.. slideconf:: :slide_classes: appear ============================================================================== Bond Prices and Yields ============================================================================== Bond Basics ============================================================================== A bond is a financial asset used to facilitate borrowing and lending. .. rst-class:: to-build - A borrower has an obligation to make pre-specified payments to the lender on specific dates. .. rst-class:: to-build - Bonds are generally referred to as fixed-income securities. .. rst-class:: to-build - Historically, their payments have been fixed. .. rst-class:: to-build - In general, they can have variable payments, but those payments are determined by a formula. .. rst-class:: to-build - At the end of the bond's maturity, the borrower pays the *face value*, or *par value* of the bond to the lender. Coupons ============================================================================== Typical coupon bonds require the borrower to make semiannual coupon payments to the lender. .. rst-class:: to-build - The coupon rate is the total annual amount paid in coupons divided by the face value. .. rst-class:: to-build - Zero-coupon bonds pay no coupons - they simply pay the face value at maturity. .. rst-class:: to-build - The only way for such assets to be marketable is for their sale value to be below the face value. Bond Example ============================================================================== Suppose a semiannual coupon bond has a face value of $1000 and a coupon rate of 8%. .. rst-class:: to-build - The total amount of coupons paid annually is .. rst-class:: to-build .. math:: 0.08*\$1000 = \$80. .. rst-class:: to-build - Since coupons are paid semiannually, this amount is divided by two. .. rst-class:: to-build - So a $40 coupon is paid every six months and $1000 is paid at maturity. Government Bonds ============================================================================== U.S. Government debt assets fall into three categories. .. rst-class:: to-build - Treasury bills (T-bills). These pay no coupons and mature in one year or less from the time of issue. .. rst-class:: to-build - Treasury notes. These typically pay semiannual coupons and mature in 2 to 10 years from the time of issue (typical maturities are 2, 5 and 10 years). .. rst-class:: to-build - Treasury bonds. These typically pay semiannual coupons and mature between 10 to 30 years from the time of issue. Government Bonds ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Bonds/bod34698_1001_lg.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/Bonds/bod34698_1001_lg.jpg :width: 6in Corporate Bonds ============================================================================== U.S. corporations also issue bonds in order to borrow money. These debt instruments are generally quite similar to their government counterparts. .. math:: \qquad .. ifslides:: .. image:: /_static/Bonds/bod34698_1002_lg.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/Bonds/bod34698_1002_lg.jpg :width: 6in Treasury Inflation Protected Securities ============================================================================== Typical bonds offer nominal returns that don't account for inflation. .. rst-class:: to-build - TIPS are bonds whose face values (and hence their coupons) are indexed to the general level of prices. .. rst-class:: to-build - If inflation is equal to 2% over the course of a year, the face value of the bond will also rise by 2%. .. rst-class:: to-build - The result is a bond with no inflation risk and which should offer a rate of return that is equal to the real, risk-free rate. TIPS Example ============================================================================== Consider the following bond. .. rst-class:: to-build - Face value = $1000. .. rst-class:: to-build - Coupon rate = 4%. .. rst-class:: to-build - Annual coupon, with three years to maturity. .. rst-class:: to-build - Inflation will be 2%, 3% and 1% for the next three years. TIPS Example ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Bonds/table10_1_lg.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/Bonds/table10_1_lg.jpg :width: 6in TIPS Example ============================================================================== We can compute the nominal and real rates of return. .. rst-class:: to-build .. math:: r_{nom} & = \frac{\text{Interest} + \text{Price Appreciation}}{\text{Initial Price}} .. rst-class:: to-build .. math:: & = \frac{40.80 + 20}{1000} \qquad \qquad \enspace \; .. rst-class:: to-build .. math:: & = 0.0608 \qquad \qquad \quad \quad \; \, TIPS Example ============================================================================== .. math:: r_{real} & = \frac{1+\text{Nominal Return}}{1+\text{Inflation}} - 1 .. rst-class:: to-build .. math:: & = \frac{1.0608}{1.02} - 1 \qquad \quad \: .. rst-class:: to-build .. math:: & = 0.04. \qquad \qquad \quad \enspace TIPS Example ============================================================================== The real rate of return is simply equal to the coupon rate. .. rst-class:: to-build - Inflation has been eliminated since the bond's face value has implicitly incorporated it into the value by tying its face value to increases in prices. STRIPS ============================================================================== STRIPS stands for "Separate Trading of Registered Interest and Principle of Securities". .. rst-class:: to-build - It is a U.S. Treasury program which identifies bond payments as a separate securities. .. rst-class:: to-build - A coupon paying bond can be "stripped" into multiple assets - each coupon and the face value being marketed as individual zero-coupon bonds. .. rst-class:: to-build - A 10 year, semiannual bond is comprised of 20 coupon payments and a final payment of the face value. .. rst-class:: to-build - It could be stripped into 21 separate, zero-coupon assets with varying maturities. Discounting Cash Flows ============================================================================== A bond's value should be equal to the net present value of its cash flows. .. rst-class:: to-build - Each payment should be discounted by the product of interest rates over the period of interest. .. rst-class:: to-build - For a coupon payment in four years .. rst-class:: to-build .. math:: PV & = \frac{C}{(1+r_1)(1+r_2)(1+r_3)(1+r_4)}. .. rst-class:: to-build - :math:`r_i` is the interest rate over year :math:`i`. Discounting Cash Flows ============================================================================== Assuming the interest rate is constant over all periods (:math:`r_i = r, \forall i`) .. rst-class:: to-build .. math:: PV & = \frac{C}{(1+r)^4}. Geometric Series ============================================================================== The following mathematical result will be useful for deriving the value of a bond: .. rst-class:: to-build - For :math:`|x| < 1`, .. rst-class:: to-build .. math:: \sum_{i=0}^{\infty} a \, x^i & = \frac{a}{1-x}. Bond Pricing ============================================================================== Let :math:`F` = Face Value, :math:`C` = Coupon Payment and :math:`V` = Bond Value. .. rst-class:: to-build - Then, .. rst-class:: to-build .. math:: V & = \frac{C}{1+r} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \ldots + \frac{C}{(1+r)^T} + \frac{F}{(1+r)^T} .. rst-class:: to-build .. math:: & = \sum_{i=1}^T \frac{C}{(1+r)^i} + \frac{F}{(1+r)^T} \qquad \qquad \qquad \qquad \qquad \quad \enspace \, .. rst-class:: to-build .. math:: & = \sum_{i=1}^{\infty} \frac{C}{(1+r)^i} - \sum_{i=T+1}^{\infty} \frac{C}{(1+r)^i} + \frac{F}{(1+r)^T} \qquad \qquad \quad \enspace Bond Pricing ============================================================================== .. math:: & = \sum_{i=1}^{\infty} \frac{C}{(1+r)^i} - \frac{1}{(1+r)^T} \sum_{i=1}^{\infty} \frac{C}{(1+r)^i} + \frac{F}{(1+r)^T} \qquad .. rst-class:: to-build .. math:: \hspace{0.25in} & = \left(\sum_{i=1}^{\infty} \frac{C}{(1+r)^i}\right) \left(1 - \frac{1}{(1+r)^T}\right) + \frac{F}{(1+r)^T} \qquad \qquad \, .. rst-class:: to-build .. math:: & = \frac{C}{1+r} \left(\sum_{i=0}^{\infty} \frac{1}{(1+r)^i}\right) \left(1 - \frac{1}{(1+r)^T}\right) + \frac{F}{(1+r)^T} \, \, .. rst-class:: to-build .. math:: & = \frac{C}{1+r} \left(\frac{1}{1-\frac{1}{1+r}}\right) \left(1 - \frac{1}{(1+r)^T}\right) + \frac{F}{(1+r)^T} \qquad \, Bond Pricing ============================================================================== .. math:: & = \frac{C}{1+r} \left(\frac{1}{\frac{1+r-1}{1+r}}\right) \left(1 - \frac{1}{(1+r)^T}\right) + \frac{F}{(1+r)^T} \; \; .. rst-class:: to-build .. math:: \hspace{0.25in} & = \frac{C}{1+r} \frac{1+r}{r} \left(1 - \frac{1}{(1+r)^T}\right) + \frac{F}{(1+r)^T} \qquad \; .. rst-class:: to-build .. math:: & = \frac{C}{r} \left(1 - \frac{1}{(1+r)^T}\right) + \frac{F}{(1+r)^T}. \qquad \qquad .. rst-class:: to-build We made use of the geometric series result, recognizing :math:`0 < x = \frac{1}{1+r} < 1`. Bond Value Formula ============================================================================== The bond price formula can be decomposed as .. rst-class:: to-build .. math:: V & = \frac{C}{r} \left(1 - \frac{1}{(1+r)^T}\right) + \frac{F}{(1+r)^T} \qquad \qquad \quad \enspace :label: bondprice .. rst-class:: to-build .. math:: & = C \times \underbrace{\left[\frac{1}{r} \left(1 - \frac{1}{(1+r)^T}\right)\right]}_{\text{Annuity factor}(r,T)} + F \times \underbrace{\left[\frac{1}{(1+r)^T}\right]}_{\text{PV factor}(r,T)}. .. rst-class:: to-build - :math:`\text{Annuity factor}(r,T)` is the present value of a $1 annuity that lasts for :math:`T` periods with interest rate :math:`r`. .. rst-class:: to-build - :math:`\text{PV factor}(r,T)` is the present value of $1 paid in :math:`T` periods. Value vs. Price ============================================================================== So far we have only computed the *present value* of the bond. .. rst-class:: to-build - What is its price? .. rst-class:: to-build - The bond price should be equal to the present value of payments. .. rst-class:: to-build - If the price was lower, you could buy the bond for :math:`P < V` and receive the promised cash flow: your net gain would be :math:`V - P`. .. rst-class:: to-build - If the price was higher, you could short the bond for :math:`P > V` and pay the promised cash flow: your net gain would be :math:`P - V`. Bond Pricing Example ============================================================================== Consider a bond with the following characteristics. .. rst-class:: to-build - :math:`F = \$1000`. .. rst-class:: to-build - 30 years to maturity. .. rst-class:: to-build - 8% coupon rate. .. rst-class:: to-build - Semiannual coupons. .. rst-class:: to-build Suppose the nominal interest rate is constant at 8% for the next 30 years. Note that .. rst-class:: to-build - The coupon payment is $40 every six months. .. rst-class:: to-build - There are 60, six-month time periods. Bond Pricing Example ============================================================================== The bond price is .. rst-class:: to-build .. math:: P & = \frac{\$40}{0.04}\left(1 - \frac{1}{1.04^{60}}\right) + \frac{\$1000}{1.04^{60}} = \$1000. .. rst-class:: to-build - If the annual interest rate equals the coupon rate, :math:`P = F`. .. rst-class:: to-build If :math:`r = 0.10`, then .. rst-class:: to-build .. math:: P & = \frac{\$40}{0.05}\left(1 - \frac{1}{1.05^{60}}\right) + \frac{\$1000}{1.05^{60}} = \$810.71. Prices and Yields ============================================================================== The return for holding a bond is typically referred to as a *yield*. .. rst-class:: to-build - A central feature of bonds is that prices and yields are negatively related. .. rst-class:: to-build - Since a bond promises a fixed payment at maturity, you want to purchase for a very low price. Prices and Yields ============================================================================== Think of a zero-coupon bond that costs :math:`P` and pays :math:`F` at maturity. .. rst-class:: to-build - The yield, :math:`y`, is the value such that :math:`P \times (1+y) = F`: .. rst-class:: to-build .. math:: y & = \frac{F}{P} - 1. .. rst-class:: to-build - Note that :math:`y` is a net return (not gross). .. rst-class:: to-build - Since the bond price is in the denominator, :math:`y` and :math:`P` have an inverse relationship: higher prices mean lower yields. .. rst-class:: to-build - This relationship holds for coupon bonds as well (as seen in the bond pricing formula in Equation :eq:`bondprice`). Prices and Yields ============================================================================== Consider a zero-coupon bond with a face value of $1000 and one period until maturity. .. rst-class:: to-build - If :math:`P = \$990`, the yield is .. rst-class:: to-build .. math:: y & = \frac{\$1000}{\$990} - 1 = 0.01010. .. rst-class:: to-build - If :math:`P = \$995`, the yield is .. rst-class:: to-build .. math:: y & = \frac{\$1000}{\$995} - 1 = 0.005025. Coupon Rate and Yield ============================================================================== We saw above that if the market interest rate :math:`r` is equal to the coupon rate, :math:`P = F`. .. rst-class:: to-build - Consider two competing investments: put your money in a savings account that pays :math:`r` each period, or buy a bond. .. rst-class:: to-build - If the coupon rate is exactly equal to :math:`r`, then the bond exactly compensates you for market return and does not need to provide price appreciation. Coupon Rate and Yield ============================================================================== - If the coupon rate is less than :math:`r`, the bond does not compensate you enough - to be competitive, it must sell at a discount (less than the face value). .. rst-class:: to-build - If the coupon rate is greater than :math:`r`, the bond overcompensates you - to be competitive, it can sell at a premium (more than the face value). Coupon Rate and Yield ============================================================================== Why would a borrower sell a bond at a discount when :math:`r` is greater than the coupon rate? .. rst-class:: to-build - Because nobody would buy the bond unless the price formula holds. Coupon Rate and Yield ============================================================================== Later, the bond price will fluctuate on the secondary market. .. rst-class:: to-build - If the market rate rises, investors will sell the bond (because of its low coupon). .. rst-class:: to-build - The price will fall until the overall return is commensurate with the market return. .. rst-class:: to-build - If the market rate falls, investors will buy the bond (because of its high coupon). .. rst-class:: to-build - The price will rise until the overall return is commensurate with the market return. Interest Rate Risk ============================================================================== Unfortunately, interest rates don't remain constant. .. rst-class:: to-build - Suppose you purchase a bond for the face value when the coupon rate and market interest rates are equal. .. rst-class:: to-build - If the market rate suddenly rises, you will have your money tied up in an investment paying a rate that is too low (opportunity cost). .. rst-class:: to-build - If you sell the bond on the secondary market, the price will be lower than the face (your purchase price) because :math:`r` is greater than the coupon rate. Interest Rate Risk ============================================================================== The opposite would be true if the market rate falls. .. rst-class:: to-build - As a result, bond holders are subject to interest rate fluctuations, or interest rate risk. Maturity Risk ============================================================================== Interest rate risk is exacerbated over longer maturities. .. rst-class:: to-build - Suppose the market interest rate moves after you purchase two bonds with 10 and 30 years to maturity. .. rst-class:: to-build - If the market rate rises you will suffer a greater loss on the 30 year bond since your money is tied to it for a longer period. .. rst-class:: to-build - As a result, prices of longer maturity bonds are more sensitive to interest rate movements. Maturity Risk ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/Bonds/table10_2_lg.jpg :width: 8in :align: center .. ifnotslides:: .. image:: /_static/Bonds/table10_2_lg.jpg :width: 6in Bond Prices Over Time ============================================================================== As a bond approaches maturity, its price will approach the face value. .. rst-class:: to-build - Fewer payments are subject to interest rate fluctuations. .. rst-class:: to-build - This is true whether the bond sells at a premium or discount. Bond Prices Over Time ============================================================================== Consider buying a bond the day before maturity. .. rst-class:: to-build - If a coupon is paid at maturity, you only receive a prorated share of the coupon (one day's worth). .. rst-class:: to-build - Net of the prorated coupon, you should pay only the slightest discount to face value. .. rst-class:: to-build - The slight discount arises because you can invest your money overnight and earn a small amount of interest.