In the mortgage banking industry, property owners who owe more than their properties are worth are described as being underwater. If the value of a real estate asset is less than the associated mortgage debt, when owners sell their properties they are left owing money on the mortgage.

Similarly, after the Nasdaq crash, many employee stock options were referred to as being underwater because the stock price had fallen below the exercise price, leaving employees with no incentive to exercise their options.

Mortgaged real estate and employee call options are both leveraged investments, which can be a powerful tool for building wealth when asset prices rise, but in a falling market, an investor's equity can be quickly wiped out. Despite this, these investments still have some value based on the possibility that asset prices can recover from their previous losses.

In this article, we'll follow a leveraged investment as it builds equity, loses value, goes underwater and then recovers. An understanding of this process is critical for investors who use mortgage debt, margin debt, long-term call options, or other similar financial derivatives to build wealth.

Equity Formation
Leverage aims to use borrowed money to build equity by investing it at a higher rate. For example, if we can borrow \$10,000 at 5% and invest it at 10%, we can make the difference between the investment gains and the interest, or \$500, as long as the opportunity lasts.

Einstein once quipped that the most powerful force in the universe is compound interest. Large profits are made when the assets in a leveraged investment compound at a higher rate than the debt over a long period of time. The above investment compounded for 10 years will generate \$9,648.

 Assets \$10,000 x (1+10%)^10 = \$10,000 x 2.5937 = \$25,937 Debt \$10,000 x (1+5%)^10 = \$10,000 x 1.6289 = \$16,289 Equity \$25,937 - \$16,289 = \$9,648

The compounding investment gains provide an extra \$5,937 (\$25, 937 - \$20,000) during the 10-year life of the investment and compound interest adds an additional \$1,289 (16,289 - \$15,000) of debt.

Note that if the investor is able to pay the interest out of pocket over the life of the investment, he or she can prevent the interest from compounding and save money. For example, this might mean regularly paying the interest on a margin balance.

The Leverage Ratio and Volatility
Leveraged investments have a starting equity and a specific leverage ratio based on the amount of equity compared to assets. For example, \$1,000 of an investor's equity could be supplemented with \$2,000 of borrowed capital to create an investment with three times the leverage.

The leverage ratio is useful shorthand for calculating percentage changes in equity based on a percentage change in assets. For example, if our underlying investment gains 10%, the equity in our three-times leveraged investment should increase by 30%. However, the leverage ratio doesn't factor in the cost of debt and isn't necessarily accurate for long time periods.

The volatility of the underlying assets can be multiplied by the leverage ratio to find the volatility of the equity. For example, a three-times leveraged investment will have three times the volatility of the same unleveraged investment.

Increased volatility is what pushes leveraged investments underwater. Every volatile investment has a chance to lose value, and when the volatility increases, scenarios that reduce or wipe out an investor's equity become much more likely.

Example: Negative Compounding
Suppose that we make a three-times leveraged \$1,000 investment in an index fund with roughly 10% annual returns and 15% annual volatility. The interest rate on borrowed money is 5%.

On average, we would to gain \$200 in the first year ((\$3,000 x 10%) - (\$2,000 x 5%)), based on \$300 worth of capital gains minus \$100 in interest. However, an extremely wide range of returns is possible for our investment. We can predict possible scenarios using a standard distribution of returns based on statistical probabilities.

For example, 68% of the time we would expect our asset returns to be within one standard deviation of 10%, or between -5% to +25%. That leads to a range of equity returns of -25% to +65%. That's a lot of unpredictability. Also, 34% of the time the return on our investment would be outside that range, potentially returning +110% or -70% in the first year.

 - % Asset Returns % Equity Returns Expected Result +10% +20% 68% of results (1 std dev) -5% to +25% -25% to +65% 95% of results (2 std dev) -20% to +40% -70% to +110% 99.7% of results (3 std dev) -35% to +55% -115% to +155%

While it is still unlikely that all of our equity could be wiped out in the first year, it is easy to see how it could happen after a few years of poor investment returns. Given the above probabilities and using Monte Carlo methods, we can calculate that our investment would be underwater about 4% of the time after five years, and 3% of the time after 20 years.

Let's say that our fund loses 20% in the first year, which results in a 70% loss of equity. Of this loss, 60% is due to the 20% fall multiplied by three-times leverage, and 10% is a result of interest payments, although in this example we'll accumulate our interest.

 - Starting Point After 20% Drop Assets \$3,000 \$2,400 (fell \$600) Debt \$2,000 \$2,100 (rose \$100) Equity \$1,000 \$300 Leverage Ratio \$3,000/\$1,000 = 3 times \$2,400/\$300 = 8 times Expected Asset Gains \$3,000 x 10% = \$300 \$2,400 x 10% = \$240 Interest Due \$2,000 x 5% = \$100 \$2,100 x 5% = \$105 Expected Profit \$300 - \$100 = \$200 \$240 - \$105 = \$135 Expected ROE \$200/\$1000 = +20% \$135/\$300 = +45% Volatility of Equity 15% x 3 = 45% 15% x 8 = 120%

After the 20% loss, our leverage ratio increases from three-times to eight-times because we now have less equity in the investment compared to total assets. This sharply increases our expected return on equity and our expected volatility.

Even after this loss our investment could still recover, but given the razor-thin equity, it is also very possible that it could be pushed underwater. Although the probabilities slightly favor recovery, either scenario is likely,

Recovery
The gradual compounding of our investment gains is the tool that creates equity in a long-term leveraged investment. However, the volatility of the risky asset also has the potential to negatively compound the investment. Situations in which the asset is compounding at a negative rate but the debt is compounding at a positive rate lead to underwater investments.

However, in every time period our assets are still more likely to compound positively than negatively. This makes it statistically likely that our leveraged investment will recover, given time. This is true even when the expected return is low or negative, because as the leveraged investment accumulates assets, the expected return will eventually turn positive and the losses will be erased.

Still, this could take a long time. In the above example, if the level of debt ever gets to the point where it is 50% higher than the level of assets, then the investment is at least 10 years away from profitability, on average, and future negative returns would make this recovery period even longer.

SEE: 5 Factors To Watch In A Housing Recovery

The Bottom Line
Even though leverage is often seen as a "get rich quick" tool for short-term speculators, it is clear that leveraged investments reward both patience and thoughtful diversification.

There is always reason for optimism in a leveraged investment. Whenever investment returns are expected to accumulate faster than interest payments, simply holding the investment for a long period of time should, on average, create large amounts of equity. This is still true even after years of heavy losses.

And although we've shown a situation in which a string of poor investment returns pushes a leveraged investment underwater, note that the reverse is also possible. A series of above-average returns can help a leveraged investment build vast amounts of equity very quickly.

When a leveraged investment gains a lot of equity, the appreciated assets continue to compound over time, but the leverage ratio falls, which makes the investment both more valuable and less volatile. This is a very good situation for an investor to be in, and the high returns generated can make up for several underwater investments.

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