Actuarial Education Services

The Practical Part FM Seminar
Text Box: What is Arbitrage Anyway and Why Is the Forward Price Less than the Price the Stock is Supposed to Grow To? Note: This is a little esoteric, and may well not help you solve any particular question. We are posting so that the way the forward prices work will make some sense. By the way, we also have an (admittedly a little esoteric) definition of arbitrage, too. We’ll cover a wider gamut in future tips and will mostly stress items that are more directly related to exam questions. With this tip, and the next one, we’re just warming up. Illustration of the problem: This is a little bit of a cumbersome idea to put in a short title, but let me illustrate with an example. 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. Why? We’ll answer this question. Note that along the way we may talk about some things that are beyond the syllabus. Note that we’re presenting more so we can show you what’s going on behind the scenes and so that the formulas you do have to know make logical sense. In this case, you’re better off taking this to help you put the pieces together than using this as a study guide. What is arbitrage? One answer comes from the world of arbitrage. This theory begins with an investment $X spread across some set of securities set. This aggregate of securities or portfolio (Portfolio A) has some random set of returns {Y(ω)}, where the ω’s represent all possible (relevant) states of the market one year (or some other time) from now. The no-arbitrage principle states that if a portfolio of different securities (Portfolio B) also has returns equal to Y(ω) at each possible future state of the market, Portfolio B also has to cost $X . If B costs $X+$Z, you could buy A for $X, and sell B for $X + $Z, pocketing $Z. One year later, whichever event/state of the market ω´ happens, you get $Y(ω´) from A and must pay $Y(ω´) to whomever you sold B to. So, you receive $Z now and you pay $Y(ω´)-$Y(ω´)=0 at the end of the year. In event, whatever (ω´) happens with the market over the year, at the end of the year the payoff from A exactly equals what you have to pay to whoever bought B. This arbitraging in effect gave you $Z for nothing. The theory is that as soon as an arbitrage opportunity comes up, investors will flock in and bid up the price of A to match that of B. And usually, something like that does happen. What is the arbitrage problem with a $112 forward price? Let’s say holding the stock and selling it one year forward is portfolio A. Let’s say holding a risk free bond maturing in one year for $112 is portfolio B. So, the cost of A, is the current stock price, $100. The cost of B is $112/1.05=$106.67. If the forward price of the stock is $112, I simply pay $100 for the stock now and receive $112 one year hence in portfolio A (whatever the state of the market ‘ω’ is) by entering into the forward contract. Similarly, I sell B for $106.67 to finance the purchase of A (pocketing $6.67 as arbitrage profit). A pays off for $112 in any ω and B costs me $112 in any ω. So, the $112 forward price allows arbitrage. On the other hand, if the forward price of the stock is $105, I simply pay $100 for the stock now and receive $105 one year hence in portfolio A (whatever the state of the market ‘ω’ is) by entering into the forward contract. Similarly, for B, I now sell B for $100 to finance the purchase of A (generating no arbitrage profit). A pays off for $105 in any ω and B costs me $105 in any ω. So, both portfolios are equally priced and the $105 forward price does not allow arbitrage. If the forward price is different than the real expected price, how do you keep it all straight? You could argue that this is a version of ‘how can you keep your lies straight’ since the forward prices are not the real expected prices. We’ll give a brief summary of how next week. Then we move on to tips that will give you more direct help.