.. slideconf:: :slide_classes: appear ============================================================================== The Term Structure of Interest Rates ============================================================================== Yield Curve ============================================================================== Bonds of different maturities often have different yields to maturity. .. rst-class:: to-build - The relationship between yield and maturity is summarized graphically in the *yield curve*. .. rst-class:: to-build - Consider several examples below. :math:`\qquad` .. ifslides:: .. rst-class:: to-build .. image:: /_static/TermStructure/bod34698_1012_lg.jpg :width: 6in :align: center .. ifnotslides:: .. image:: /_static/TermStructure/bod34698_1012_lg.jpg :width: 6in Yield Curve Slope ============================================================================== An upward sloping yield curve is evidence that short-term interest rates are going to rise. .. rst-class:: to-build - Consider two investment strategies. .. rst-class:: to-build - Buy and hold a two-year zero-coupon bond, offering 6\% return each year. .. rst-class:: to-build - Buy a one-year bond today, offering a 5\% return over the coming year, and roll the investment into another one-year zero-coupon bond a year from now, offering an interest rate of :math:`r_2`. .. rst-class:: to-build - These investments should be equivalent. Why? Yield Curve Slope ============================================================================== Suppose you begin with \$890 to invest (the price of a two-year zero-coupon bond with 6\% YTM. .. rst-class:: to-build - Equating the returns to each strategy gives: .. rst-class:: to-build .. math:: \$890 \times (1.06)^2 & = \$890 \times (1.05) \times (1+r_2) .. rst-class:: to-build .. math:: \Rightarrow 1 + r_2 & = \frac{1.06^2}{1.05} = 1.0701 .. rst-class:: to-build .. math:: \Rightarrow r_2 & = 0.0701. Two Investment Strategies ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/TermStructure/pg484_1.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/TermStructure/pg484_1.jpg :width: 6in Spot Rates and Short Rates ============================================================================== We distinguish between two types of interest rates. .. rst-class:: to-build - Spot rate: the rate offered *today* on zero-coupon bonds of different maturities. .. rst-class:: to-build - In the previous example, the one-year spot rate is 5\% and the two year spot rate is 6\%. .. rst-class:: to-build - Short rate: the rate for given time interval (one year) offered at different points in time. .. rst-class:: to-build - In the previous example, the first-year short rate is 5\% (same as the spot!) and the second-year short rate is 7.01\%. Spot Rates and Short Rates ============================================================================== The spot rate for a given period should be the geometric average of short rates over that interval. .. rst-class:: to-build - Let :math:`y_2` be the two-year spot rate. .. rst-class:: to-build - Let :math:`r_1` and :math:`r_2` be the first-year and second-year short rates. .. rst-class:: to-build - Don't forget that :math:`y_1 = r_1`. .. rst-class:: to-build .. math:: (1+y_2)^2 & = (1+r_1)\times(1+r_2) .. rst-class:: to-build .. math:: \Rightarrow 1+y_2 & = \sqrt{(1+r_1) \times (1+r_2)}. Spot Rates and Short Rates ============================================================================== So, if the yield curve slopes up (:math:`y_2 > y_1 = r_1`), we conclude that short-term rates will rise (:math:`r_2 > r_1`). .. rst-class:: to-build - Reverse reasoning holds for a downward sloping yield curve. Spot Rate and Short Rate Example ============================================================================== Assume the following spot rates and short rates: .. rst-class:: to-build - Spots: :math:`y_1 = 0.05`, :math:`y_2 = 0.06` and :math:`y_3 = 0.07`. .. rst-class:: to-build - Shorts: :math:`r_1 = y_1` and :math:`r_2 = 0.0701`. .. rst-class:: to-build - What is the three-year short rate, :math:`r_3`? Spot Rate and Short Rate Example ============================================================================== Buying a three-year zero-coupon bond should be identical to buying a two-year zero and rolling into a one-year zero. .. rst-class:: to-build .. math:: (1+y_3)^3 & = (1+y_2)^2 \times (1+r_3) .. rst-class:: to-build .. math:: \Rightarrow 1.07^3 & = 1.06^2 \times (1+r_3) .. rst-class:: to-build .. math:: \Rightarrow r_3 & = \frac{1.07^3}{1.06^2} - 1 = 0.09025. Spot Rate and Short Rate Example ============================================================================== We know .. rst-class:: to-build .. math:: (1+y_2)^2 & = (1+r_1) \times (1+r_2). .. rst-class:: to-build So the full decomposition is .. rst-class:: to-build .. math:: (1+y_3)^3 & = (1+y_2)^2 \times (1+r_3) .. rst-class:: to-build .. math:: & = (1+r_1) \times (1+r_2) \times (1+r_3) .. rst-class:: to-build .. math:: \Rightarrow 1.07^3 & = 1.05 \times 1.0701 \times 1.09025. Spot Rate and Short Rate Example ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/TermStructure/pg486_3.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/TermStructure/pg486_3.jpg :width: 6in General Short Rates ============================================================================== We can generalize the previous results. .. rst-class:: to-build - Investing in an :math:`n` period zero-coupon bond should be the same as investing in an :math:`n-1` zero and rolling into a one-period zero at time :math:`n-1`. .. rst-class:: to-build .. math:: (1+y_n)^n & = (1+y_{n-1})^{n-1} \times (1+r_n) .. rst-class:: to-build .. math:: \Rightarrow 1+r_n & = \frac{(1+y_n)^n}{(1+y_{n-1})^{n-1}}. Forward Rates ============================================================================== In the development above, we assumed no uncertainty. .. rst-class:: to-build - All future rates were known at time zero. .. rst-class:: to-build - In reality, we don't have perfect knowledge of time :math:`n` short rates at time zero. Forward Rates ============================================================================== To distinguish between actual short rates that occur in the future, we define the forward rate to be .. rst-class:: to-build .. math:: \Rightarrow 1+f_n & = \frac{(1+y_n)^n}{(1+y_{n-1})^{n-1}}. .. rst-class:: to-build - The time :math:`t=n` forward rate is the break-even interest rate that equates the returns of an n-period zero-coupon bond with an :math:`(n-1)` -period zero rolled into a one-period zero. .. rst-class:: to-build - It may not be equal to the expected future short rate. Expectations Hypothesis ============================================================================== The *Expectations Hypothesis* of the yield curve says that expected short rates equal forward rates: .. rst-class:: to-build .. math:: E[r_n] & = f_n .. rst-class:: to-build .. math:: \Rightarrow (1+y_n)^n & = (1+y_{n-1})^{n-1}(1+E[r_n]). .. rst-class:: to-build - If the yield curve slopes upward, short rates are expected to rise: :math:`E[r_n] > E[r_{n-1}] > r_1 = y_1`. .. rst-class:: to-build - If the yield curve slopes downward, short rates are expected to fall: :math:`E[r_n] < E[r_{n-1}] < r_1 = y_1`. Liquidity Preference Theory ============================================================================== According to the *Liquidity Preference Theory* of the yield curve, investors must be compensated for holding longer-term bonds. .. rst-class:: to-build - Longer-term bonds are subject to greater risk, and so investors should demand a premium for holding them. .. rst-class:: to-build - In reality, a *premium* means that investors will only buy them for a lower price (which means greater yield). Liquidity Preference Theory ============================================================================== The Liquidity Preference Theory can be expressed as forward rates being equal to expected short rates plus a premium, :math:`\phi`: .. rst-class:: to-build .. math:: f_n & = E[r_n] + \phi .. rst-class:: to-build .. math:: \Rightarrow (1+y_n)^n & = (1+y_{n-1})^{n-1}(1+E[r_n] + \phi). .. rst-class:: to-build - According to this theory, expected short rates *can be* constant if the yield curve is upward sloping. .. rst-class:: to-build - If the yield curve is downward sloping, expected short rates must be falling. Why? Liquidity Preference Example ============================================================================== Suppose you buy a two-year bond and that .. rst-class:: to-build - Short rates for the next two years are constant at 8\%: :math:`r_1 = E[r_2] = 0.08`. .. rst-class:: to-build - The liquidity premium for year two is 1\%: :math:`\phi = 0.01`. Liquidity Preference Example ============================================================================== What is the yield to maturity of the two year bond? .. rst-class:: to-build .. math:: (1+y_2)^2 & = (1+r_1)(1+f_2) \hspace{1.25in} .. rst-class:: to-build .. math:: & = (1+r_1)(1+E[r_2]+\phi) .. rst-class:: to-build .. math:: & = (1.08)(1.09) \hspace{0.8in} .. rst-class:: to-build .. math:: \Rightarrow 1+y_2 & = \sqrt{1.08 \times 1.09} \hspace{1.4in} .. rst-class:: to-build .. math:: & = 1.085. \hspace{1.2in} .. rst-class:: to-build - So the yield curve slopes up (:math:`y_2 > y_1`) even though expected short rates are constant. Expectations Hypothesis Example ============================================================================== However, if there is no liquidity premium .. rst-class:: to-build .. math:: (1+y_2)^2 & = (1+r_1)(1+f_2) \hspace{0.95in} .. rst-class:: to-build .. math:: & = (1+r_1)(1+E[r_2]) \hspace{0.02in} .. rst-class:: to-build .. math:: & = (1.08)(1.08) \hspace{0.5in} .. rst-class:: to-build .. math:: \Rightarrow 1+y_2 & = \sqrt{1.08 \times 1.08} \hspace{1.1in} .. rst-class:: to-build .. math:: & = 1.08. \hspace{1in} .. rst-class:: to-build - Now the yield curve is flat. Implications of the Theories ============================================================================== The slope of the yield curve *always* determines whether *forward rates* are rising or falling. .. rst-class:: to-build - :math:`y_2 > y_1` means :math:`f_2 > f_1` (by the definition of forward rates!). .. rst-class:: to-build - If the Expectations Hypothesis holds, :math:`E[r_2] = f_2`, so :math:`y_2 > y_1` means :math:`E[r_2] > r_1 = f_1 = y_1`. Implications of the Theories ============================================================================== - If the Liquidity Preference Theory holds, we have no guarantee that :math:`E[r_2] > r_1` if :math:`y_2 > y_1`. .. rst-class:: to-build - Short rates could be constant with a moderate liquidity premium. .. rst-class:: to-build - Short rates could be rising some, with a small liquidity premium. .. rst-class:: to-build - Short rates could be falling, with a large liquidity premium. .. rst-class:: to-build - What about if :math:`y_2 < y_1`? Exp. Hypothesis vs. Liquidity Pref Theory ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/TermStructure/bod34698_1014_lg.jpg :width: 4.8in :align: center .. ifnotslides:: .. image:: /_static/TermStructure/bod34698_1014_lg.jpg :width: 6in Historical Term Spread ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/TermStructure/bod34698_1015_lg.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/TermStructure/bod34698_1015_lg.jpg :width: 6in