzzle #2: Follow-up Questions

**Other Puzzles:** Puzzle #1, Puzzle #3, Puzzle#4

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