# Mortgage Math and Increasing Asset Prices

The point of this little exercise is to illustrate how the decline of interest rates increases the value of long-term cash flows and inflate the value of assets associated with those cash flows.  It is an admittedly counter intuitive assertion that a decline in current interest rates increases the future value of cash flows. And it is rooted in the unfamiliar mathematical mechanics of division, namely dividing by fractions (decimals/percentages).  In a division problem, if the top number (numerator) is held constant and the bottom number (denominator) becomes smaller the product becomes larger. For example 10 divided by 0.10 (10%) equals 100, while 10 divided by 0.01 (1%) equals 1000, a ten-fold increase.

To illustrate how a decline in interest rates inflates asset values, let us look at the math of a common financial transaction, with which many of us are familiar, by turning to home prices and mortgages.  The table below shows a home valued at \$100,000 on which there is a 30-year mortgage at 6%, the monthly payment of which is \$600. At the end of 30 years, the homeowner will have paid the \$100,000 principal amount and \$115,000 in interest expense for a total stream of payments of \$215,000.  Now let us say interest rates fall in half (3%), the monthly payment falls to \$422, interest expense over the life of the loan is now \$51,000 producing a total stream of payments of \$151,000.  The difference between \$215,000 and \$151,000 is a significant \$64,000.  This \$64,000 does not simply evaporate– the market would not let that happen. This value instead is transferred to the value of the asset.  The house price can appreciate to a level of \$140,000 at which the buyers can still receive a mortgage that is within their monthly budget.  At the \$140,000 price the interest expense is \$72,488 producing a total value of \$212,488, roughly the same amount at the 6% interest rate (the invisible hand of the market).

These same dynamics apply to other asset classes such as the stock market.  Although the terms differ, instead of interest rates we talk of discount rates and net present values of future cash flows, the net effect is the same.  The cash flows are divided by a small number producing a larger value and just like the above illustration the asset value increases, but in this case it is a stock rather than a house price.

This is not meant to be a course in corporate finance, but to illustrate the impact of falling interest rates and how they inflate asset values.  For additional illustrative purposes, if we look at five years of cash flows (\$1 million per year) with a discount rate of 6% and 3%, we will see that the value of the cumulative cash flows increase by \$357,000 (8.7%).  The magic of math increases the value of the cash flows and by extension the asset producing those cash flows by 8.7%. This gives the reader a sense of why the value of stocks like Apple, as well as house prices, have increased so dramatically in 2020. The market has placed a premium on stocks that can grow their future cash flows.  In the table above the yearly cash flow is held constant – imagine if cash flows grow 20% each year.  The increase is even more dramatic.  The net present value of the five years of cash flow increase to \$6.7 million! The combination of growth and falling interest rates is a powerful propellant driving asset values higher.

Of course, there is another story here also.  When interest rates rise, this story flips on itself and higher rates bring down the value of assets.
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