Bond Prices and YieldsBond Prices and Yields

Part 2 Part 2

Yield CurveYield Curve

• A curve can be developed that shows various annualized yields-to-maturity, y, against times-to-maturity, T, for bonds from the same issuing entity or from issuing entities with the same risk

y

0 T

2

Current US Treasury Yield CurveCurrent US Treasury Yield Curve

3www.treasury.gov

Bond Credit RatingBond Credit Rating

From Investopedia

4

Yield CurveYield Curve

• A corporate bond, say a AA rated bond, might have the following yield curve

y

0 T

AA Corporate Bond Yield Curve

U.S. Treasury Debt Yield Curve

risk premium or credit spread

5

Zero Coupon Yield CurveZero Coupon Yield Curve

• An especially useful version of the yield curve plots zero coupon bond yields against times-to-maturity

• A zero coupon yield curve depicts pure interest rates with no ambiguity due to coupon reinvestment risk

• These curves are not observable since no bonds with time to maturity greater than one year are issued, but can be constructed from coupon bond yield curves

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Liquidity RiskLiquidity Risk

• Liquidity risk results from a bond issue having few buyers and sellerso The issue is ‘illiquid’ or not ‘liquid’

• Older U.S. Treasuries (referred to as ‘off the run’) can trade with a liquidity rate premium compared with newer issued U.S. Treasuries (referred to as ‘on the run’)

y = f(risk free time value of money, credit risk, liquidity risk)

7

Price – Yield Curve Price – Yield Curve

F=$1000c=7% semiannualT=4.5 yrs

Illustrates how price changes as yield-to-maturity changes for a particular bond ( c, m, N, and F are constant)

Each (p,y) point results from a DCF

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Homework 13Homework 13

• Plot the price – yield curve for the following bondo Par value: $100o c = 3%o m = 2 o N = 5, T = 5.0o Plot P for yields, y, varying from .5% to 6.0%

• Chose a y increment that results in a smooth curve

• Submit a knitr pdf with echoed code, plot, and markdown descriptions

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Determine the Fair Price of a Bond Determine the Fair Price of a Bond

• In this case c, N, m and the zero coupon yield curve, zi, are known

• Compute the fair value, P

zi

0 Ti for zero coupon bonds

ti for bond cash flows

CFi Cash flow diagram

Zero coupon bond yield curve

10

P

Determine the Fair Price of a Bond Determine the Fair Price of a Bond

F=$1000c=7% semiannualT=4.5 yrs

With the following zero coupon yield curve

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Homework 14 Homework 14

• Calculate the fair price of the bond from the previous slide

• Print a table with cash flows, discount factors, and discounted cash flows

• Submit a knitr pdf with echoed code, discount factor plot, and markdown descriptions

12

Amortizing Bond or LoanAmortizing Bond or Loan

• Bond principal is repaid periodically, not all at maturity• Examples are home and automobile loans - amortizing loan

• Given the nominal annual interest rate, r, m, P, and N, what is the monthly payment, C?

• C : monthly paymento Includes principal repayment and interest

• N : number of years• m : number of compounding periods per year

(12 for home and auto loans)• r : nominal fixed interest rate for the loan• P : loan principal (the mortgage amount) • Solve for C using Excel Goal Seek, R uniroot, …

o Find the value of C that equates the left and right hand sides

13

Amortizing BondAmortizing Bond

14

F

F

C

Homework 15Homework 15

• You wish to borrow $300,000 at 6.5% annual fixed with monthly payments for 30 years

• What is your monthly payment? • What is your total payout over 30 years?• How much total interest will you pay?• Submit a knitr pdf with echoed code and

detailed markdown descriptions

15

FormulasFormulas

• Annuityo An annuity is a finite sequence of fixed payments, C. If

the nominal yield is y, then the present value, P, is

o The formula can be rearranged to compute the fixed payment, C, if the present value, P, is known

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FormulasFormulas

• Annuity Example o Home mortgage example

• $300,000 loan at 6.5% fixed rate compounded monthly for 30 years

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FormulasFormulas

• Bondso Annuity for coupon payment plus the discounted par

value

o Example: F=$1000, c=7% semi-annual, T=4.5 yrs, y (annual nominal yield) = 8%

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FormulasFormulas

• Bondso Bond with fractional initial period

19

d-e=139 e=227

previous coupon

next coupon

6/30/15 6/30/16 6/30/17 6/30/18 6/30/19 6/30/20 6/30/21

F = $1000c = 5%C= $50y = 6%e = 227 daysd = 366 daysN, M = 6m =1

Now11/16/201

5

CF+C

FormulasFormulas

• Bond with fractional initial period

20

d-e=139 e=227

previous coupon

next coupon

6/30/15 6/30/16 6/30/17 6/30/18 6/30/19 6/30/20 6/30/21

F = $1000c = 5%C= $50y = 6%e = 227 daysd = 366 daysN, M = 6m =1

Now11/16/201

5

CF+C

FormulasFormulas

• Bond with fractional initial period

21

d-e=139 e=227

previous coupon

next coupon

6/30/15 6/30/16 6/30/17 6/30/18 6/30/19 6/30/20 6/30/21

F = $1000c = 5%C= $50y = 6%e = 227 daysd = 366 daysN, M = 6m =1

Now11/16/201

5

CF+C