NOTES 2: The Yield Curve
Overview of the Yield Curve
The yield curve is a fancy name for the relationship between the interest rates (yields) paid on securities of differing maturity lengths. In principle, you could construct yield curves for any type of debt securities. We tend to focus on government securities. For the purpose of this note, we will discuss the relationship between short-term government treasuries (1 year T-bills) and longer-term government treasuries (i.e.2year, 5-year, or 10-year Treasury bonds).
Remember from Notes 1 that nominal interest rates on a security (i) are approximately equal to real interest rates on the security (r) plus expected inflation (πe).:
i = r + πe (1)
If this were a finance class, I would add an additional term, a risk premium (ρ), to the nominal interest rate such that:
i = r + πe + ρ
In our finance classes, this term could measure default premiums, refinancing premiums, or liquidity premiums. For government treasuries, none of these risks are overly relevant (we generally believe the U.S. federal government is not going to default on its debts, there is no asymmetric call option which makes refinancing salient, the market is very liquid, etc.). So, I have set ρ = 0 (for simplicity). If we believe these risks are important, wewill need to carry around the ρ term throughout the subsequent analysis. From 1996 to 2006, the market for U.S. Treasuries became more liquid resulting in a decline in ρ. This caused the yield curve to flatten slightly. The reason for this is that foreign governments (like China) have started holding (and trading) 10-year securities more frequently than they did in the past.
For notation, we have to realize that interest rates are time and maturity specific. For example, i can be a nominal interest rate on a 1 year security starting today. That is distinct from the nominal interest rate on a five year security starting today or the nominal rate on a 1 year security starting tomorrow. In terms of our notation, we will refer to it,m as being a security starting in period t andcoming to maturity in period m. So, a 1 year T-bill starting today will have a nominal interest rate denoted i0,1 while a 5 year Treasury starting today will have a nominal interest rate denoted i0,5. These are securities originated today (in period 0) and maturing in periods 1 and 5, respectively. Likewise, a 1 year T-bill starting one year from now will have a nominal interest rate denoted i1,2 (starting in period 1 and maturing in period 2).
Arbitrage implies a direct relationship between securities of similar risk starting in the same period of differing maturities. Consider a two period world. Suppose I want to invest $1 today for a total of two periods. If that was the case, I have two options available to me. I could purchase a $1 two-year Treasury starting today (with interest rate i0,2) and keep my money in it for two periods or I could invest my $1 in a one year security starting today (with interest rate i0,1) and then I could reinvestment the proceeds into another one year security starting one year from now (with interest rate i1,2). These options, from my investment purposes, are exactly the same. I start with $1 today and I want to know how much I am going to have after two periods. Because of arbitrage, I should be indifferent between these two options. If one option dominated, everyone would invest in that option driving down the returns to that option. Put another way, arbitrage guarantees that:
(1+i0,2)2 = (1+i0,1)(1+E(i1,2))(2)
where E(.) is an expectations operator such that E(ii,2) is the expected future nominal interest rates between periods 1 and 2. The return to investing the $1 in a two year security starting today would be (1+i0,2)2 <I would get the annual interest rate i0,2 for two periods–hence the squaring>.
The same logic holds for securities of longer duration. For example, I should be indifferent between holding a five year government treasury (starting today) and buying a 1 year security (starting today) and rolling the proceeds over into four subsequent one year securities. In other words:
(1+i0,5)5 = (1+i0,1)(1+E(i1,2))(1+E(i2,3))(1+E(i3,4))(1+E(i4,5))
More generally,
(1+i0,m)m = (1+i0,1)…(1+E(im-1,m))(3)
Given equations (1) and (3), we now have a complete mechanism for understanding the yield curve. Let’s continue with a two period example. Suppose, r0,1 = 2% and πe0,1 = 2% such that a one year government security starting today has a nominal interest rate (i0,1) = 4%. Suppose further that I expect r1,2 = 2% and πe1,2 = 4% such thata one year government security starting one year from now has an expected nominal interest rate E(i1,2) = 6%. Given this information, what should the nominal interest rate be on a two year government security starting today (i0,2). Given (2), we get:
(1+i0,2)2 = (1+i0,1)(1+E(i1,2))
(1+i0,2) = ((1+i0,1)(1+E(i1,2)))0.5
i0,2 = ((1+i0,1)(1+E(i1,2)))0.5 - 1
i0,2 = ((1+0.04)(1+0.06))0.5 - 1
i0,2 =5%
In our simple analysis, we could draw a yield curve:
Yield
6.5%
6.0%
5.5%
5.0%
4.5%
4.0%
3.5%
Treasury
1year2year Maturity
This above yield curve is based on the following assumptions – fixed real one year interest rates of 2% over the next two years (r0,1 = r1,2 = 2%), expected inflation over the next one year (πe0,1) = 2%, and expected inflation over the following year (πe1,2) = 4%.
What causes the yield curve to shift or change shape?
There has been lots of talk recently about a flattening of the yield curve. Given our above discussion, it is evident what would cause a yield curve to flatten (or, more generally shift). Before turning to what causes a yield curve to flatten, let’s ask: what can cause a yield curve to shift?
In our above analysis, we supposed that annual real interest rates were currently 2% and, moreover, were expected them to stay at 2% over the subsequent year (i.e., r0,1 = r1,2 = 2%). Suppose, instead that real interest rates were expected to increase to 3% per year for all current and future periods (i.e., r0,1 = r1,2 = 3%). If we assume that expectations of inflations rates were unchanged, theni0,1 = 5% (3% + 2%) and i0,2 = 6% (using (3)). Graphically, we can represent the new yield curve as:
Yield
6.5%
(new yield curve at higher r)
6.0%
5.5%
5.0%(old yield curve at original r)
4.5%
4.0%
3.5%
Treasury
1year2year Maturity
In summary, permanent increases in real interest rates (r) starting today will shift the yield curve upwards. Likewise, permanent increases in expected inflation rates starting today will also shift the yield curve upwards. You should be able to prove this to yourself (suppose πe0,1 increased from 2% to 4% and πe1,2 increased from 4% to 6% - what would the new yield curve look like?)
The real question is: what causes the yield curve to flatten (become steeper) over time? Moreover, under what conditions will the yield curve become inverted (downward sloping)? The answer hinges exclusively on changes in our expectations of future real interest rates and changes in our expectations of future expected inflation rates.
Suppose nothing changes with respect to our current expectations (starting today) of one year real interest rates and current inflation rates such that r0,1 = 2% and πe0,1 = 2%. However, suppose we think that inflation in the futurewill be the same as it is today (further suppose we expect real interest rates (r1,2) to remain at 2%). This is exactly the same analysis as we did above, except that expected future inflation (πe1,2) is now lower.
In that case, one year nominal interest rates starting today (i0,1) will remain constant at 4%. However, now two year nominal interest rates starting today (i0,1) will be 4%. You should be able to prove that yourselves. Graphically, we can represent the new yield curve:
Yield
6.5% Drawn for r0,1 = r1,2 = 2% and πe0,1 = 2%
6.0%
5.5%
5.0%(old yield curve with πe1,2 = 4%)
4.5%
(new yield curve with πe1,2 = 2%)
4.0%
3.5%(new yield curve with πe1,2 = 1%)
Treasury
1year2year Maturity
If we expected future inflation to be lower than today’s inflation, the yield curve will be inverted (sloping downward), all else equal. If everything above stayed the same (r0,1 = r1,2 = 2% and πe0,1 = 2%) and πe1,2 = 1%, i0,2 = 3.5% (prove that to yourself).
In other words, accelerating inflation rates will lead to upward sloping yield curves and falling inflation rates will lead to an inverted yield curve (all else equal).
We could have done the whole analysis above changing future expected real rates (holding expected inflation rates at their original levels). The results would be the same. When expected future real interest rates (r1,2) are higher than current real interest rates (r0,1), the yield curve would get steeper. Likewise, when expected future real interest rates are lower than current real interest rates, the yield curve will be flatter, possibly inverted. As we will see later in class, expected future recessions (or periods of low interest rates) will cause expectations of future real interest rates to fall.
Summary:
The yield curve slopes up when expected real interest rates in the future are higher than real interest rates today and when inflation rates in the future are expected to be higher than inflation rates today. Likewise, if expected inflation or expected real interest rates in the future are lower than inflation and real interest rates today, the yield curve will be flat (or possibly inverted).
Later in the class, we will talk about what causes low (falling) inflation or low (falling) real interest rates. In short, both are usually associated with periods of low aggregate demand. So, if we are expecting recessions in the future, the yield curve is usually flat. If we expect booms in the future, the yield curve is usually upward sloping.
Note: This hinges on short term securities being substitutes for long term securities (which allows arbitrage to hold). Sometimes the press talks about the yield curve being driven by foreigners’ (like China’s) demand for long term U.S. securities. This relationship hinges on short term securities being a poor substitute for long term securities - if so, arbitrage would equate the returns across securities. There is a discussion of this in the literature - the separating markets hypothesis - that says long term securities are not perfect substitutes for short term securities. I do not believe this is relevant for government treasuries because the market is so deep and so liquid and any one player is small relative to the overall market.
Why do macro economists care about the yield curve?
Basically, by looking at the yield curve, we can back out the market’s expectation of future economic conditions (r and πe). This is another important input into our policy models. We care whether the market believes that future inflation will be high or whether future inflation will be low. The yield curve gives us some insight into this.
Why should the business community care about the yield curve?
For the same reason macroeconomists care about the yield curve, the business community should care about the yield curve. It gives insight into what the market expects of future business conditions. Moreover, it is often a convenient way to back out changes in expectations of inflation. In reality, the real interest rate does not change that much. Most of the action is on the expected inflation side. So, if that is the case, assuming real interest rates are constant will allow individuals and firms to back out the market’s expectation of future inflation exactly.
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