This piece is the second in a three-part series on investment leverage (see the first article in the series here).   It is a longer learning article and as such is meant to have an impact on your day to day investing knowledge and behavior.  In our first part of this series, I talked about defining leverage, why investors use it and how to think about leverage in a series of financial transaction examples.  This week I am moving on to how to measure leverage.  There is no one accepted way to do this.  Traders often use option “greeks” which I will touch on below.  But often those measures don’t do a good job of giving the leverage a real “value” in the investment in question.  How much “oomph” do I get from applying leverage here in different price return scenarios?  Is it “worth the risk” to apply leverage to this position?  We will tackle these kinds of questions below.  Next week we will talk about leverage in a portfolio context.  Onward!

### Simple Ways of Measuring Option Investment Leverage

There are several single-point, easily calculable numbers to measure option-based investment leverage. There are uses for these simple measures of leverage, but unfortunately, for reasons I will discuss, the simple numbers are not enough to help an investor intelligently manage a portfolio containing option positions.

The two simple measures are lambda and notional exposure. Both are explained in the following sections.

#### Lambda

The standard measure investors use to determine the leverage in an option position is one called lambda. Lambda—sometimes known as percent delta—is a derivative of the delta factor and is found using the following equation: Let’s look at an actual example. The other day, I bought a deep in-the-money (ITM) long-tenor call option struck at \$20 when the stock was trading at \$30.50. The delta of the option at that time was 0.8707, and the price was \$11. The leverage in my option position was calculated as follows: What this figure of 2.4 is telling us is that when I bought that option, if the price of the underlying moved by 1 percent, the value of my position would move by about 2.4 percent. This is not a hard and fast number—a change in price of either the stock or the option (as a result of a change in volatility or time value or whatever) will change the delta, and the lambda will change based on those things.

Because investment leverage comes about by changing the amount of your own capital that is at risk vis-à-vis the total size of the investment, you can imagine that “moneyness” has a large influence on lambda. Let’s take a look at how investment leverage changes for in-the-money (ITM), at-the-money (ATM), and out-of-the-money (OTM) options. The stock underlying the following options was trading at \$31.25 when these data were taken, so I’m showing the \$29 and \$32 strikes as ATM: When an option is deep ITM, as in the case of the \$20-strike call, we are making a significant expenditure of our own capital compared with the size of the investment. Buying a call option struck at \$20, we are—as explained in the preceding section—effectively borrowing an amount equal to the \$20 strike price. In addition to this, we are spending \$11.50 in premium. Of this amount, \$11.25 is intrinsic value, and \$0.25 is time value. We can look at the time value portion as the prepaid interestwe discussed in the preceding section, and we can even calculate the interest rate implied by this price (this option had 189 days left before expiration, implying an annual interest charge of 2.4 percent, for example). This prepaid interest can be offset partially or fully by profit realized on the position, but it can never be recaptured so must be considered a sunk cost. Time value always decays independent of the price changes of the underlying, so although an upward movement in the stock will offset the money spent on time value, the amount spent on time value is never recoverable.

The remaining \$11.25 of the premium paid for a \$20-strike call option is intrinsic value. Buying intrinsic value means that we are exposing our own capital to the risk of an unrealized loss if the stock falls below \$31.25. Lambda is directly related to the amount of capital we are exposing to an unrealized loss versus the size of the “loan” from the option, so because we are risking \$11.25 of our own capital and borrowing \$20 with the option (a high capital-to-loan proportion), our investment leverage measured by lambda is a relatively low 2.50.

Now direct your attention to a far OTM call option—the one struck at \$39. If we invest in the \$39-strike options, we are again effectively taking out a \$39 contingent loan to buy the shares. Again, we take the time-value portion of the option’s price—in this case the entire premium of \$1.28—to be the prepaid interest (an implied annualized rate of 6.3 percent) and note that we are exposing none of our own capital to the risk of an unrealized loss. Because we are subjecting none of our own capital in this investment and taking out a large loan, our investment leverage soars to a very high value of 15.63. This implies that a 1 percentage point move in the underlying stock will boost our investment return by over 15 percent!

Obviously, these calculations tell us that our investment returns are going to be much more volatile for small changes in the underlying’s price when buying far OTM options than when buying far ITM options. This is fine information for someone interested in more speculative strategies—if a speculator has the sense that a stock will rise quickly, he or she could, rather than buying the stock, buy OTM options, and if the stock went up fast enough and soon enough offset any drop of implied volatility and time decay, he or she would pocket a nice, highly levered profit.

However, there are several factors that limit the usefulness of lambda. First, because delta is not a constant, the leverage factor does not stay put as the stock moves around. For someone who intends to hold a position for a longer time, then, lambda provides little information regarding how the position will perform over their investment horizon.

In addition, reading the preceding descriptions of lambda, it is obvious that this measure deals exclusively with the percentage change in the option’s value. Although everyone (especially fly-by-night investment newsletter editors) likes to tout their percentage returns, we know from our earlier investigations of leverage that percentage returns are only part of the story of successful investing. Let’s see why using the three investments I mentioned earlier—an ITM call struck at \$20, an OTM call struck at \$39, and a long stock position at \$31.

I believe that there is a good chance that this stock is worth north of \$40—in the \$43 range, to be precise (my worst-case valuation was \$30, and my best-case valuation was in the mid-\$50 range). If I am right, and if this stock hits the \$43 mark just as my options expire, what do I stand to gain from each of these investments?

Let’s take a look. This table means that in the case of the \$20-strike call, we spent \$11.50 to win gross proceeds of \$23.00 (= \$43 – \$20) and a profit net of investment of \$11.50. Netting \$11.50 on an \$11.50 investment generates a percentage profit of 100 percent.

Looking at this chart, the first thing you are liable to notice is the “Percent Profit” column. That 2,122 percent return looks like you might see advertised on an option tout service, doesn’t it? Yes, that percentage return is wonderful, until you realize that the absolute value of your dollar winnings will not allow you to buy a latte at Starbuck’s. Likewise, the 100 percent return on the \$20-strike options looks heads and shoulders better than the measly 38 percent on the shares, until you again realize that the latter is still giving you more money by a quarter.

Recall the definition of leverage as a way of “boosting investment returns calculated as a percentage,” and recall that in my previous discussion of financial leverage, I mentioned that the absolute dollar value is always highest in the unlevered case. The fact is that many people get excited about stratospheric percentage returns, but stratospheric percentage returns only matter if a significant chunk of your portfolio is exposed to those returns!

Lambda is a good measure to show how sensitive percentage returns are to a move in the stock price, but it is useless when trying to understanding what the portfolio effects of those returns will be on an absolute basis.

#### Notional Exposure

Look back at the preceding table. Let’s say that we wanted to make lambda more useful in understanding portfolio effects by seeing how many contracts we would need to buy to match the absolute return of the underlying stock. Because our expected dollar return of one of the \$39-strike calls only makes up about a third of the absolute return of the straight stock investment (\$3.82 / \$11.75 = 32.5% ≈ 1/3), it follows that if we wanted to make the same dollar return by investing in these call options that we expect to make by buying the shares, we would have to buy three of the call options for every share we wanted to buy. Recalling that options are transacted in contract sizes of 100 shares, we know that if we were willing to buy 100 shares of Oracle’s stock, we would have to buy options implying control over 300 shares to generate the same absolute profit for our portfolio.

I call this implied control figure notional exposure. Continuing with the \$39-strike example, we can see that the measure of our leverage on the basis of notional exposure is 3:1. The value of the notional exposure is calculated by multiplying it by the strike; in this case, the notional exposure of 300 shares multiplied by the strike price of \$39 gives a notional value for the contracts of \$11,700. This value is called the notional amount of the option position.

Some people calculate a leverage figure by dividing the notional amount by the total cost of the options. In our example, we would pay \$18 per contract for three contracts, so leverage measured in this way would work out to be 217(= \$11,700 ÷ \$54). I actually do not believe this last measure of leverage to be very helpful, but notional control will become important when we talk about the leverage of short-call spreads later in this series.

These simple methods of measuring leverage have their place in analyzing option investment strategies, but in order to really master leverage, we must move to understanding leverage in the context of portfolio management.