The Effect of Options on Portfolio Risk

The Effect of Options on Portfolio Risk

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his article discusses options investing within a portfolio from the perspective of risk.

We will use the concepts of total return and scenario analysis to help us to look past considering only the profit or loss on particular option trades. Rather than analyzing the dollars made or lost, we are interested in the return of the investment. This is the first step towards understanding the quality of an investment rather than just whether or not one made or lost money. We can use these ideas to measure the risk or diversifying benefits of a particular option investment as well as having a way to compare between alternatives.

We will consider only basic strategies and make some simplifying assumptions. This is done to keep calculations easy and provide some intuition, rather than getting caught up in a lot of mathematics that won't contribute to a basic understanding.

In this analysis, we will consider four types of investments; stock, cash, a call option and a put option.

As a reminder, a call option is a contract giving its owner the right to buy a share of stock at a pre-determined price. If the stock price increases, the call is worth more money, and if the stock does not appreciate by the options expiration, it becomes worthless.

A put option is a contract that gives someone the right to sell a share of stock at a pre-determined price. If the stock appreciates, the put declines in value and will eventually expire worthless. If the stock price declines, the put will increase in value.

Conversely, investors can take short positions in these types of contracts, just as they can short stock, with the expected results. If someone sells a call and the stock declines in value, they retain the premium, just as a put seller would if the price goes up. In the other cases, where the stock declines and an investor is short a put, or the stock goes up and you are short a call, the seller must make good on the contract.

As part of the analysis, we consider an investment over a set of scenarios. For our purposes we will consider how a portfolio will behave over a one month period under a variety of option related strategies.

As our baseline strategy, we consider a $100.00 portfolio invested solely in stock, and we consider three equally likely possible scenarios. There is some logic to why these three scenarios are chosen, but let's leave that for another article.

Scenario	Final Stock Price	Portfolio Return
1	114.1	14.1%
2	100.0	0.0%
3	85.9	-14.1%

If we believed each of these scenarios to be equally likely, the average return of the three scenarios would be 0%. The 'risk' of this strategy is measured by calculating the standard deviation of the three return possibilities and multiplying by a factor to produce an 'annualized' number. In this case, the risk calculated is 40%. The scenarios and probabilities chosen insure in these examples that the expected return is always 0%. This is just done to simplify the analysis. In practice, investors might have different views about the likelihood of these scenarios. For now though, we make this assumption so that we can focus only on the risk of the portfolio.

This number, 40%, is our benchmark for risk. It tells us how much 'risk' is in this stock portfolio. Put another way, it tells us how much risk the investor is willing to undertake in the hope of realizing a high return. Now let's consider the effect of options.

The four types of option strategies we want to consider are long and short positions in puts and calls, each overlaid on the above stock holding of $100.00. For our example both the put and call are assumed to each cost $4.71 for each $100.00 of stock covered. We will assume that the option premium is either borrowed or lent and unwound at the end of the month, without any interest charge. For example, the strategy named Long Call refers to a portfolio of stock with a long call option added. The premium required to buy this call is borrowed, and repaid at the end of the month. Based on this we have the following payouts for the options in each strategy and scenario.

OPTION PAYOUT
Scenario	Stock Price	Long Call	Long Put	Short Call	Short Put
1	114.1	14.1	0.0	-14.1	0.0
2	100.0	0.0	0.0	0.0	0.0
3	85.9	0.0	14.1	0.0	-14.1

For each of these four strategies, we now table the value of the portfolio at the end of the holding period. This is just equal to the value of the stock, plus the payout of the option plus the premium received, where it is understood that the premium and payout could be negative.

PORTFOLIO VALUE
Scenario	Stock Price	Long Call	Long Put	Short Call	Short Put
1	114.1	123.6	109.4	104.7	118.9
2	100.0	95.3	95.3	104.7	104.7
3	85.9	81.1	95.3	90.6	76.4

Based on these values, we calculate the return of the portfolio for each strategy and scenario. Once those values have been calculated, we can calculate the standard deviation of these returns, which we equate to the risk of the portfolio.

TOTAL RETURN
Scenario	Stock	Long Call	Long Put	Short Call	Short Put
1	14.1%	23.6%	9.4%	4.7%	18.9%
2	0.0%	-4.7%	-4.7%	4.7%	4.7%
3	-14.1%	18.9%	-4.7%	-9.4%	-23.6%
RISK	40%	61%	23%	23%	61%

In two of the cases, where an investor buys a put or sells a call, the risk measure has declined. This is why these types of strategies are referred to as 'hedges'. In the other two, the risk has gone up, quite significantly. How much higher is this? We can answer this by considering two other strategies. The first is called Leverage, where we buy an additional $50.0 worth of stock on margin. The other, called Divest, sells half of the stock position. Skipping a couple of steps, the returns and risk for these scenarios is tabled below:

TOTAL RETURN
Scenario	Stock	Leverage	Divest
1	14.1%	21.2	7.1
2	0.0%	0.0	0.0
3	-14.1%	-21.2	-7.1
RISK	40%	60%	20%

Note that the option strategies that were hedges lowered the risk to the level it would have if only half the stock was held. The other two option strategies increased the risk of the portfolio to the same level as it would be if another half of the portfolio stock was purchased on margin.

Of course, the return profiles for these strategies are all different, and the investor alone must decide which is preferable, but from the perspective of risk, total return analyses such as this allow one to place these strategies on a comparable basis.