## Description

The Treasury Yield Curve

## Treasury bills, notes, and bonds with different times until maturity generally have different yields to maturity. For example, the interest rate on a 13-week T-bill might be 2%, while the rate on a 5-year T-note is 3% and the rate on a 30-year T-bond is 4% at the same time. The graphic relation between YTM and term to maturity is known as the *yield curve* or the *term structure of interest rates*. If yields on long-term bonds exceed (are less than) yields on short-term bonds, the yield curve is said to be upward (downward) sloping. A downward sloping yield curve is also called an *inverted* curve. If long and short-term yields are about the same, the yield curve is flat. The yield curve could be hump-shaped, with both short and long rates below intermediate-term rates, or bowl-shaped. Historically, the yield curve slopes upward more often than it slopes downward.

To analyze the yield curve, Wall Street professionals use yields on Treasury Bills and Strips, which are zero-coupon bonds. The yield to maturity on a coupon bond is really a blend of returns from the multiple coupons and principal received at various times; the yield to maturity on a zero-coupon bond is a ‘pure’ annualized return for that maturity. So, for the remainder of this chapter, when we use the term “yield” or “rate”, we mean the yield to maturity on a zero-coupon Treasury security.

Let’s start with an example. Suppose the 1-year yield is 6% and the 2-year yield is 7%. Why might these two rates differ? We know the four key factors in analyzing investments are expected return, risk, liquidity, and taxation. Taking these factors in reverse order, taxation is not a big issue here, as each investment is a Treasury and is taxed the same way. Liquidity is important. By definition, the shorter-maturity bond more quickly converts to cash than the longer-maturity bond. Risk is definitely important. Treasuries don’t have default risk, but they do have price risk and reinvestment risk, which depend on the investor’s time horizon. They also have inflation risk. We discuss liquidity and risk a little later in the chapter.

What about expected return? If I didn’t care about differences in risk or liquidity, would I always buy the 2-year bond in this situation, with its higher yield?

No, not necessarily. Your decision depends on what you expect the 1-year rate will be one year from now. Suppose you think the 1-year rate will be 9% in a year. In that case, you should buy the 1-year bond now. Over a two-year horizon, reinvesting all interest earned, for each $1 invested now, you expect to have

(1.06)(1.09) = 1.1554

From investing $1 in the 2-year bond, you will have only

(1.07)^{2} = 1.1449

Alternatively, suppose you think the 1-year rate will be 7% in a year. Then, you should buy the 2-year bond now. Over a two-year horizon, for each $1 invested now in the one-year bond, you expect to have only

(1.06)(1.07) = 1.1342

From investing $1 in the 2-year bond, you will have

(1.07)^{2} = 1.1449

What is the “break-even” expected 1-year rate, denoted f, so an investor that didn’t care about liquidity and was risk-neutral would be indifferent between one- and two-year bonds?

Set (1.06)(1+f) = (1.07)^{2}

(1+f) = (1.07)^{2}/(1.06)

f = 8.0094%

This break-even rate is defined as the *forward rate* between years one and two.

To generalize, define y_{T} as the zero-coupon bond yield with maturity of T years. Strategy (1): you invest $1 at the T-year rate, so you will have (1+y_{T})^{T} at time T. Strategy (2): you invest $1 at the (T-1)-year rate, so you will have (1+y_{T-1})^{T-1} at time T-1, and then you invest the proceeds for another year at the one-year rate at that time. The forward rate f_{T} is the one-year interest rate between dates T-1 and T that makes the payoff from Strategy (2) the same as the payoff from Strategy (1). So, by definition,

(1+y_{T-1})^{T-1}(1+f_{T}) = (1+y_{T})^{T}

(1+f_{T}) = (1+y_{T})^{T}/(1+y_{T-1})^{T-1}

f_{T} = (1+y_{T})^{T}/(1+y_{T-1})^{T-1} – 1

Example: If the 3-year yield is 4% and the 4-year yield is 5%, then the forward rate between years 3 and 4 is

(1.04)^{3}(1+f_{4}) = (1.05)^{4}

(1+f_{4}) = 1.08058

f_{4} = 8.058%

By purchasing the 4-year rather than the 3-year bond, I am essentially earning a 4% return each year for the first 3 years, and then earning an 8.058% return over the last year of the 4-year horizon.

To better understand what the forward rate represents, suppose I have decided that I want to lend money for at least 4 years before spending the proceeds. I could buy a 4-year bond, or could buy the 3-year bond. The forward rate of 8.058% in this example is the rate of interest I can “lock in” for the fourth year by purchasing the 4-year bond, instead of purchasing the 3-year bond and then buying a one-year bond after 3 years. Ignoring risk and liquidity for now, presumably I would buy the 3-year bond if I thought the one-year rate 3 years from now would exceed 8.058%. I would buy the 4-year bond if I thought the one-year rate 3 years from now would be less than 8.058%.

Example: If the 2-year yield is 3% and the 3-year yield is 3.5%, then the forward rate between years 2 and 3 is:

(1.03)^{2}(1+f_{3}) = (1.035)^{3}

(1+f_{3}) = 1.045073

f_{3} = 4.5073%

The Expectations Hypothesis of the Yield Curve

The expectations hypothesis states that yields to maturity on long-term bonds are determined __solely by__ market expectations of future short-term interest rates. Under this theory, if the yield curve slopes upward, that means investors believe future short-term rates will be higher than current short-term rates. If the yield curve is flat, investors expect short-term rates to stay the same as they are now. If the yield curve slopes downward, that means investors believe future short-term rates will be lower than current short-term rates. Liquidity and risk don’t matter under this theory; only expected return does.

## To see the logic of the expectations hypothesis, suppose 1-year T-bills now pay 3%, and investors generally expect T-bill rates to rise over the next five years. Would most investors buy a 5-year T-note yielding the same 3%, locking in a 3% return for five years? Probably not. The 5-year T-note would have to pay more than 3%, because otherwise everyone would buy the T-bills, and then expect to be able to reinvest at rates above 3% after the T-bills mature. If everyone buys the T-bills, the price rises, reducing the yield. If no one buys the 5-year note, its price would then fall, increasing its yield. Using this logic, when investors expect short-term rates to rise in the future, we end up with an upward sloping yield curve.

In terms of forward rates, the expectations hypothesis states that the one-year forward rate f_{T} is equal to the market’s expected one-year actual or ‘spot’ rate between T-1 and T. In other words, the compound rate of return from investing in a long-term bond maturing at time T is the same as the __expected__ compound rate of return from rolling over a series of T one-year bonds through time T.

Example: Suppose the 1-year yield is 2%, and the 2-year yield is 2.25%. Suppose the 3-year yield is 2.5%. Under the expectations hypothesis, what does “the market” believe that the 1-year rate will be in one year? What does “the market” believe the 1-year rate will be in two years?

(1+y_{1})^{1}(1+f_{2}) = (1+y_{2})^{2}

(1+.02)^{1}(1+ f_{2}) = (1+.0225)^{2}

(1+ f_{2}) = 1.0455/1.02

f_{2} = 2.501%

Under the expectations hypothesis, the forward rate of 2.5% is the market’s expectation of the one-year rate between times 1 and 2.

(1+y_{2})^{2}(1+f_{3}) = (1+y_{3})^{3}

(1.0225)^{2}(1 + f_{3}) = (1.025)^{3}

1+f_{3} = 1.03, so f_{3} = 3%

Under the expectations hypothesis, the forward rate of 3% is the market’s expectation of the one-year rate between times 2 and 3.

Example: Suppose that 1-year yield is 5% and the 2-year yield is 4%. If you expect the 1-year rate to be 3.5% in one year, and you do not care about any differences in liquidity and risk between the two investments, should you buy the 1-year or the 2-year bond now?

(1.05)(1 + f_{2}) = (1.04)^{2}

1+f_{2} = 1.0301, so f_{2} = 3.01%

If you think the 1-year rate in one year will be 3.5%, buy the 1-year bond now.

In Practice, When Will Investors Expect Short-Term Rates to Rise or Fall in the Future?

The expectations hypothesis says that the yield curve slopes upwards when “the market” (investors as a group) expect short-term rates to rise in the future. When will that happen?

Expectations of economic growth: If investors expect economic growth to accelerate, then they will generally expect short-term rates to rise in the future. In a strong economy, consumers increase purchases of durable goods like cars, trucks, and houses, which generally require financing. Similarly, strong economic growth encourages business investment, financed by borrowing to a large extent. When individuals and business attempt to borrow more to buy “big ticket” items, interest rates get pushed up.

The slope of the yield curve is one of the ten leading economic indicators (LEI) in the LEI index each month, calculated by The Conference Board, an organization of business economists. The steeper the (upward) slope of the yield curve, the greater the chance of stronger economic growth in the future.

Expectations of inflation: If investors expect the inflation rate to rise in the future, then they will generally expect short-term rates to rise along with it. To preserve the same pre-tax real interest rate, short-term nominal rates would need to increase one-for-one with inflation.

To summarize, under the expectations hypothesis, the yield curve will have a steep upward slope when investors believe the economy will improve and/or inflation will increase in the future. The yield curve will invert when a recession, possibly combined with lower inflation, is predicted. If future economic growth and inflation are expected to be about the same as they are now, the yield curve will be flat.

Is the Expectations Hypothesis Valid? Empirical Evidence from Historical Returns

Under the expectations hypothesis, long-term yields are simply averages of current and expected short-term yields. So, as long as investor expectations are rational and there is no major long-term trend in market interest rates over time, then the average return earned from investing in short-term bonds should be very close to the average return earned from investing in long-term bonds. What has actually happened in the United States over the past 90 years?

On January 1 of each year over 1928-2018, if an investor had bought a 3-month T-bill and then rolled it over on April 1, July 1, and October 1, their average annual arithmetic return was 3.43%. Over the same period, if the investor had bought a 10-year Treasury bond on January 1 and then sold it at the end of each year, the average annual arithmetic holding-period return was 5.10%.

Very little of this difference can be explained by capital gains (or losses) on the 10-year bonds, because market interest rates only fell a little over the period (3.17% at the end of 1927, versus 2.69% at the end of 2018). So, unless investors were wrong again and again and again, always predicting short-term interest rates would rise much more than they ever did, the expectations hypothesis is not consistent with the historical data. Expectations of future short-term interest rates can’t be the only thing that affects the yield curve. There must be a risk or liquidity premium earned from buying 10-year T-bonds rather than buying 3-month T-bills.

The Liquidity Preference Theory of the Yield Curve

## The liquidity preference theory takes it for granted that at any given point in time, expectations of future short-term interest rates are an important factor in the shape of the yield curve. But the liquidity preference theory also postulates that investors demand a yield spread or *term premium* for purchasing long-term rather than short-term bonds, because short-term bonds are inherently more liquid. By definition, short-term bonds return principal to the owner faster than do long-term bonds, so are more “cash-like”. If investors prefer liquidity over illiquidity, which they do, then long-term interest rates will be pushed up. Under the liquidity preference theory, even if investors expected short-term interest rates to remain at their current level forever, the yield curve would not be flat, but would be upward sloping, to account for the lower liquidity of long-term bonds.

This theory seems quite reasonable if one is comparing a one-month T-bill to a two-year T-note, for example. The T-Bill repays principal in just four weeks, so has a major advantage in liquidity. But what about a 2-year T-note versus a 10-year T-note, or versus a 30-year T-bond? Can liquidity preference itself really explain any significant difference in yields between 2- and 30-year bonds, given that neither investment pays back its principal very quickly? Is the difference in liquidity between 3-month T-bills and 10-year T-bonds important enough to explain a more than 1.5% difference in average annual returns over time, when for professional traders, the bid-ask spread for 10-year Treasuries is only about 1/32 of 1% of par? Perhaps we should also think about the risks of investing in long-term bonds instead of short-term bonds.

The Slope of the Yield Curve, and the Risks of Investing in Long-Term Versus Short-Term Bonds

There are three risks to think about when investing in Treasury securities, price risk, reinvestment risk, and inflation risk.

Price risk is the risk that the price of a bond will be low at a time you might want to sell it. There is no price risk if you hold a bond until maturity. But if you might need to sell before maturity, we know that bond prices fall when market interest rates rise. Price risk is higher for long-term than for short-term bonds, and is measured by (modified) duration. Price risk is of most importance to investors that have a relatively short time horizon, those planning to liquidate and spend their investment proceeds soon.

Reinvestment risk is the risk that market interest rates will be low when you want to reinvest the principal from a maturing bond, or reinvest the coupons from a bond you continue to own. Reinvestment risk is higher for short-term than for long-term bonds. Reinvestment risk is of most importance to investors that have a relatively long time horizon, those planning to liquidate and spend their investments in the distant future.

Inflation risk is the risk that consumer prices will rise, reducing the purchasing power of the principal and coupons on your bond. In practice, market interest rates generally rise or fall roughly one-for-one with the inflation rate. Inflation risk affects all bond investors, but is particularly troublesome for long-term, fixed rate bond investors. A fixed interest coupon does not “keep up” with inflation. So the longer the bond’s maturity, the longer it will be until the coupon earned adjusts to a change in the inflation rate. By investing in short-term bonds and rolling over at maturity, the investor’s income will generally keep up with inflation over time, maintaining purchasing power. The shorter the maturity, the quicker the adjustment in the interest rate paid on the bond for changes in inflation.

Yield Curve Summary

At any particular point in time, the Treasury yield curve is heavily influenced by market expectations of future short-term interest rates. To the extent that T-Bill rates are expected to be higher (lower) in the future than they are now, the yield curve will have more of an upward (downward) slope. Investors typically expect T-Bill rates to increase in the future when they expect a pickup in economic growth, or a higher inflation rate.

Long-term bonds are inherently less liquid than short-term bonds. People prefer more liquid investments to less liquid investments, all else equal. So, long-term bonds will have a positive liquidity premium in their yields.

Long-term bonds have more price risk and more inflation risk than short-term bonds. Their only advantage is less reinvestment risk, for those investors that have long time horizons. If most investors have relatively short time horizons, long-term bonds will have a positive risk premium in their yields.

Historically, long-term Treasury Bonds have earned significantly higher average returns than Treasury Bills. This evidence suggests that investors as a group demand a term premium for the lower liquidity, higher price risk, and higher inflation risk inherent in long-term bonds.

Questions

https://www.wsj.com/market-data/bonds

- If 3-year Treasury strips have a YTM of 3.4% and 4-year Treasury strips have a YTM of 3.1%, what is the forward rate between years 3 and 4? Under the expectations hypothesis, what does this forward rate represent?
- Suppose the 4-year yield is 3.2%, and the 5-year yield is 3%. Under the expectations hypothesis, what is the implied market expectation of the 1-year rate four years from now?
- Under the expectations hypothesis, if the yield curve is flat at 4%, then most investors believe that future short-term interest rates will be ______ 4%. Under the liquidity preference theory, if the yield curve is flat at 4%, then most investors believe that future short-term interest rates will be ______ 4%.

https://fred.stlouisfed.org/series/T10Y3M

- How is the slope of the yield curve used as a leading economic indicator?

https://www.clevelandfed.org/our-research/indicators-and-data/yield-curve-and-gdp-growth.aspx

http://stockcharts.com/freecharts/yieldcurve.php mid-2000, late-2002, mid-2007, early-2009, now

- Suppose the yield curve is flat. If you have no particular opinion on whether short-term interest rates will go up or go down, what are the good things about investing in short-term bonds? The good things about investing in long-term bonds?
- If most investors have relatively short time horizons, and investors are
__not__concerned about inflation risk, would you expect the yield curve to be upward or downward sloping most of the time? What if most investors have relatively long time horizons? - If investors
__are__concerned about inflation risk (as they should be), would you expect the yield curve to be upward or downward sloping most of the time? - TIPS, just like regular Treasury Bills, Notes and Bonds, have their own yield curve. Why should the TIPS yield curve have less of an upward slope than the regular yield curve?

https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield

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