Bonds Prices and Yields

Bonds  Corporations and government entities can raise capital by selling bonds  Long term liability (accounting)  Debt capital (finance)  The bond has  Principal, par, or face value: F  Price: P  Yield: y (actually “yield to maturity” and the discount rate)  Maturity date, time to maturity, term, or tenor: T Date at which the bond principal, F, is returned to investors  In the case of a coupon bond (as opposed to a zero coupon bond) Coupon rate: c (annual, simple, nominal rate) Annual payment frequency: m; or period  t In the U.S. semiannual coupons is typical: m = 2 or  t =.5 2

Zero Coupon Bonds  ZCBs do not pay a coupon  The return and ‘yield’ (rate) is due to the purchase price at a discount to face value  U.S. Treasury bills (T – bills) are zero coupon bonds U.S. Treasury bills  Time-to-maturity at issue is 4, 13, 26, 52 weeks  Face value $100 to $5,000,000  A ZCB yield is the interest rate, and the discount rate denoted z 3 F P t=0 t=T

Zero Coupon Bond  For T ≤ 1 year: where z is the annual simple rate or yield  For T > 1 year where z is the annualized effective rate or yield If a bond has a term of a year or less, simple interest is used, otherwise compound annual interest is used by convention 4 F P t=0 t=T

Zero Coupon Bond Example  The face value is $1000, the market price is $850, and the time to maturity is 3.5 years. What is the annualized yield ?  The face value is $1000, the market price is $975, and the time-to- maturity is 0.5 years. What is the annualized yield? 5

Coupon Bond P = current price C = coupon payment F = face or par value t=0.0 t=  t t=2∙  t t=M∙  t=T i=0 i=1 i=2 i=M t 0 =0.0 t 1 =  t t 2 =2  t t M = M∙  t =T 6

Coupon Payment  Bond coupon cash flows, C, are defined by a nominal, simple coupon rate, c, and a compounding frequency per year, m, or coupon period measured in years,  t  The total cash flow at time t i, CF i, is defined as: 7 T=num of years (floating) N=num of years (integer) m=periods per year In this course, generally M=N  m 360= 30  12

Coupon Bond Yield  Yield to maturity is the actual yield achieved for a coupon bond if  The bond is held to maturity, and  Each coupon payment is reinvested at a rate of return of y through time T The risk that coupons cannot be reinvented at a rate greater than or equal to y due to market conditions is called “reinvestment risk”  The yield to maturity is the investor’s expected return on investment and is thus the issuer’s rate cost  It’s the issuer’s cost of debt, k D, for the bond  The yield reflects both the time value of money and the credit worthiness of the borrower  The expected variance in the cash flow is reflected in the yield 8

Bond Price  The discount rate y is the yield to maturity or simply the yield on a coupon bond  It’s an internal rate of return that sets the discounted cash flow on the right hand side to the market price of the bond, P, on the left hand side 9 y is the nominal annual yield to maturity in this formula with integer periods y is effective annual yield to maturity in this formula with discrete real time periods

 For a fractional initial coupon period: t 1 < ∆t Fractional Initial Time Period For a bond with semi-annual coupons, assume that the next coupon payment is in 3 months. The coupon payments occur at t 0 =0.0, t 1 =0.25, t 2 =0.75, t 3 =1.25, t 4 = 1.75, … i=0 i=1 i=2 i=M t 0 =0.0 t 1 t 2 =t 1 +  t t M = T C = coupon payment F = face or par value 10

Zero Coupon Bonds Again  A bond dealer can split a coupon bond into ZCBs  one for the principal and  one for each coupon  This is called ‘stripping’ the bond  The advantage of a ZCB is that there is no reinvestment risk  For a ZCB, the yield, y, is the zero coupon rate denoted as z 11

Bond Equation Applications  Find the yield-to-maturity, y, from a known market price, P  Solve for y (nominal, y, or effective, y ‘bar’)  Solve for the roots of a nonlinear equation In this course use Excel Goal Seek  Example: Compute both the effective and nominal yield for a bond with $1000 face value, current market price of $800, coupon rate of 7% paid semiannually, and 4.5 years to maturity. 12

Bond Equation Applications 13

Bond Equation Applications  Convert the nominal yield to the effective yield  Find market price from a known yield  For the bond in the last example, what is the price? Given an effective annual yield of 12% or A nominal annual yield of 12% 14

Bond Equation Applications 15

Bond Equation Applications  For the bond with a 12% effective yield and price $ at time 0, here’s a plot of price as time progress from 0 to 4.5 years assuming a constant yield of 12% 16

Corporate Credit Rating From Investopedia 17 AAA companies

Reinvestment Risk  Consider a $1000 bond with a coupon rate of 10% paid annually for 10 years. Initially, the yield is 11%, the price is $941.11, and the yield curve is flat. Prior to the payment of the next coupon, we consider three scenarios 1. the yield curve shifts parallel down to 9% 2. the yield curve remains flat at 11% 3. the yield curve shifts parallel up to 12% What are the actual yields? 18

Plot price v. yields to maturity F=$1000 c=7% semiannual T=4.5 yrs Bond “price – yield” or P-y curve Illustrates how price changes as yield-to-maturity changes for a particular bond ( c, m, M, and F are constant) Each point represents a DCF calculation 19

Home Mortgage Calculation  Given the nominal interest rate, m=12, P, and N, what is the monthly payment, C?  C : monthly payment  Includes principal repayment and interest – there is no return of principal “F”  N : number of years  m : number of compounding periods per year (12 for home loans)  y : nominal fixed interest rate for the loannominal fixed interest rate for the loan  P : loan principal (the mortgage amount)  Solve for C using Excel Goal Seek  Find the value of C that equates the left and right hand sides 20

Mortgage Example  You wish to borrow $300,000 at 6.5% fixed for 30 years.  The following excel table shows the calculations for the first 12 months and the last 5 months.  The monthly payment of $1896 is determined using goal seek to force the sum of the last column to $300,000.  Note that you will pay out $682,633 in principal and interest  $300,000 in principal  $382,633 in interest 21

Mortgage Example 22

Perpetuity 23 Now in the case that M=∞ C is constant and of course y < 1 This is a perpetuity If a nominal annual rate, y, is used then P C i Example: How much money do you need to invest, P, to pay out $1 per year forever if the pay out rate is 10% (effective) per year?

Annuity 24 Now how much money do you need to invest at 10% to receive a $1 / year payout for M years ? That’s an annuity (a perpetuity would pay out forever) P C i M M+1 Annuity: Present Value Annuity: Payout

Annuity 25 Now how much money do you need to invest at 10% to receive a $1 / year payout for M years ? That’s an annuity (a perpetuity would pay out forever) M=20 years C=$1 Y=10% P=$8.51

Annuities 26

Closed Form Formulas  Annuity  Home mortgage annuity formula example  Bonds  Annuity for coupon payment plus the discounted face value 27

Closed Form Formulas  Bonds  Example of bond w/ F=$1000, c=7% semi-annual, T=4.5yrs, y annual nominal = %  Bond with fractional initial period 28

Closed Form Formulas last coupon next coupon e=64 days d = 365 days e/d=.175 8/15/08 8/15/09 8/15/10 8/15/11 8/15/12 8/15/13 8/15/14 6/12/09 F=$100 y=4% annual c=5% annual y & c are effective & nominal Clean and Dirty Price example (p. 7.10) using closed form 29