Bond Valuation

The fundamental principle of bond valuation is that the bond's value is equal to
the present value of its expected (future) cash flows. The valuation process
involves the following three steps:

1. Estimate the expected cash flows.
2. Determine the appropriate interest rate or interest rates that should be used
to discount the cash flows.
3. Calculate the present value of the expected cash flows found in step one by
using the interest rate or interest rates determined in step two.

A bond's value is measured by its sale price, but a business can
estimate a bond's price before issuance by calculating its
present value.

 When calculating the present value of a bond, use the market rate as the
discount rate.
 Regardless of whether the bond is sold at a premium or discount, a company
must list a "bond payable" liability equal to the face value of the bond.
 If the MARKET rate is greater than the bond's contract rate, the bond will be
sold at a discount. If the MARKET rate is less than the bond's contract rate,
the bond will be sold at a premium.
TERMS
 contract rate
Another term for coupon rate, this is the amount of interest the business will
pay on the principal of the bond.
 market rate
The interest rate associated with other bonds that have a similar risk factor.
 market interest rate
the interest rate determined through various INVESTMENT systems, such
as the stock market or the bond market



Bond Valuation
A business must record a liability in its records when it issues a series of bonds.
The value of the liability the business will record must equal the amount of
money or goods it receives when it issues the bond. Whether the amount the
business will receive equals its face value depends on the difference between
the bond's contract rate and the MARKET rate of interest at the time the
bond is issued .



Determining Appropriate Interest Rates
The minimum interest rate that an investor should accept is the yield for a risk-
free bond (a Treasury bond for a U.S. investor). The Treasury security that is
most often used is the on-the-run issue because it reflects the latest yields and
is the most liquid.

For non-Treasury bonds, such as corporate bonds, the rate or yield that would
be required would be the on-the-run government security rate plus a premium
that accounts for the additional risks that come with non-Treasury bonds.

As for the maturity, an investor could just use the final maturity date of the
issue compared to the Treasury security. However, because each cash flow is
unique in its timing, it would be better to use the maturity that matches each of
the individual cash flows.

Computing a Bond's Value
First, we need to find the present value (PV) of the bond's future cash flows.
The present value is the amount that would have to be INVESTED today to
generate that future cash flow. PV is dependent on the timing of the cash flow
and the interest rate used to calculate the present value. To figure out the
value, the PV of each individual cash flow must be found. Then, just add the
figures together to determine the bond's price.



PV at time T = expected cash flows in period T / (1 + I) to
the T power

After you calculate the expected cash flows, you will need to add the individual
cash flows:




Value = present value @ T1 + present value @ T2 +
present value @T
n


Let's throw some numbers around to further illustrate this concept.

Example: The Value of a Bond
Bond GHJ matures in five years with a coupon rate of 7% and a maturity value
of $1,000. For simplicity's sake, let's assume that the bond pays annually and
the discount rate is 5%.

The cash flow for each of the years is as follows:
Year One = $70

Year Two = $70

Year Three = $70

Year Four = $70

Year Five = $1,070

Thus, the PV of the cash flows is as follows:

Year One = $70 / (1.05) to the 1
st
power = $66.67
Year Two = $70 / (1.05) to the 2
nd
power = $ 63.49
Year Three = $70 / (1.05) to the 3
rd
power = $ 60.47
Year Four = $70 / (1.05) to the 4
th
power = $ 57.59
Year Five = $1,070 / (1.05) to the 5
th
power = $ 838.37

Now to find the value of the bond:
Value = $66.67 + $63.49 + $60.47 + $57.59 + $838.37
Value = $1,086.59

How Does the Value of a Bond Change?

As rates increase or decrease, the discount rate that is used also changes. Let's
change the discount rate in the above example to 10% to see how it affects the
bond's value.

Example: The Value of a Bond when Discount Rates Change
PV of the cash flows is:
Year One = $70 / (1.10) to the 1
st
power = $ 63.63
Year Two = $70 / (1.10) to the 2
nd
power = $ 57.85
Year Three = $70 / (1.10) to the 3
rd
power = $ 52.63
Year Four = $70 / (1.10) to the 4
th
power = $ 47.81
Year Five = $1,070 / (1.10) to the 5
th
power = $ 664.60

Value = 63.63 + 57.85 + 52.63 + 47.81 + 664.60 = $ 886.52


 As we can see from the above examples, an important property of PV is
that for a given discount rate, the older a cash flow value is, the lower its
present value.
 We can also compute the change in value from an increase in the
discount rate used in our example. The change = $1,086.59 - $886.52 =
$200.07.
 Another property of PV is that the higher the discount rate, the lower the
value of a bond; the lower the discount rate, the higher the value of the
bond.
Look Out!
If the discount rate is higher than the coupon rate the PV
will be less than par. If the discount rate is lower than the
coupon rate, the PV will be higher than par value.

How Does a Bond's Price Change as it Approaches its Maturity Date?
As a bond moves closer to its maturity date, its price will move closer to par.
There are three possible scenarios:

1.If a bond is at a premium, the price will decline over time toward its par
value.
2. If a bond is at a discount, the price will increase over time toward its par
value.
3. If a bond is at par, its price will remain the same.

To show how this works, let's use our original example of the 7% bond, but now
let's assume that a year has passed and the discount rate remains the same at
5%.

Example: Price Changes Over Time
Let's compute the new value to see how the price moves closer to par. You
should also be able to see how the amount by which the bond price changes is
attributed to it being closer to its maturity date.

PV of the cash flows is:
Year One = $70 / (1.05) to the 1
st
power = $66.67
Year Two = $70 / (1.05) to the 2
nd
power = $ 63.49
Year Three = $70 / (1.05) to the 3
rd
power = $ 60.47
Year Four = $1,070 / (1.05) to the 4
th
power = $880.29


Value = $66.67 + $63.49 + $60.47 + $880.29 = $1,070.92

As the price of the bond decreases, it moves closer to its par value. The amount
of change attributed to the year's difference is $15.67.

An individual can also decompose the change that results when a bond
approaches its maturity date and the discount rate changes. This is
accomplished by first taking the net change in the price that reflects the change
in maturity, then adding it to the change in the discount rate. The two figures
should equal the overall change in the bond's price.

Computing the Value of a Zero-coupon Bond
A zero-coupon bond may be the easiest of securities to value because there is
only one cash flow - the maturity value.

The formula to calculate the value of a zero coupon bond that matures N years
from now is as follows:



Maturity value / (1 + I) to the power of the number of years
* 2
Where I is the semi-annual discount rate.

Example: The Value of a Zero-Coupon Bond
For illustration purposes, let's look at a zero coupon with a maturity of three
years and a maturity value of $1,000 discounted at 7%.

I = 0.035 (.07 / 2)
N = 3

Value of a Zero-Coupon Bond

= $1,000 / (1.035) to the 6
th
power (3*2)
= $1,000 / 1.229255
= $813.50

Arbitrage-free Valuation Approach
Under a traditional approach to valuing a bond, it is typical to view the security
as a single package of cash flows, discounting the entire issue with one discount
rate. Under thearbitrage-free valuation approach, the issue is instead viewed as
various zero-coupon bonds that should be valued individually and added
together to determine value. The reason this is the correct way to value a bond
is that it does not allow a risk-free profit to be generated by "stripping" the
security and selling the parts at a higher price than purchasing the security in
the MARKET .

As an example, a five-year bond that pays semi-annual interest would have 11
separate cash flows and would be valued using the appropriate yield on the
curve that matches its maturity. So the MARKETS implement this approach by
determining the theoretical rate the U.S. Treasury would have to pay on a zero-
coupon treasury for each maturity. The investor then determines the value of all
the different payments using the theoretical rate and adds them together. This
zero-coupon rate is the Treasury spot rate. The value of the bond based on
the spot rates is the arbitrage-free value.

Determining Whether a Bond Is Under or Over Valued
What you need to be able to do is value a bond like we have done before using
the more traditional method of applying one discount rate to the security. The
twist here, however, is that instead of using one rate, you will use whatever
rate the spot curve has that coordinates with the proper maturity. You will then
add the values up as you did previously to get the value of the bond.

You will then be given a MARKET price to compare to the value that you
derived from your work. If the market price is above your figure, then the bond
is undervalued and you should buy the issue. If the market price is below your
price, then the bond is overvalued and you should sell the issue.

How Bond Coupon Rates and MARKET Rates Affect Bond Price
If a bond's coupon rate is above the yield required by the market, the bond
will TRADE above its par value or at a premium. This will occur because
investors will be willing to pay a higher price to achieve the additional yield. As
investors continue to buy the bond, the yield will decrease until it
reaches MARKET equilibrium. Remember that as yields decrease, bond
prices rise.


 If a bond's coupon rate is below the yield required by the market, the
bond will TRADE below its par value or at a discount. This happens
because investors will not buy this bond at par when other issues are
offering higher coupon rates, so yields will have to increase, which means
the bond price will drop to induce investors to purchase these
bonds. Remember that as yields increase, bond prices fall.
http://www.learningmarkets.com/understanding-bond-yields/ (Video)
http://www.rbi.org.in/scripts/FAQView.aspx?Id=79

RISKS:

It is the risk that bond prices will fall as interest rates rise. By buying a bond, the bondholder has
committed to receiving a fixed rate of return for a fixed period. Should the market interest rate rise from
the date of the bond's purchase, the bond's price will fall accordingly. The bond will then be trading at a
discount to reflect the lower return that an investor will make on the bond.Market interest rates are a
function of several factors such as the demand for, and supply of, money in the
economy, the inflation rate, the stage that the business cycle is in as well as the
government's monetary and fiscal policies. However, interest rate risk is not the
only risk of INVESTING in bonds; fixed-income investments pose four
additional types of risk for investors:

Reinvestment Risk
The risk that the proceeds from a bond will be reinvested at a lower rate than
the bond originally provided. For example, imagine that an investor bought a
$1,000 bond that had an annual coupon of 12%. Each year the investor
receives $120 (12%*$1,000), which can be reinvested back into another bond.
But imagine that over time the MARKET rate falls to 1%. Suddenly, that $120
received from the bond can only be reinvested at 1%, instead of the 12% rate
of the original bond.
Call Risk
The risk that a bond will be called by its issuer. Callable bonds have call
provisions, which allow the bond issuer to purchase the bond back from the
bondholders and retire the issue. This is usually done when interest rates have
fallen substantially since the issue date. Call provisions allow the issuer to retire
the old, high-rate bonds and sell low-rate bonds in a bid to lower debt costs.

Default Risk
The risk that the bond's issuer will be unable to pay the contractual interest or
principal on the bond in a timely manner, or at all. Credit ratings services such
as Moody's,Standard & Poor's and Fitch give credit ratings to bond issues, which
helps to give investors an idea of how likely it is that a payment default will
occur. For example, most federal governments have very high
credit ratings (AAA); they can raise taxes or print MONEY to pay debts, making
default unlikely. However, small, emerging companies have some of the worst
credit (BB and lower). They are much more likely to default on their bond
payments, in which case bondholders will likely lose all or most of
their INVESTMENT .

Inflation Risk
The risk that the rate of price increases in the economy deteriorates the returns
associated with the bond. This has the greatest effect on fixed bonds, which
have a set interest rate from inception. For example, if an investor purchases a
5% fixed bond and then inflation rises to 10% a year, the bondholder will lose
money on the INVESTMENT because the purchasing power of the proceeds has
been greatly diminished. The interest rates of floating-rate bonds (floaters) are
adjusted periodically to match inflation rates, limiting investors' exposure to
inflation risk.

BOND YIELDS:

What is the relationship between yield and price of a bond?
If interest rates or MARKET yields rise, the price of a bond falls. Conversely, if interest rates or market
yields decline, the price of the bond rises. In other words, the yield of a bond is inversely related to its
price. The relationship between yield to maturity and coupon rate of bond may be stated as follows:
 When the MARKET price of the bond is less than the face value, i.e., the bond sells at a
discount, YTM > current yield > coupon yield.
 When the MARKET price of the bond is more than its face value, i.e., the bond sells at a
premium, coupon yield > current yield > YTM.
 When the MARKET price of the bond is equal to its face value, i.e., the bond sells at par, YTM
= current yield = coupon yield.
24. How is the yield of a bond calculated?
24.1 An investor who purchases a bond can expect to receive a return from one or more of the following
sources:
 The coupon interest payments made by the issuer;
 Any capital gain (or capital loss) when the bond is sold; and
 Income from reinvestment of the interest payments that is interest-on-interest.
The three yield measures commonly used by investors to measure the potential return from INVESTING
in a bond are briefly described below:
i) Coupon Yield
24.2 The coupon yield is simply the coupon payment as a percentage of the face value. Coupon yield
refers to nominal interest payable on a fixed income security like Government security. This is the fixed
return the Government (i.e., the issuer) commits to pay to the investor. Coupon yield thus does not reflect
the impact of interest rate movement and inflation on the nominal interest that the Government pays.
Coupon yield = Coupon Payment / Face Value
Illustration:
Coupon: 8.24
Face Value: Rs.100
MARKET Value: Rs.103.00
Coupon yield = 8.24/100 = 8.24%
ii) Current Yield
24.3 The current yield is simply the coupon payment as a percentage of the bond’s purchase price; in
other words, it is the return a holder of the bond gets against its purchase price which may be more or
less than the face value or the par value. The current yield does not take into account the reinvestment of
the interest income received periodically.
Current yield = (Annual coupon rate / Purchase price)X100
Illustration:
The current yield for a 10 year 8.24% coupon bond selling for Rs.103.00 per Rs.100 par value is
calculated below:
Annual coupon interest = 8.24% x Rs.100 = Rs.8.24
Current yield = (8.24/Rs.103)X100 = 8.00%
The current yield considers only the coupon interest and ignores other sources of return that will affect an
investor’s return.
iii) Yield to Maturity
24.4 Yield to Maturity (YTM) is the expected rate of return on a bond if it is held until its maturity. The
price of a bond is simply the sum of the present values of all its remaining cash flows. Present value is
calculated by discounting each cash flow at a rate; this rate is the YTM. Thus YTM is the discount rate
which equates the present value of the future cash flows from a bond to its current MARKET price. In
other words, it is the internal rate of return on the bond. The calculation of YTM involves a trial-and-error
procedure. A calculator or software can be used to obtain a bond’s yield-to-maturity easily (please see
the Box III).
Box III
YTM Calculation
YTM could be calculated manually as well as using functions in any standard spread sheet like MS Excel.
Manual (Trial and Error) Method
Manual or trial and error method is complicated because Government securities have many cash flows
running into future. This is explained by taking an example below.
Take a two year security bearing a coupon of 8% and a price of say Rs. 102 per face value of Rs. 100;
the YTM could be calculated by solving for ‘r’ below. Typically it involves trial and error by taking a value
for ‘r’ and solving the equation and if the right hand side is more than 102, take a higher value of ‘r’ and
solve again. Linear interpolation technique may also be used to find out exact ‘r’ once we have two ‘r’
values so that the price value is more than 102 for one and less than 102 for the other value.
102 = 4/(1+r/2)
1
+ 4/(1+r/2)
2
+ 4/(1+r/2)
3
+ 104/(1+r/2)
4

Spread Sheet Method using MS Excel
In the MS Excel programme, the following function could be used for calculating the yield of periodically
coupon paying securities, given the price.
YIELD (settlement,maturity,rate,price,redemption,frequency,basis)
Wherein;
Settlement is the security's settlement date. The security settlement date is the date on which the security
and FUNDS are exchanged.Maturity is the security's maturity date. The maturity date is the date when
the security expires.
Rate is the security's annual coupon rate.
Price is the security's price per Rs.100 face value.
Redemption is the security's redemption value per Rs.100 face value.
Frequency is the number of coupon payments per year. (2 for Government bonds in India)
Basis is the type of day count basis to use. (4 for Government bonds in India which uses 30/360 basis)