MATT BRIGIDA
Associate Professor of Finance (SUNY Polytechnic Institute) & Financial Education Advisor, Milken Institute
Throughout this presentation we will refer to bills, notes, and bonds as collectively "bonds" for simplicity.
Is the annualized yield on a default risk-free zero-coupon bond for each maturity.
The yield curve is the annualized yield on coupon bonds, and so this does not tell us the exact return/discount rate. Remember coupon bonds have reinvestment risk.
Since asset prices are the sum of discounted cash flows, and the term structure gives us the rate at which to discount these cash flows.
We'll soon see that we can estimate future zero-coupon rates given the term structure. If those future rates we estimate are low, they may be an indicator of an upcoming recession.
What follows are three common theories that seek to explain why the term structure may take a particular shape.
Say, for example, you are going to invest in zero-coupon default-free bonds for 2 years. Pure expectations says you will earn the same return if you invest in a two year bond, or you invest in a 1 year bond, and reinvest your money in 1 year in another 1 year bond.
where $r_{2,0}$ is the 2 year rate today, $r_{1,0}$ is the 1 year rate today, and $f_{1,1}$ is the expected 1 year rate in 1 year.
This is the same regardless of maturity. So pure expectations would say a 10 year zero coupon will pay the same as investing in a 3 year zero coupon, and in 3 years reinvesting in a 7 year zero coupon bond (or a 6 year and in 6 years reinvesting in a 4 year bond).
$$(1 + r_{10,0})^{10} = (1 + r_{3,0})^3(1 + f_{7,3})^7$$or
$$(1 + r_{10,0})^{10} = (1 + r_{6,0})^6(1 + f_{4,6})^4$$Take the last equation on the previous slide. We can solve it for f:
$$f_{4,6} = \left(\frac{(1 + r_{10,0})^{10}}{(1 + r_{6,0})^6}\right)^{1/4} - 1$$Forward Rate: %
Rearranging $(1 + r_{2,0})^2 = (1 + r_{1,0})(1 + f_{1,1})$ for $r_{0,2}$ affords:
$$r_{0,2}=\sqrt{(1 + r_{1,0})(1 + f_{1,1})} - 1$$Which shows the rate on an n year bond is the geometric average of the expected 1 year rates over the next n years.
Two-Year Rate: %
Bonds with a greater maturity generally have higher levels of interest rate risk. So it is more risky to buy a 2 year bond than buy a 1 year bond, and reinvest in another 1 year bond in 1 year. If both strategies earn the same yield, why would anyone buy the 2 year bond?
The Liquidity Premium theory says the 2 year bond should return more than successively investing in 1 year bonds. It doesn't say much however.
$$(1 + r_{2,0})^2 = (1 + r_{1,0})(1 + f_{1,1}) + LP_2$$where $LP_2$ is the liquidity premium on a 2 year bond.
Since interest rate risk increases in maturity, we have:
$$LP_1 < LP_2 < LP_3 < .... LP_n$$The theory gets its name because shorter-term bonds are also more liquid (they become cash sooner without loss of value) and thus require less of a premium.
The next slide contains a plot of the term structure by pure expectations, with an added liquidity premium.
Often, the yield curve exhibits a shape that isn't explainible by the pure expectation or liquidity premium theories, and this is where segmented markets fits in.
The idea is that certain investors provide and demand funds at separate points on the term structure. The term structure may behave differently at these various points as determined by the investors at each maturity.
Segmented markets may explain why points on the term structure may behave independently---the long end may increase with no change in short term yield for example.
Once we have the term structure, we can use it as a base rate to value corporate and other bonds. We adjust the term structure rate for the particular bonds: