Tonight’s article will be less mathematical, and more qualitative than last night’s article. Last night I did say:

The relationship of the VIX to the S&P 500 is an interesting one, one that I have studied for the past nine years. Over that time, I have used the relationships to:

• Design investment strategies for insurance companies selling Equity Indexed Annuities.
• Estimate the betas of common stocks. (Not that I believe in MPT…)
• Trade corporate bonds.
• Gauge the overall risk cycle, in concert with other indicators.

Investing to Back Equity Indexed Annuities

Let me talk about these applications. Equity indexed annuities [EIAs] are tricky to design an investment strategy for. Typically they contain long guarantees, say ten years or so, where the minimum payoff must be guaranteed. That payoff is typically 90% of the initial premium plus 3% compound interest. The optimal strategy invests 80% or so of the money to immunize that guarantee, while using the other 20% to invest short to pay for option premiums that match the payoff pattern promised in the EIA.

But here’s the problem. The forward market for 1-year implied volatility doesn’t exist in any deep way, so the insurance company decides that it will have to take its chances, and assume that volatility will mean revert over longer periods of time. Also, they try to build in enough policy flexibility that they can less favorable option terms to policyholders during times of high volatility (at the risk of higher lapsation). Certain bits of actuarial smoothing in the reserves can also be useful in assuming mean reversion. But what happens if volatility rises high and stays there a while? Unfortunately, that tends to be the same time when credit spreads are wide, because option implied volatility is positively correlated with credit spreads. So, at the time that the strategy needs the most help, option costs are high (or payouts are chintzy and lapse rates go up), and corporate bond prcies sag due to wider spreads.
If the insurance company can handle the lack of incremental income, investing in higher credit quality instruments in tight spread low implied volatility environments can mitigate the risks. The benefit to such a strategy is that in a higher spread and implied volatility environment, you can do a down-in-credit trade (lower credit quality) at the time that it is being rewarded. This takes real courage and foresight on the part of the insurance management team to manage this way, but it pays off in the long run. (Which is why the strategy doesn’t usually get used…)

Corporate Bonds Generally

On my monitor screens, I keep three things front and center: the S&P 500, the long bond, and the VIX. This served me well in 2001-2003 when I was a corporate bond manager. After 9/11, we did a massive down-in-credit trade, buying all of the industries that were out of favor because of fears of terrorism. This is the only time I can remember when our client, who never said “no” to incremental yield, told us to hold back. We were almost done anyway, but in the depths of November 2001, we questioned our own sanity. Then, as implied volatility fell, credit spreads did as well, and the prices of our bonds rose, so in the spring of 2002, we reversed the trade and then some. We were in great position for the double bottoms that happened in July and October of 2002. We played the risk cycle well. Following equity volatility aided structural management of the corporate bonds.
When implied volatility was so high, and volatile, I would use the S&P 500 and the VIX to aid the timing of my trades. When the S&P was falling, and the VIX rising, and the long (Treasury) bond rising in price and falling in yield, I would wait until the S&P 500 would level off, and the VIX begin to fall a little. Then I would buy the corporates that I had been targeting. I would get good executions because of the dourness of the day, but more often than not, the market would turn an hour after I bought as corporate spreads would begin to tighten in response to the better tone from the equity markets and implied volatility.Though we were a qualitative credit analysis shop, I would have analysts review companies when their stock price had fallen by more than 30% since the purchase of the bond, and where the equity’s implied volatility had risen by more than 30%. This test flagged Enron, and others, before they collapsed. Not everything that fits those parameters is a sell, but they are all to be reviewed.

One aspect of the bond market that outsiders don’t know about is that when it gets frothy, deals come and go rapidly. Some get announced and close in as little as seven minutes. Speed is critical at such times, but so is judgment on how fairly the deal is priced. When the market is that hot, a corporate bond manager does not have time to ask the credit analyst what he thinks about a given company. I developed what I called the one-minute drill, which when I explained it to my analysts, fascinated them. Using a Bloomberg terminal, I would check the equity price movement over the last twelve months (red flag — down a lot), equity implied volatility (red flag — up a lot), balance sheet (how much leverage, and what is the trend?), income statement (red flag — losing money), cash flow statement (red flag — negative cash flow from operations), and the credit ratings and their outlooks. I can do that in 30 seconds to a minute at most, giving me ample time to place an order, even if the deal closes seven minutes from its announcement. I told my analysts that I trusted their opinions, but in the few weeks that we might hold the bonds while they were working out their opinion, if I didn’t have any red flags, it was safe to hold the bonds for a month off of the one minute drill. If the analyst didn’t like the bonds, typically I would kick them out for a small gain.
For those with access to RealMoney, I recommend my articles on bonds and implied volatility. Changes in Corporate Bonds, Part 1 , and Changes in Corporate Bonds, Part 2.

Estimating Beta

You can estimate beta using the VIX. Here’s how: start with the Capital Asset Pricing Model (ugh), and apply a variance operator to each side. After simplification, the eventual result will be (math available on request):
What this means is that the actual volatility of the individual stock is equal to the square of its beta times the actual volatility of the market portfolio, plus the firm-specific variance. Now, one can estimate this relationship using a non-linear optimizer (Solver in Excel can do it), regressing actual market volatility on the volatility of the individual firm, allowing for no intercept term, and constraining the errors to be positive, because firm specific variance can’t be negative. In place of actual volatility, implied volatility can be used, because the two are closely related.
I played around with this relationship and found that it yielded estimates of beta that I thought were reasonable. It’s a lot more work than the ordinary calculation, though. The estimate might be more stable than that from using returns.

Gauging the Overall Risk Cycle

When I look at systemic risk, the VIX plays a big role. Other option volatilities are valuable as well, bond volatilities, swaption volatilities, currency volatilities, etc. Also playing a role are credit spreads in the fixed income markets. Together, these help me analyze how much risk is being perceived in the market as a whole. I don’t have a single summary measure at present; different variables are important at different points in the cycle.

Profitable Trading Rules

I think that there are no lack of profitable trading rules for the S&P 500 from the VIX. But you have to choose your poison. Absolute rules tend to have few signals, and require holding for some time, but are quite profitable. Here’s an example: Buy the S&P 500 when the VIX goes over 40, and sell when it drops below 15. Relative rules tend to have more signals, and don’t require long holding periods, but are modestly profitable on average, with more losing trades. Example: buy when the VIX is over its 50-day moving average by 50%, and sell when it is less than the 50-day moving average.

I don’t use any of those rules for my investing, but I do watch the VIX out of the corner of my eye to help me decide when conditions are are more or less favorable to put on more risk. Along with my other variables for tracking the risk cycle, it can aid your investment performance as well.

I’ve estimated a number of mean-reverting models in my time. I had one of the best dynamic full yield curve models around in the mid-90s. The investment department at Provident Mutual said it was the first model that was not artificially constrained that behaved like the yield curve that they knew.

In yesterday’s article, I mentioned that I could give math behind estimating mean-reversion of volatility. In order to do the regression to estimate mean-reversion, we use a lognormal process, because volatility can’t be negative.

Taking the logs of both sides:

Alpha is the drift term, that will help us calculate the mean reversion level, beta is the daily mean reversion speed, and epsilon is a standard normal disturbance term.  Assume that there are no more random shocks, so that the volatility returns to its equilibrium level, which implies:

Substituting into the log-transformed equation, we get:

where V-bar is the mean reversion level for volatility. From there, the solution is straightforward:

From my regression, alpha equals 0.046000, and beta is 0.98436. That implies a mean reversion target of 18.937, and that volatility moves 1.564% toward the mean reversion target. One last note: the standard deviation of the error term was 6.3383%, which helps show that in the short run, the volatility of implied volatility is a larger effect than mean-reversion. But in the long-run, mean-reversion is more powerful, because with the law of large numbers, the average of all the disturbances gets closer to zero.

Apologies to those who tried to look at my graphs and couldn’t.  It’s fixed now.  The difficulty was something embedded in the HTML code, so I had to go through and manually edit to get each one.  Posting pictures at my blog has been less than easy, though I would like to do more of it.

Partly due to the lack of a Bloomberg terminal, but also because of my diminished interest in the insurance industry, I will be discontinuing my earnings review of the insurance space.

Also, for some my my main pages, I had disabled the ability to comment, but it seems as if something has happened to disable commentary on all of them.  For the Amateurs’ Page, if you have a question, just e-mail me, and I’ll post it.  Thanks, and keep the feedback coming; it helps me become better at serving you.

I have often wondered what I would do if I no longer had my trading restrictions, and could invest in financial stocks freely. I no longer have to wonder. As of this morning, I have bought shares in Safety Insurance, Aspen Insurance, Lincoln National, and (what else?) Assurant. They replace the following names in my portfolio: Allstate (waving a fond farewell… but I’ll be back), Noble Corp, and SPX Corp. I will be exiting Lyondell when the arb premium gets a bit tighter. Also, I had two rebalancing buys Tuesday Morning — I added to positions in Lithia Motors and Cemex.

Why do I no longer have restrictions? My employer and I parted ways amicably. I would recommend their services to anyone who wanted to invest in such a hedge fund, but I increasingly found that wanted to do something closer to what I like in investment management. Much as I like analyzing the insurance industry, I’m better at managing broad market equity and bond assets.

So, for the first time in four years, I’m looking at business opportunities. I’ve got a number of them that I am considering, including writing more, setting up my own investment management shop, or working with someone else that I might be compatible with. Do you have any suggestions for me? At this point I haven’t eliminated anything, so if you do have an idea, please e-mail me.

This is bittersweet for me, because I genuinely liked the people that I worked with. That said, I was planning on leaving no later than March 2009 (because of vesting), so this accelerates what was already being considered. This blog was initially developed for this possibility; I just did not expect it to become live only five months after the start.

PS — One of the losses that I feel immediately is the loss of my Bloomberg terminal. I’m going through withdrawal. I’ve used Bloomberg terminals for 15 years now, and given the variety of work that I have done, I know all eleven of the yellow keys to some degree, and I am an expert on eight of them (all except Money Markets, Munis, and Preferred Stocks). I am fitfully learning to do without, but am looking forward to having one back, because I can do snazzy things with it.

Full Disclosure: long AHL SAFT LNC AIZ LAD CX LYO