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February 23, 2004

Buy-and-Hold For the Duration?

John P. Hussman, Ph.D.
All rights reserved and actively enforced.

[Just a note: In January, the expense ratio of the Strategic Growth Fund was lowered again, to 1.33%. The expense ratio is affected by the amount of fund assets, fee breakpoints and other factors, and may increase or decrease over time. ]

Duration is an extremely useful statistic, but is largely ignored by investors and even financial planners. That's probably because it isn't a simple calculation, and it's not always interpreted correctly. Let's take a look.

Duration as an accurate measure of maturity

First, hold your arm out sideways from your body, and imagine that you're holding a pencil that's about 10 miles long. Now raise and lower your arm a little. If there was a tiny little guy sitting right on your shoulder, or just a few inches out onto your arm, he would ride up and down a bit, but he would be fairly stable. If the little guy was sitting on your hand as you move your arm up and down, his ride would be a bit more exciting. Of course, if the poor little fella was sitting at the end of that pencil, he'd be screaming and holding on for dear life at the slightest move in your arm.

That's duration. It measures that "distance" more accurately than the stated maturity of a security, and also measures the sensitivity of the security price to changes in its yield. When interest rates change, securities with short durations (such as Treasury bills) don't fluctuate much in price. But securities with long durations (such as 30-year zero coupon bonds) can swing wildly.

Let's do a few real-life examples. If you hate math, you might want to ignore the calculations and focus on the concepts – this stuff is really important.

Consider a 5-year zero-coupon bond with a 10% yield to maturity. If the face value to be received in 5 years is $100, the current price will be ($100/1.10^5 = ) $62.09. Duration in this case is simple. Since the bond delivers a single payment 5 years from now, and nothing before that, the bond has a duration of 5 years. That's the “maturity” interpretation of duration.

Duration as a measure of price sensitivity

Now let's look at the “price sensitivity” interpretation. If you divide the duration by (1+interest rate), we have something called the “modified duration”, which tells you how much your bond's price will change if interest rates change by 1%.

The modified duration of our 5-year zero coupon bond is 5/1.10 = 4.545. You can calculate that if interest rates drop to 9%, the price of the bond rises by 4.67%, to $64.99. If interest rates rise to 11%, the price of the bond falls by 4.42% to $59.35. On average, then, a 1% change in interest rates affects the bond price by [(4.67+4.42)/2 = ] 4.545%, which is just its modified duration. Pretty neat, huh?

As a side note, notice that the change in the bond price due to a decline in interest rates was different than the change due to an advance. That feature is called “convexity.” So we've already got two concepts for the price of one.

In contrast to a zero-coupon bond, a coupon bond delivers a whole stream of payments over its lifetime. For that bond, the duration will always be less than its maturity (for instance, the current 30-year Treasury bond has a duration of only about 15 years). You find the duration of a coupon bond by calculating the average date that you receive those payments, where the weight you give to each year is equal to the proportion of the total present value that you receive in that year.

For instance, suppose you have a bond yielding 10% that pays $100 a year from now, and $100 two years from now. The price of that bond is ($100/(1.10)+$100/(1.10)^2 = ) $173.55. The present value of the first payment is ($100/(1.10) = ) $90.91, which is 52.4% of the total present value. The present value of the second payment is ($100/(1.10)^2 = ) $82.64, which is 47.6% of the total present value. So the duration of the bond is .524 x 1 year + .476 x 2 years = 1.476 years. That's the “average” date that you get your payments (in terms of present value).

[Geek's note: mathematically, duration is defined as the elasticity of the security price with respect to changes in the gross rate of return, which is [-dP/P] / [dk/(1+k)], where k is the capitalization rate of the security. When bonds have call features and the like, the calculation is usually done numerically rather than analytically.]

Matching the portfolio duration to the investment horizon

One of the fascinating aspects of duration is that it also assists with financial planning. Suppose that interest rates vary over time, and we want to create a portfolio that will have the most predictable value at some particular date in the future, say 20 years from now, regardless of where interest rates go. It turns out that the portfolio with the least amount of uncertainty is a portfolio with a duration that matches the investment horizon – in this case, 20 years.

The easiest way to see this is to note that a 20-year zero coupon bond will deliver exactly the same amount of future value 20-years from now, regardless of where interest rates move in the interim. But even for coupon bonds, that basic result holds, because you'll be able to reinvest your coupon payments over time in a way that ends up stabilizing your final wealth.

So we have a basic financial planning concept. If a buy-and-hold investor with no particular view about market conditions or future returns wishes to have a fairly predictable amount of wealth at some future date, that investor should hold a portfolio with a duration that is roughly equal to the investment horizon.

Now, this result doesn't apply for investors who aren't inclined to buy-and-hold, or who believe that the market's return/risk profile changes over time (for example, the Hussman Funds systematically vary their portfolio duration depending on the Market Climates we observe). But for buy-and-hold diehards, the way to create a portfolio with the most predictable future value is to set the duration of their portfolio equal to their investment horizon.

Stocks have a very long duration

While the concept of duration is generally used in bond market analysis, stocks also have a duration. In the case of stocks, duration measures the percentage change in stock prices in response to a 1% change in the long-term return that stocks are priced to deliver (see Estimating the Long-Term Return on Stocks). For the stock market as a whole, the modified duration is simply the price/dividend ratio, which for the S&P 500 is currently about 62. The duration itself is about 67 (the precise figure depends on the exact return that you believe stocks are priced to deliver).

[Geek's note: to see this, consider the dividend discount model P = D/(k-g). Differentiate with respect to k to get dP/dk = -D/(k-g)^2. Divide through by price, which is D/(k-g), and then substitute P/D for 1/(k-g). Notice that this result is independent of g. For stocks that don't pay a predictable stream of dividends, you have to calculate duration explicitly from the stream of expected free cash flows, but for blue-chip indices, the price/dividend ratio is an excellent proxy for modified duration.]

Buy-and-hold ain't what it used to be

Here's where the rubber hits the road. Historically, the price/dividend multiple on the S&P 500 has hovered near 25 (even as recently as 1990), meaning that investors with a 20 or 30-year investment horizon could comfortably invest their entire portfolios in stocks, and feel that they were still being fairly conservative.

Not anymore. At a price/dividend ratio (and modified duration) of 62 on the S&P 500, only a handful of investors can count on a predictable retirement from a fully invested stock portfolio. Right now, those investors are drinking juice from their sippy cups and watching Barney.

What this also means is that even buy-and-hold stock market investors with 20-year horizons are sensitively dependent on the path that stock values take between now and the particular date that they retire.

Consider the typical Wall Street investment allocation for buy-and-hold investors: 60% stocks, 30% bonds, 10% cash. Since the overall duration of the U.S. bond market is about 8 years, and cash has a duration of zero, the standard allocation implicitly assumes an investment horizon of roughly [.6 x 67 + .3 x 8 + .1 x 0 = ] 42.6 years. At present, that advice might be suitable for buy-and-hold investors just entering the work force. But it's nowhere near appropriate for buy-and-hold investors close to retirement.

The bottom line is that portfolio duration matters. For passive, buy-and-hold investors, a fully invested portfolio in stocks has a duration of over 60 years.

Duration is an important consideration at the Hussman Funds, and we believe that it can be managed explicitly and beneficially. Unlike passive buy-and-hold investors, we also believe that the return per unit of market risk can vary depending on observable conditions, and we take proportionately more risk when those conditions appear favorable.

In general, the lower the valuation of the stock market (and consequently, the lower its duration), the more market risk we are typically willing to take in the Strategic Growth Fund. In our view, taking a buy-and-hold position in stocks regardless of valuations and durations is inconsistent with the most basic principles of financial planning.

In the Strategic Total Return Fund, we also vary the duration of the portfolio depending on the Market Climate we observe. Again, lower valuations in bonds translate into shorter durations, so greater exposure to bond market risk becomes appropriate in those conditions. In addition, we periodically increase our portfolio duration based on market action that suggests a broad willingness among bond market investors to take risk.

Market Climate

As of last week, the Market Climate for stocks remained characterized by unusually unfavorable valuations and modestly favorable market action. That said, market action is becoming an increasing concern. It's notable that despite recent highs in the major indices such as the Dow, an increasing number of measures have failed. For example, the Russell 2000, the Dow Transportation average, the number of stocks hitting new highs, the McClellan Oscillator (a measure of market breadth), and overall trading volume have all displayed divergent market action in recent weeks. In short, the market has displayed increasing negative divergences on weak volume, despite new highs in the major indices. That's not a favorable sign, and those observations are mirrored in our proprietary measures as well. We liquidated the call option side of our “straddle” on market strength early last week, and added to our “contingent” put option position.

At present, the Strategic Growth Fund remains fully invested in a widely diversified portfolio of stocks, with about half of those holdings hedged against the impact of market fluctuations. We also hold a position in out-of-the-money put options (valued at about 1% of net assets) sufficient to hedge an additional 35% of the portfolio in the event of substantial market weakness. So while we would expect to participate in further market advances, should they occur, it's reasonable to say that our level of defense rose distinctly last week. That's not any sort of short-term statement about market direction, however. Given that stocks have been weak for several sessions, it would not be surprising to see a strong and potentially sharp corrective rally. Unfortunately, we don't have enough evidence to take market risk on that prospect.

It's certainly possible that the early difficulties we've seen will “clear.” In that event, we would revert back to a roughly 50% hedged position, and would liquidate the contingent put options we've established. Again, that put position is currently valued at about 1% of net assets.

As of last week, the Market Climate for bonds remained characterized by modestly unfavorable valuations and modestly unfavorable market action. In the Strategic Total Return Fund, the portfolio duration remains just under 2.5 years, meaning that a 1% (100 basis point) change in interest rates would be expected to impact the Fund by about 2.5% on the basis of bond price fluctuations. We added modestly to our holdings of precious metals shares on Friday's price weakness (which is not a forecast of near-term direction, but instead a response to a small change in valuations), and continue to hold a substantial position in Treasury Inflation Protected Securities, along with moderate positions in utility shares and government agency notes.

New from Bill Hester: Stocks For The (Really, Really) Long-Run

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