Finance 3310
Lecture Notes
Lecture 6
I. Bond valuation
a) definition of terms
b) basic valuation model
c) yield to maturity
d) interest rate risk
e) miscellaneous
BONDS:
Definition of Terms
1. Bond - a long-term security which promises to
make fixed interest payments and a principal payment at maturity.
Also called:
debt security
fixed income security
2. Par value - the stated or face amount of bond
(usually $1,000 or $5,000)
3. Maturity date - date on which par amount is repaid
4. Coupon interest rate - stated or promised annual
rate of interest on the bond. Most bonds make semi-annual coupon
payments. The coupon payment in this case is the coupon rate x
par amount ÷ 2. The coupon rate simply defines the bond's cash
flows.
i) 0 coupon bond - As the name implies, these bonds have no coupon payments. They are created by investment banks who take government bonds and split the principal and interest into separate securities. (P only and I only in WSJ)
ii) contrast w/ market rate or yield to maturity
iii) market rate usually = coupon rate at issue date
5. call provisions - give the issuer the right to buy back the bond at a predetermined price prior to maturity. When are issuers likely to 'buy back' a previously issued bond?
Next we want to look at bond valuation. This task is made
easier by noting that the value of anything is simply the value
today of that anything's future cash flows. Thus, the value of a
bond is simply the present value of the bond's future cash flows.
Bond Valuation
A bond is just a stream of fixed cash
flows
* principal or par or face amount
* interest payments
0 --------I------- I-------- I-------- I------- I-------- I------ I, P
|______|______|______|______|______|______|______|
where I = coupon payment (coupon rate x par amount) and P =
par amount.
* The coupon interest payment is just an annuity
* The principal payment is a single cash flow.
* discount rate ==> Rd = investors' required return on the
bond; same as market rate or yield on bond.
Thus, we find the present value of an annuity and the present value of a single cash flow, and then just add the two together. Later, we will do it all at once on the calculator.
Mathematically:
Bond Value =
T7.4 Ex 1: Assume interest is paid annually:
What is cost of bond if i = 100, n = 20, P = 1000 & Rd =
10%?
i) PV(annuity) 100, 20, 10% = 851.36
ii) PV of $1,000 20 yrs from now = 148.64
iii) cost = value = 851.36 + 148.64 = $1,000
This bond is said to be 'selling at par' or selling 'flat.'
T7.5 Ex 2: Same bond as above except that now Rd = 12%.
i) PV(annuity) 100,20,12% = 746.94
ii) PV of 1,000 20 yrs from now = 103.66
iii) cost = value = 746.94 + 103.66 = 850.60
This bond is selling at a discount ==> cost <
par
Why?? The reason is, the bond is paying 10%, but the market
demands 12%. Stated somewhat differently, this bond is priced to
yield 12%. That is, if you pay 850.60 for the bond and rates do
not change, you would earn 12%. (prove this: hold the bond for
one year and then sell it, assuming rates remain at 12%. What is
the rate of return on your investment?)
T7.6 Ex 3: Same bond as above except that now Rd = 8%.
i) PV(annuity) 100,20,8% = 981.81
ii) PV of 1,000 in 20 yrs = 214.55
iii) cost = value = 1,196.36
Here, cost > par ===> bond selling at premium
Why? The bond is paying 10% but market only demands 8%. This bond is 'priced to yield' 8%.
Instead of finding the present value of an annuity and adding
to it the present value of a single cash flow, it turns out we
can do this all at once on our calculator. Consider the first
example above. We key into the calculator what we know (I, P, Rd
& n) and solve for what we don't know (PV).
FV = P = 1,000
Pmt = I = 80
I = Rd = 10%
n = 20
PV = ? (value of bond)
Suppose that instead of annual coupons, the bonds have
semi-annual coupons. Rework examples 1 through 3 assuming
semi-annual coupons. (1,000; 849.53; 1,197.93)
Summary of key points
1. If market rate, Rd, = coupon rate, then cost = par .
Bond is selling flat.
2. If market rate > coupon rate, then
==> cost < par
==> Bond is said to be selling at a discount
3. If market rate < coupon rate, then
==> cost > par
==> Bond is said to be selling at a premium
4. a) An increase in market rates ==> bond prices fall
(T7.7)
b) A decrease in market rates ==> bond prices rise
Why are bond prices and interest rates inversely related? Very
simply, because of the present value equation:
PV = CF/(1+R)n : As R increases, PV decreases
5. The market value approaches par over time.
Why? Because the bond pays the (promised) par amount at maturity.
bonds selling at a premium ==> mkt gets lower
bonds selling at a discount ==> mkt gets higher
The actual time path of bond prices could involve both
discounts and premiums depending on the movement in market
interest rates.
Zero coupon:
1. no interest payment
2. Par is future value
3. Sells at discount (why?) Also called 'deep discount' bonds
4. Cost is just PV of future par amount
Yield to Maturity (YTM)
The yield to maturity is:
1. The rate earned if bond is held to maturity.
2. The rate which discounts all future cash flows to their
current value (price)
3. The 'market' rate for the bond in question
Given the current price of the bond and the bond's expected
future cash flows (par & interest), what rate do you expect
to earn on the bond if you buy it at its current price and hold
it to maturity (and rates don't change again).
Technically:
price = 
We know the price, I, and P and want to find the Rd that
solves the above equation.
How do you find it?
a) trial and error
We know if the bond is selling at a discount ytm > coupon
And, if the bond is selling at a premium ytm < coupon
ex: Suppose the price of a bond which matures in 10 years with
a coupon of 8% is 875.00. What is YTM??
1. Since selling at a discount, YTM > coupon
2. Try 10% PV(I) = 531.42 , PV(P) = 385.54 -----> Cost = 916.63
3. Try a higher rate; say 11%
PV(I) = 471.14 PV(P) = 352.18 ----> Cost = 823.22
4. Now, try something in between; say 10.5%
PV(I) = 481.18 PV(P) = 368.44 ----> Cost = 849.63
5. Repeat; rate is between 10 and 10.5%
b) Calculator
Again, key in what you know and solve for what you don't know.
As an example, suppose the price of a 10% coupon (annual), 20
year maturity bond is $1,196.36. What is the YTM? (note: this is
example 3 above)
FV = 1,000
PV = (1,196.36)
n = 20
Pmt = 100 (10% x 1,000)
i = YTM = ? (8%)
Suppose the bond pays interest semi-annually. For example,
suppose the above bond pays interest semi-annually and the price
is 1,197.93. What is the YTM?
FV = 1,000
PV = 1,197.93
n = 20
Pmt = 100
i = ? (4%) ==> note: this is a semi-annual rate; there are two ways to convert it to an annual rate:
1) 4% x 2 = 8% APR (simple interest)
2) (1 + 4%)2 - 1 = 8.16% EAR (compound interest)
Interest Rate Risk
Before discussing interest rate risk, it will be useful to
introduce the concept of risk.. Risk, in general,
represents the chance that your actual or realized return turns
out to be much different than what you expected. (The nominal
yield on a Treasury Bill is considered riskless. Why?)
When you buy a bond, the YTM is your expected return if
the bond is held to maturity. If interest rates change, your
actual return will be different than the YTM. The same is true
even if the bond is not held to maturity. There are two sources
or kinds of interest rate risk: one is principal risk, the
other is reinvestment risk.
Principal risk - the risk that interest rates will rise and decrease the value of your bond
Reinvestment risk - the risk that interest rates
will drop so that you must reinvest interim cash flows at a lower
rate than the YTM (ytm assumes reinvestment of interest at YTM)
Summary Comments about Interest Rate Risk
1. Everything else equal, bonds with longer maturities are
more sensitive to changes interest rates. T7.9 , T7.10
2. Everything else equal, bonds with lower coupons are more
sensitive to changes in interest rates. (T7.10)
3. Notice that principal risk and reinvestment risk work in
opposite directions.
4. It may be possible that the two completely offset each
other. (immunization)
5. The 'measure' of this sensitivity is called duration.
6. Zero coupon bonds can be useful for eliminating interest
rate risk.
Other Characteristics of Debt
Debt is not an ownership interest in the firm and creditors
usually do not have voting power. Interest on bonds is deductible
for tax purposes. (Dividends to shareholders are not.) The
written agreement between the the corporation (borrower) and its
creditors is called an indenture. The indenture contains
basic terms of the bond (coupon, principal amount, repayment
times, etc.), a description of any property used as collateral,
any call provisions and protective covenants. Call provisions
give the issuer the right to repurchase part or all of the issue
prior to maturity. Protective covenants are legal
restrictions on corporate behavior that either specify things the
corporation must do (positive covenants) or things the
corporation may not do (negative covenants).
Another risk associated with bonds is default risk.
Default risk is the risk that borrowers fail to make all the
promised payments. Technical default can occur if the issuer
violates any of the debt covenants (contractual agreements) of
the bond. Rating agencies (S&P, Moody's) rate the default
risk of bonds. (see T7.13) The highest rated bonds are called
investment grade. Bonds rated below investment grade are called junk
bonds.
The biggest issuer of bonds is the U.S. government. State and
local governments also issue bonds. These bonds are municipal
bonds. The interest on municipal bonds is exempt from federal
taxes. Investors will compare after tax yields of taxable bonds
with yields on municipal bonds.
Inflation and Interest Rates
In General: R = r + ¼ + Rp
Where:
R = nominal rate of interest
r = Real Rate Of Interest (S & D Capital)
¼ = Expected Inflation Rate (to Protect Real)
Rp = Risk Premium (Reward For Bearing Risk)
Nominal Rate Of Return: This is the rate you already
know how to calculate. It is the percentage change in dollars.
1 + R = FV/PV
FV = PV * (1+R)
Real Rate Of Return: This is the percentage
change in goods.
Real Terms - Ie, In Terms Of Goods
1 + r =
=
1 + r =
Inflation =
1 + Inflation =
(1 + r) * (1 + ¼) = (1 + R) ==> R = r + ¼ + R¼
We drop the last term: ------> R = r + ¼ ( Fisher equation)
Suppose Nominal Did Not Include ¼:
Ex 1: R = 3%; R=3% Pt+1 = $1.06 Pt = $1.00
Start W/ $1.00; Can consume now or save. If save, will have
$1.00 X 1.03 = $1.03 in 1 year. What Is actual real rate of
return?
1 + r =
= 
===> R = - 3%
Ex. 2 (Cash)
What Is The Real Rate Of Return On Cash?
R = 0 ===> R = -¼
Determination Of Real Rate Of Interest (See
Irving Fisher, The Theory of Interest)
• Time Preference For Consumption
• Investment Opportunities
• Supply And Demand For Capital
Risk Premium. The risk premium may consist of the following
sources of risk:
• default risk
• liquidity risk
• interest rate risk (maturity risk)
Term Structure of Interest Rates
The term structure of interest rates is the relation between
YTM and time to maturity for a set of zero coupon bonds. The yield
curve is a graph of the of the term structure. (There is a
technical difference.)
We are interested in why the yield curve, or term structure,
has a particular shape. In other words, why is the yield curve
either upward or downward sloping? As stated previously, interest
rates consist of real rates, expected inflation and a risk
premium. We will consider US government bonds, which we assume to
be free of default-risk. In this case the risk premium is an interest
rate risk premium, reflecting that the longer the maturity
the greater is the interest rate risk. (This is called the
maturity risk premium in some books.) Expected inflation results
in nominal rates containing an inflation premium. The
inflation premium is intended to protect lenders' purchasing
power. Real rates, which are discussed above, do not move much
through time. Thus, changes in interest rates are primarily a
result of changes in expected inflation. The inflation premium
may rise or fall depending on expected future inflation. If
inflation were expected to remain constant, then longer-term
rates might still be higher because of the interest rate risk
premium.
Summarizing some of the previous discussion, interest rates on
bonds consist of the following 'pieces':
• Real rate
• Inflation risk premium
• Interest rate risk premium (maturity premium)
• Default risk premium
• Liquidity risk premium
