Actuarial Education Services

The Practical Part FM Seminar
Text Box: Risk-Adjusted Pricing or ‘How Do We Keep All Our Lies Straight?’ Note From Joe: There’s a couple of aspects of this tip that I thought I discuss briefly. First, I know the title sounds a little inflammatory, especially since risk-adjusted pricing is not lying-witness it’s broad acceptance in the marketplace. But I’ll talk below about why you might think you have to keep the pricing straight. Secondly, I said all along that we wanted to push this toward more practical tips with the next tip. And we will do just that, include a tip with direct practical value on exam questions, next week. Illustration: You have a stock presently priced at $100/share, the stock price is expected to grow at the current market growth rate of 12% (assuming no dividends at present), and the current risk-free rate for a one year investment is 5%. But, the one year forward price is $105, not $112. If the one year forward price for the stock doesn’t match it’s expected value, and my option pricing is based on a the statistical distribution of prices one year from now, what do I use for the mean value of the price one year out? If I use $105 for the mean I won’t match reality, especially if the stock does rise to $112. And what probability do I assign to a possible stock price of $120? If there is a method to make all of this work how can you keep track of all the ways it differs from reality? That’s why I use the phrase ‘How do we keep all our lies straight?’ Like last week, we’ll answer this question and along the way we may talk about some things that are beyond the syllabus. Once again we’re presenting something beyond the exam material to show you what’s going on. You’re better off taking this to help you put the pieces together than using this as a study guide. Risk-Adjusted Pricing The basic concept here is that the first dollar you own (said another way, the last dollar you have left) is much more important to you than, say your one millionth dollar. The first dollar (plus some others) buys you food and shelter for the night. The one millionth means you have, say a 70 inch tv instead of a 65 inch model. So, the first dollar affects your life a lot more. Option pricing is similar, is a simplified model the first dollar of the stock price means a lot more to investors than the 100th. So, the possible stock price of $1 gets it’s true expected probability weight times an added multiplier reflecting the extra value of the first dollar. The possible stock price of $100 gets it’s true expected probability weight times a reduction multiplier reflecting the reduced psychic value of the 100th dollar of the price. Does it Work? There are a whole set of financial math theorems that relate to this. One is that as long as the marketplace is working correctly, the expected future value of a stock under risk-adjusted (also called risk-neutral) pricing will be the current price inflated at the risk-free rate. So that matches the $105 forward price in our example. Also, note that the risk adjustments are multipliers and they are the same, at all times, for a given set of possible stock prices. So, once the distribution starts to converge to, say, $112, (i.e., the option nears expiry and the stock price moves towards $112) the risk-adjusted pricing will converge to $112 as well. Scratching the Surface This is a pretty bare bones explanation. And I’ve tried to just sketch the outline to make it clear why a forward price of $105 given an expected future stock price of $112 can actually make real sense. If this still sounds like a process of just keeping the lies straight please email me at joeboor@embarqmail.com and we’ll try an expanded version.