INVESTMENTS:
Analysis and Management
Third Canadian Edition
W. Sean Cleary
Charles P. Jones
Prepared by
Khalil Torabzadeh
University of Lethbridge
Chapter 11
Bond Yields and Prices
Learning Objectives
• Calculate the price of a bond.
• Explain the bond valuation process.
• Calculate major bond yield measures, including
yield to maturity, yield to call, and horizon return.
• Account for changes in bond prices.
• Explain and apply the concept of duration.
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Bond Valuation Principle
• Intrinsic value



Is an estimated value
Present value of the expected future cash flows
Required to compute intrinsic value
•
•
•
Expected future cash flows
Timing of expected cash flows
Discount rate, or required rate of return by
investors
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Bond Valuation
• Value of a coupon bond:
n
Ct
F
P

t
n
(1

r)
(1

r)
t 1
Where
P = The price of bond today (time period 0)
C = the regular coupons or interest
payments
F = the face value of the bond
n = the number of periods to maturity
r = the appropriate discount rate or market
yield
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Bond Valuation
• Biggest problem is determining the discount
rate or required yield, r
• Required yield is the current market rate earned
on comparable bonds with the same maturity
and credit risk
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Bond Yields
• The basic Components of Interest Rates:
o

Short-term riskless rate
Provides foundation for other rates
RF ≈ RR + EI
Where
RF = short-term T- bill rate
RR = the real risk-free rate of interest
EI = the expected rate of inflation
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Interest Rates
• Maturity differentials

Term structure of interest rates
•
Accounts for the relationship between time and
yield for bonds the same in every other respect
• Risk premium


Yield spread or yield differential
Associated with issuer’s particular situation
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Measures of Bond Yields
• They include:
o
o
o
o
Current Yield
Yield to Maturity
Yield to Call
Realized (horizon) Yield
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Current Yield
• Defined as the ratio of the coupon interest to
the current market price, C/P
• Uses the current market price instead of the
face amount of a bond ($1,000)
• Not a true measure of the return – does not
account for the difference between bond’s
purchase piece and eventual redemption at
par value
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Yield to Maturity
• Yield to maturity (YTM)



Rate of return on bonds most often quoted for
investors
Promised compound rate of return received
from a bond purchased at the current market
price and held to maturity
Equates the present value of the expected
future cash flows to the initial investment
•
Similar to internal rate of return
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Yield to Maturity
• Solve for YTM (semi-annual coupons):
C t /2
MV
P

t
2t
(1  YTM/2)
t 1 (1  YTM/2)
2n
• Investors earn the YTM if the bond is held to
maturity and all coupons are reinvested at
YTM
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Yield to Call
• Yield to a specified call date and call price
• Substitute number of periods until first call
date for and call price for face value (semiannual coupons)
• Applies to callable bonds
C t /2
CP
P

t
2c
(1  YTC/2)
t 1 (1  YTC/2)
2c
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Reinvestment Risk
For: (1) longer-term bonds
(2) bonds with higher coupon rates
(i.e., have more money to reinvest)
NO reinvestment risk for “Zeroes”
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Realized Yield
• Rate of return actually earned on a bond given
the reinvestment of the coupons at varying rates
• Can only be calculated after investment period is
over
• Horizon return analysis

Bond returns based on assumptions about
reinvestment rates
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Bond Price Changes
• Over time, bond prices that differ from face value
must change
• Bond prices move inversely to market yields
• The change in bond prices due to a yield
change is directly related to time to maturity
and inversely related to coupon rate
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Bond Price Changes
• Holding maturity constant, a
rate decrease will raise
prices a greater percent
than a corresponding
increase in rates will lower
prices
Market yield
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Measuring Bond Price Volatility:
Duration
• Important considerations



Different effects of yield changes on the prices
and rates of return for different bonds
Maturity inadequate measure of a bond’s
economic lifetime
A measure is needed that accounts for both size
and timing of cash flows
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Duration
• A measure of a bond’s lifetime, stated in years,
that accounts for the entire pattern (both size
and timing) of the cash flows over the life of the
bond
• The weighted average maturity of a bond’s cash
flows

Weights determined by present value of cash
flows
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Calculating Duration
• Need to time-weight present value of cash flows
from bond
PV(CFt )
D 
t
t 1Market Price
n
• Duration depends on three factors



Maturity of the bond
Coupon payments
Yield to maturity
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Duration Relationships
• Duration increases with time to maturity, but at
a decreasing rate


For coupon paying bonds, duration is always
less than maturity
For zero coupon-bonds, duration equals time to
maturity
• Duration increases with lower coupons
• Duration increases with lower yield to maturity
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Why is Duration Important?
• Allows comparison of effective lives of bonds
that differ in maturity, coupon
• Used in bond management strategies,
particularly immunization
• Measures bond price sensitivity to interest rate
movements, which is very important in any bond
analysis
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Estimating Price Changes Using
Duration
• Modified Duration = D* = D/(1+r)
• D* can be used to calculate the bond’s
percentage price change for a given change
in interest rates
• It works well for “small” changes in interest
rates and parallel shifts in the term structure
of interest rates.
-D
%  in bond price 
r
(1  r)
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Convexity
• Refers to the degree to which duration changes
as the yield to maturity changes

Price-yield relationship is convex
• Duration equation assumes a linear relationship
between price and yield
• Convexity largest for low coupon, long-maturity
bonds, and low yield to maturity
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Duration Conclusions
• To obtain maximum price volatility, investors
should choose bonds with the longest duration
• Duration is additive

Portfolio duration is just a weighted average of
each individual bond’s duration
• Duration measures volatility, which is one of the
aspect of risk in bonds
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Appendix 11A
Treasury Bill Yields and Prices
• T-bill are sold in Canada on a discount basis
rBEY
Face  P 365


 100
P
n
P
Face
(1  rBEY
n

)
365
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Appendix 11B Effective Duration
B  B
EffectiveDuration 
2 B0 (y )
• For option-free bonds

Effective Duration = Modified Duration
• For bonds with embedded options

Effective Duration ≠ Modified Duration
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Appendix 11B Effective Convexity
V  V  2V0
EffectiveConvexity 
2
2V0 (y )
• Percentage change in bond price
= Duration effect + Convexity effect
= (- Effective duration) (Δy) + (Convexity) (Δy)2
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Appendix 11C Convertible Bonds
• Convertible Bonds

Bonds that are convertible into a specified
number of C/S
• Terminology
Par Value (M)
Conversion Price (CP) 
Conversion Ratio (CR)
M
ie. How many shares per bond
So, CR 
CP
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Appendix 11C Convertible Bonds
Where “r” is the required rate of return on
identical (similar) non-convertible bonds.
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Appendix 11C Convertible Bonds
Conversion Premium =
Market price of convertible – Conversion value
Minimum (Floor) Value = Maximum (straight
bond value; conversion value)
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Why Issue/Buy Convertibles?
• Investors


income (interest) but at lower rate
participate in share price appreciation
• Firm



“lower” coupons
delayed equity financing (when share price
rises)
usually have call feature attached
Cleary Jones/Investments: Analysis and Management, 3rd Canadian Edition, Chapter 11
Copyright
Copyright © 2009 John Wiley & Sons Canada, Ltd. All rights
reserved. Reproduction or translation of this work beyond that
permitted by Access Copyright (The Canadian Copyright
Licensing Agency) is unlawful. Requests for further information
should be addressed to the Permissions Department, John Wiley
& Sons Canada, Ltd. The purchaser may make back-up copies
for his or her own use only and not for distribution or resale. The
author and the publisher assume no responsibility for errors,
omissions, or damages caused by the use of these programs or
from the use of the information contained herein.