Building a Better Model for Valuing Bonds

Bonds

Bond investors want to know two things: how much risk they face, and whether they’ll be paid enough to take it.

But posing those questions is a lot easier than answering them, and analysts have devised various ways to tackle the problem, none of them perfect. While it’s fairly easy to compare a specific bond’s current yield, or interest earnings, to safer alternatives like short-term Treasury bills or bank savings, it’s obviously impossible to nail down all future risks. Will inflation spike? Will war break out? Will there be a financial catastrophe? Will bonds issued next year pay more than bonds sold this year?

No one knows for sure.

Still, the combined views of all investors offer some guidance, and that wisdom is distilled in the yield curve — a graph with yields on the vertical axis and maturities on the horizontal. (Maturity is the period the bond will pay interest before the investor’s money is returned.) The yield curve reveals a lot about how investors see risks and rewards today, or five, 10 or 30 years down the road.

But because those views are constantly changing, next year’s yield curve can look very different from today’s. It could be steeper, flatter or even “inverted,” with short-term bonds paying more than long-term ones. These changes reveal even more about the bond market’s inner workings.

To tease out more insight, Wharton finance professor Philipp Illeditsch and two colleagues from the London Business School have devised a new analytical model that looks at changes over time in volatility — ups and downs in bond prices and yields — and the ever-varying risk premium, or extra payment investors require for taking on a certain level of risk. Typically, analytical models focus on either volatility or changes in the risk premium, but have trouble tackling both at the same time.

“We are … introducing a new class of dynamic term structure model…. It’s like a statistical model of the yield curve.”

Our model does a better job at capturing both,” Illeditsch says, adding: “We are pretty much introducing a new class of dynamic term structure model Twitter …. It’s like a statistical model of the yield curve.”

Illeditsch and his colleagues, Peter Feldhutter and Christian Heyerdahl-Larsen, describe their work in the paper, “Risk Premia and Volatilities in a Nonlinear Term Structure Model.”

At its core, the new model assesses three key factors in the bond market: inflation, the steepness of the yield curve, and the curve’s shape – whether, for instance, it gets steeper or flatter as bond maturities get longer.

Focusing on prices and yields for U.S. Treasury bonds from 1961 to 2014, the researchers find that the yield curve contains more information than previously thought. And they conclude that bonds are more attractive than commonly believed during times of low volatility, when unexpected events are less common.

All bonds are loans from the bond purchaser to the bond issuer. A person who pays $1,000 for a 10-year bond yielding 4% will receive $40 a year in interest earnings, then get his $1,000 back after 10 years.

But if the investor wants to sell the bond before the 10 years are up, the market price will depend on prevailing conditions and other investors’ views of what will happen before the bond matures. If newer bonds pay 5% or 6%, no one will pay full price for the older 4% bond, so its price drops. But if newer bonds paid less than 4%, investors might pay more than $1,000 for the 4% bond.

In a highly simplified example, a 4% bond that originally sold for $1,000 might be worth only $500 if newer bonds were to yield 8%. That’s because the older bond’s $40-a-year payment would equal 8% of $500, the same yield one could earn on a new bond. In real life, it’s much more complicated than this and many other factors come into play, but this illustrates how changing market conditions affect the value of a bond investment.

Seeing the Future

Like all investors, those who purchase bonds rely on an assessment of potential risks and rewards over the bond’s lifetime. One risk is the chance the Federal Reserve might raise short-term interest rates, or that inflation expectations rise. And, as noted above, the bond’s value in the market can drop if newer bonds pay higher yields. The falling price of the bond could create a loss for the investor even if the interest earnings continue to flow. High volatility is another risk, because when prices and yields fluctuate faster and further, the future seems less predictable.

Though no one knows just how the future will unfold, constantly changing bond prices and yields show how investors as a whole expect risks and rewards to play out over time. Short-term bonds, like those maturing in only a few months, typically offer lower yields, or interest payments, than long-term bonds maturing in 10, 20 or 30 years, because the investor faces fewer risks over a short period. A long-term bond generally pays more because it entails more risk.

But the steepness of this yield curve changes as the view of future risks rises and falls. When future risks seem high, the curve is steep – a 10-year bond might pay 7%, and a one-year bond just 2%. When future risks seems low, this difference in yields is smaller, because investors don’t think tying their money up for the long term requires a much larger risk premium. Occasionally, the yield curve is inverted – with short-term bonds yielding more than long-term ones.

“Three factors explain over 95% of the time variation in yields.”

In considering a bond purchase, investors want to know two things, Illeditsch explains: how much risk there will be over the period the bond will be owned, and how much more investors will earn to take on that risk. Illeditsch and his colleagues set about finding a better way to assess the factors that cause the yield curve to change over time.

“Three factors explain over 95% of the time variation in yields,” he says. The most important, accounting for about 90% of that variation, is inflation. If annual inflation is 1%, a bond paying 3% is profitable in “real,” after-inflation terms. But if inflation jumps to 3%, it eats up the real interest earnings. High inflation, or fear that inflation will rise, therefore undermines bond prices.

The next key factor is the steepness of the yield curve, or how large the difference is between short- and long-term yields. With a steep curve, long-term yields are much higher, generally making long-term bonds more attractive than short-term ones. The investor earns a premium for taking long-term risk. When the curve is flatter, investors don’t earn much more to compensate for the risk of tying their money up for many years.

Finally, there is the curvature of the yield curve. Sometimes the curve is fairly straight, with the yield of a five-year bond, for example, falling on a straight line between the yields of one and 10-year bonds. At other times, the curve may get steeper as maturities get longer. And at others it flattens out as maturities lengthen. The five-year bond may yield substantially more than a one-year bond, while the 10-year bond yields about the same as the five-year.

With their new model, Illeditsch and his colleagues feel they can better understand quick changes in the markets.

In the early 1980s, for instance, yields and volatility were high, and then came down under Fed Chairman Alan Greenspan. Looking at the factors prevailing in the early 1980s, standard models don’t capture the volatility, which is essential to understanding future risks.

“But our model can capture that,” Illeditsch says.

It will take more work to turn this model into something a portfolio manager can use to guide bond investing, but Illeditsch and his colleagues believe they have taken an important step toward a simplified approach to better understanding the risks and rewards in the bond market.

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