Download bonds and their valuation

Chapter
8
VALUATION of
securities
limitations
• Valuation is not an exact science
• Consideration governing share valuation are
varied and numerous
BONDS AND
THEIR
VALUATION
Introduction

Assets can be real or financial; securities like shares and bonds are called
financial assets while physical assets like plant and machinery are called
real assets.

The concepts of return and risk, as the determinants of value, are as
fundamental and valid to the valuation of securities as to that of physical
assets.
BONDS AND
The Basic Valuation Model
THEIR
VALUATION
1. The Basic Valuation Model: The value of an security is the present value of all future
cashflows associated with it over the specified period. The expected returns are
discounted, using the required return matching with the risk of asset as the
appropriate discount rate. Symbolically,
•
Vo=
A1
A2
-------- + -------- +
(1+k) 1
(1+k) 2
An
----- + -------(1+k) n
• Where Vo = Value of security at time zero (t = 0)
• At = cash flow stream expected at the end of year t
• K = appropriate discount rate
• n=life of the asset
• Alternatively, where expected cash flows is a mixed stream
• V = [ ( A1 x PVIFk,1 ) + ( A2 x PVIFk,2 ) +- - - - - + ( An x PVIFk,n)
Where
PVIF1, PVIF2, PVIFn = present value interest factor in different period at discount rate k.
• If expected cash flow is an Annuity,
V = A * PVIFA (k,n)
Illustration 6: Assuming a discount rate of 10 percent, and the associate
d cash flows detailed
below. Compute the value of assets X and Y.
• Year
Expected cash flow
X
• 1
• 2
• 3
Rs.10,000
10,000
10,000
Y
5,000
10,000
15,000
Solution:
• Value of asset X = Rs 10,000 x PVIFA (10%,3) = Rs 10,000 x 2.4870 = Rs.
24,870
• Value of asset Y: = [(Rs.5, 000 x PVIF10%,1 ) + (Rs. 10,000 xPVIF10%,2 ) + (
Rs. 15,000 x PVIF10%,3)
= [(Rs.5, 000 x 0.909) + (Rs. 10,000 x 0.826) + (Rs. 15,000 x 0.751)
= Rs.4545+ RS.8260+Rs.11265 = Rs. 24,070
Valuation of Bonds / Debentures
A bond / debenture are a long term debt instrument used by the
government/ business/ enterprises to raise a large sum of money.
• Par value: face value
• Coupon rate and interest: Coupon is the specified interest rate.
The interest payable to the bondholder is equal to:
(par value x coupon rate).
• Maturity period: corporate bonds 3-10 years. Govt. bond have
maturity upto 20-25 years.
Most bonds
(i) pay interest half yearly at a stated coupon interest rate,
(ii) have a maturity of 10years(Maturity period refers to the number of
years after which the par value is payable to the bondholder).
(iii) Have a par
/face value of Rs 1,000 that must be repaid at maturity. (Par value i
s the value on the face of
the bond. It represents the amount the entity borrows and promises to
repay at the time of maturity).
A Basic Bond Valuation:
Value of bond is the present value of the contractual payments
its issuer is obliged to make from the beginning till maturity.
The appropriate discount ratewould be the required return matching with risk and
the prevailing interest rate.
Symbolically,
• B = I x (PVIFAkd n) + M x (PVIFkd n)
• Where,
B = value of the bond at t = 0
I = annual interest paid
n = maturity period of the bond(term of the bond)
M = Par/maturity value
Kd = required return on the bond
Illustration 7: A firm has issued 12% coupon
rate, 8year bond with a Rs, 100 par value, that pays
interest annually. The required rate of return is 14%. Compute the value of bond.
Solution:
Bo = [Rs 12 x (PVlFA14%, 8) + Rs 100 (PVlF14%, 8)]
= (Rs 12 x 4.639) + (Rs 100 x 0.351)
= Rs 90.77
Impact of required Return (RR) on Bond Value
•When the
- required Return (RR) =the coupon rate (CR) (the bond value equals the par value)
•- (RR) > (CR) , the bond value would be less than its par value, that is,
the bond would sell at a discount equal to (M-B)
•- (RR) <(CR) , the bond value would be more than its par value, that is,
the bond would sell at a premium equals to (B-M)
Yield to Maturity
The YTM is the rate of return that investors earn if they buy a bond at a
specific price and hold it until maturity.
Formula:
B = I x (PVIFAkd,n) + M x (PVIFkd,n)
• Illustration:The bonds of the Premier Company Ltd (PCL) are currently selling
at Rs.10, 800.
Assuming (i) coupon rate of interest, 10 per cent, (ii) par value, Rs 10,000,
(iii) maturity 10 years and (iv) annual interest payment, compute the YTM or what rate
of interest would an investor earn if he holds it till its maturity?
•
Solution: Substituting the values in following Equation
B = I x (PVIFAkd,n) + M x (PVIFkd,n)
Rs 10,800 = [Rs 1,000 x (PVIFAkd, 10) + Rs 10,000 x (PVIFkd, 10)
If kd= 10 per cent, that is, equal to' the coupon rate, the value of the bond would be Rs 10,000.
Since the value of the bond is Rs 10,800, the kd must be less than 10 per cent.
Using 9 per cent discount rate gets
= [Rs 1,000 x (PVIFA 9,10) + Rs 10,000 x (PVIF 9,10)
= (Rs 1,000 x 6.418) + (Rs 10,000 x 0.422) = Rs 6,418 + Rs 4,220 = Rs 10,638
Since the value of the bond (Rs 10,638) at kd = 9 per cent is less than Rs 10,800 (current market
price). Try a lower rate of discount (kd). Using 8 per cent, we get
(Rs 1,000 x 6.710) + (Rs 10,000 x 0.463)
= Rs 6,710 + Rs 4,630 = Rs 11,340
Since the bond value (Rs 11,340) is higher than the current price of Rs 10,800, the kd (YI'M)
between 8 and 9 per cent. The exact value can be found by interpolation, which is 8.77% `
2. Semi annual Interest and Bond Values:
The procedure to value bonds paying interest semiannually(half
yearly) is similar to that for compounding interest more frequently than
annually. However, here we find out the present value.
• B = I x (PVIFAkd/2, 2n) + M x (PVIFkd/2, 2n)
2
•
•
•
Illustration 10: For facts in illustration4, assume (i) the bonds of the firm pay interest
semiannually, (ii) the required stated return is 14 per cent for similar-risk bonds that also pays
half-yearly interest. Compute the value of bond.
•
•
•
Solution: Substituting the values in following Equation we get
B = I x (PVIFAkd/2, 2n) + M x (PVIFkd/2, 2n)
2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
B = (Rs 1,000 / 2) x [PVlFA14I2 x 2; 10] + Rs 10,000 x [PVlF14I2 x 2; 10]
= (Rs 1,000 / 2) x [PVlFA7, 20] + Rs 10,000 x [PVlF7, 20]
= (Rs 500 x 10.594) + (Rs 1,000 x 0.258)
= Rs 5,297 + Rs 2,580 = Rs 7,877
3. Valuation of Preference Shares: Preference shares like debentures are usually subject to
fixed rate of return/dividend. In case of no stated maturity, their valuation is similar to perpetual
bonds. Symbolically,
V
=
Dp
Kp
The valuation of redeemable preference shares is given by following equation
= Dp( PVIFAkp,n) + MV( PVIFpv,n)
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
4. Valuation of Ordinary Shares: The ordinary / Equity shareholders buy / hold shares in
expectation of periodic cash dividends and increasing share value. They would buy a share' when
it is undervalued (i.e. its true value is more than its market price) and sell it when its market price
is more than its true value (i.e. it is overvalued). The value of a share is equal to the present value
of all future dividends it is expected to provide over an infinite time horizon. Symbolically,
P
=
D1
+
D2
+- - - - - +
2
Where ,
P = Value of shares
Dt = per share dividend expected at the end of year, t
Ke = required return on share
D∞
The equation is designed to compute the value of shares with reference to the expected growth
pattern of future dividends and the appropriate discount rate. We illustrate below the
computation reference to (i) zero growth, (ii) constant growth and (iii) variable growth.
BONDS AND
THEIR
VALUATION
Valuation – Three Major Techniques

Asset Valuation Approach: Asset side of the Balance Sheet

Income Valuation Approach: Profit and Loss Statement

Market Multiple Valuation Approach: Liability side of the Balance Sheet
Valuation – Asset Based Approach

Company valuation as a special case of asset valuation

Analyze the “Asset” side of the balance sheet

Assets are where company has already spent the money and assets will
give cash flows in future

Company’s value depends upon the size and reliability of these future
cash flows
BONDS AND
THEIR
VALUATION
Occasions for Valuation








Equity analysis
Merger and Acquisition
Employment (when ESOPs are involved)
IPO, restructuring, divestiture
Exit of a Joint-Venture partner
Equity investment (VC Financing)
Loan decision by a banker (financial health estimation and default
probability estimation)
Management Buyout, internal share transfer
Company Valuation
BONDS AND
THEIR
VALUATION
Type of Companies being Valued

Listed companies – market price (consensus of active traders, quarterly
statements, guidance, forecasts)

Companies with assets which are mostly physical – depreciation,
appreciation, obsolescence

Examples – transport operator, mechanical ancillary units
Company Valuation
BONDS AND
THEIR
VALUATION
Type of Companies being Valued
Knowledge-based companies have huge intangible assets

Patents, trademarks, Copyright (Software source code)

Processes, quality and development methodology,

Goodwill, brand, customer base, relationships

Team (education, experience, skills)
Company Valuation
BONDS AND
THEIR
VALUATION
Difficulties in Valuing Early Stage Companies






Immediate earnings are negative
No past history
No comparable companies
No market prices
Asymmetric information
Management efficacy yet to be established
Company Valuation
BONDS AND
Bonds and Their Valuation
THEIR
VALUATION
Buying and selling pressures dominantly originate with active investors. And
they follow certain rules of the game which are



Rule 1 : Buy when value is more than price. This underlines the fact that
shares are under priced and it was to be a bargain to buy now and sell
when prices move up towards value.
Rule 2 : Sell when value is less than price. In a situation like this, shares
would be overpriced and it would advantageous to sell them now and
avoid less when price later moves down to the level of the value.
Rule 3 : Don’t trade when value is equal to price. This is a state when the
market price is an equilibrium is not expected to change.
BONDS AND
Valuation of Fixed Income Securities:
THEIR
VALUATION
A debenture is a legal document containing an acknowledgement of
indebtedness by a company. It contains a promise to pay a stated rate of
interest for a defined period and then to repay the principal at a given date of
maturity.

In short, a debenture is a formal legal evidence of debt and is termed as
the senior securities of a company.
BONDS AND
THEIR
VALUATION
Features of a Bond





Face Value
Interest Rate—fixed or floating
Maturity
Redemption value
Market Value
BONDS AND
REASONS FOR ISSUING BONDS





THEIR
VALUATION
To Reduce the Cost of Capital: Bonds are the cheapest source of
financing.
To Widen the Sources of Funds: By issuing bonds, the corporation can
attract funds from individual investors and especially from those investing
institutions which are reluctant or not permitted to purchase equity shares.
To Preserve Control: An increase in debt does not diminish the voting
power of present owners since bonds ordinarily carry no voting right.
To Gain the Benefit of Leverage: The presence of debt and / or
preference shares in the company's financial structure means that it is
using financial leverage. When financial leverage is used, changes in
earnings before interest and tax (EBIT) translate into the larger changes in
earnings per share.
To Effect Tax Saving: Unlike dividends on equity, the interest on bonds is
deductible in figuring up corporate income for tax purposes.
BONDS AND
THEIR
VALUATION
TYPES OF BONDS








Convertible and Non-Convertible Bonds: Convertible bonds can be one of the finest holdings for the
investor looking for both appreciation of investment and income of bond.
Collateral Trust Bonds: Instead of being secured by a pledge of tangible property, as are mortgage bonds,
collateral trust issues are secured by a pledge of intangibles, usually in the form of stocks and bonds of
corporation.
Income Bonds: Income bonds are bonds on which the payment of interest is mandatory only to the extent of
current earnings. If earnings are sufficient to pay only a portion of the interest, that portion usually is required to
be paid, but if the corporation is able to pay the unearned balance out of its cash resources, it is of course free
to do so.
Redeeemable and Irredeemable Bonds: A redeemable debenture is a bond which has been issued for a
certain period on the expiry of which its holder will be repaid the amount thereof with or without premium.
Participating Bonds: Companies with poor credit positions issue participating bonds. They have a guaranteed
rate of interest but may also participate in earnings up to an additional specified percentage.
Sinking Fund Bonds: Sinking fund bonds arise when the company decides to retire its bond issue systematically by setting aside a certain amount each year for the purpose. The payment, usually fixed annual rupees
amount or percentage instalment, is made to the sinking fund agent who is usually the trustee.
Serial Bonds: Like sinking fund bonds, serial bonds are not special types of bonds but just names given to
describe the method of repayment. Thus, any bond can be such by merely specifying it in the indenture.
Mortgage or Secured Bonds: The term mortgage generally refers to a lien on real property or buildings.
Mortgage bonds may be open-end, close-end, and limited open-end.
BONDS AND
RISK MANAGEMENT IN BONDS
THEIR
VALUATION

Default Risk: Default risk identifies the uncertainty prevalent in the
repayment of interest and principal to the bondholders.

Purchasing Power Risk: Debt instrument investors have to look at the
real rate of return, or the actual return minus the rate of inflation.
Price Risk: Investors who need their principal prior to maturity have to
rely on the available market for the securities.
Liquidity Risk: The exchange listing of debt securities does not
guarantee liquidity.
Reinvestment Risk: The maturity period of bonds are spread over a fixed
time duration.



BONDS AND
THEIR
VALUATION
Bonds Values and Yields

Bonds with maturity

Pure discount bonds

Perpetual bonds
BONDS AND
2ND
THEIR
VALUATION
Bond with Maturity
Bond value = Present value of interest + Present value of maturity value:
n
B0  
t 1
INTt
Bn

(1  kd )t (1  kd ) n
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Excel Books
BONDS AND
THEIR
VALUATION
Yield to Maturity

The yield-to-maturity (YTM) is the measure of a bond’s rate of return that
considers both the interest income and any capital gain or loss. YTM is
bond’s internal rate of return.

A perpetual bond’s yield-to-maturity:
n 
B0  
t 1
INT
INT

(1  kd )t
kd
BONDS AND
THEIR
VALUATION
Current Yield

Current yield is the annual interest divided by the bond’s current value.

Example: The annual interest is Rs. 60 on the current investment of Rs.
883.40. Therefore, the current rate of return or the current yield is:
60/883.40 = 6.8 per cent.

Current yield does not account for the capital gain or loss.
BONDS AND
THEIR
VALUATION
Yield to Call

For calculating the yield to call, the call period would be different from the
maturity period and the call (or redemption) value could be different from
the maturity value.

Example: Suppose the 10% 10-year Rs 1,000 bond is redeemable
(callable) in 5 years at a call price of Rs 1,050. The bond is currently
selling for Rs 950.The bond’s yield to call is 12.7%.
5
950  
100
t 1 1  YTC 
t

1,050
1  YTC 
5
BONDS AND
THEIR
VALUATION
Bond Value and Amortisation of Principal


A bond (debenture) may be amortised every year, i.e., repayment of principal
every year rather at maturity.
The formula for determining the value of a bond or debenture that is
amortised every year, can be written as follows:
n
CFt
B0  
t
t 1 (1  k d )

Note that cash flow, CF, includes both the interest and repayment of
the principal.
BONDS AND
THEIR
VALUATION
Pure Discount Bonds

Pure discount bond do not carry an explicit rate of interest. It provides
for the payment of a lump sum amount at a future date in exchange for
the current price of the bond. The difference between the face value of the
bond and its purchase price gives the return or YTM to the investor.
BONDS AND
2ND
THEIR
VALUATION
Pure Discount Bonds

Example: A company may issue a pure discount bond of Rs 1,000 face
value for Rs 520 today for a period of five years. The rate of interest can be
calculated as follows:
520 
1, 000
1  YTM 
5
1, 000
 1.9231
1  YTM  
520
i  1.92311/ 5  1  0.14 or 14%
5
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Excel Books
BONDS AND
THEIR
VALUATION
Pure Discount Bonds

Pure discount bonds are called deep-discount bonds or zero-interest
bonds or zero-coupon bonds.

The market interest rate, also called the market yield, is used as the
discount rate.

Value of a pure discount bond = PV of the amount on maturity:
B0 
Mn
1  kd 
n
BONDS AND
THEIR
VALUATION
Perpetual Bonds

Perpetual bonds, also called consols, has an indefinite life and therefore, it
has no maturity value. Perpetual bonds or debentures are rarely found in
practice.
BONDS AND
2ND
THEIR
VALUATION
Perpetual Bonds

Suppose that a 10 per cent Rs 1,000 bond will pay Rs 100 annual interest
into perpetuity. What would be its value of the bond if the market yield or
interest rate were 15 per cent?

The value of the bond is determined as follows:
INT 100
B0 

 Rs 667
kd
0.15
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BONDS AND
2ND
THEIR
VALUATION
Bond Values and Changes in Interest Rates

The value of the bond declines
as the market interest rate
(discount rate) increases.
The value of a 10-year, 12 per
cent Rs 1,000 bond for the
market interest rates ranging
from 0 per cent to 30 per cent.
1200.0
1000.0
Bond Value

800.0
600.0
400.0
200.0
0.0
0%
5%
10%
15%
20%
25%
30%
Interest Rate
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BONDS AND
2ND
THEIR
VALUATION
Bond Maturity and Interest Rate Risk
 The intensity of interest rate risk
would be higher on bonds with long
maturities than bonds with short
maturities.
 The differential value response to
interest rates changes between
short and long-term bonds will
always be true. Thus, two bonds of
same quality (in terms of the risk of
default) would have different
exposure to interest rate risk.
Present Value (Rs)
Discount rate (%) 5-Year bond 10-Year bond Perpetual bond
5
1,216
1,386
2,000
10
1,000
1,000
1,000
15
832
749
667
20
701
581
500
25
597
464
400
30
513
382
333
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Excel Books
BONDS AND
2ND
THEIR
VALUATION
Bond Maturity and Interest Rate Risk
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BONDS AND
THEIR
VALUATION
Bond Duration and Interest Rate Sensitivity

The longer the maturity of a bond, the higher will be its sensitivity to the
interest rate changes. Similarly, the price of a bond with low coupon rate
will be more sensitive to the interest rate changes.

However, the bond’s price sensitivity can be more accurately estimated by
its duration. A bond’s duration is measured as the weighted average of
times to each cash flow (interest payment or repayment of principal).
BONDS AND
2ND
THEIR
VALUATION
Duration of Bonds

Let us consider the 8.5 per
cent rate bond of Rs. 1,000
face value that has a current
market value of Rs. 954.74
and a YTM of 10 per cent,
and the 12 per cent rate bond
of Rs. 1,000 face value has a
current market value of Rs.
1,044.57 and a yield to
maturity of 10.8 per cent.
Table shows the calculation
of duration for the two bonds.
Year
2001
2002
2003
2004
2005
Cash Flow
85
85
85
85
1,085
Year
2001
2002
2003
2004
2005
Cash
Flow
115
115
115
115
1,115
8.5 Percent Bond
Present Value
at 10 %
77.27
70.25
63.86
58.06
673.70
943.14
11.5 Percent Bond
Present Value
at 10.2%
103.98
94.01
85.00
76.86
673.75
1,033.60
Proportion of
Bond Price
0.082
0.074
0.068
0.062
0.714
1.000
Proportion of
Bond Price x Time
0.082
0.149
0.203
0.246
3.572
4.252
Proportion of
Bond Price
0.101
0.091
0.082
0.074
0.652
1.000
Proportion of Bond
Price x Time
0.101
0.182
0.247
0.297
3.259
4.086
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Excel Books
BONDS AND
THEIR
VALUATION
Volatility

The volatility or the interest rate sensitivity of a bond is given by its duration
and YTM. A bond’s volatility, referred to as its modified duration, is given
as follows:
Volatility of a bond 

Duration
(1  YTM)
The volatilities of the 8.5 per cent and 11.5 per cent bonds are as follows:
Volatility of 8.5% bond 
4.252
 3.87
(1.100)
4.086
Volatility of 11.5% bond 
 3.69
(1.106)
BONDS AND
THEIR
VALUATION
The Term Structure of Interest Rates

Yield curve shows the relationship between the yields to maturity of bonds
and their maturities. It is also called the term structure of interest rates.

Yield Curve (Government of India Bonds)
Yield (%)
7.5%
7.18%
7.0%
6.5%
6.0%
5.90%
5.5%
Maturity
(Years )
5.0%
0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 >10
BONDS AND
THEIR
VALUATION
The Term Structure of Interest Rates


The upward sloping yield curve implies that the long-term yields are higher
than the short-term yields. This is the normal shape of the yield curve,
which is generally verified by historical evidence.
However, many economies in high-inflation periods have witnessed the
short-term yields being higher than the long-term yields. The inverted yield
curves result when the short-term rates are higher than the long-term
rates.
BONDS AND
THEIR
VALUATION
The Expectation Theory



The expectation theory supports the upward sloping yield curve since
investors always expect the short-term rates to increase in the future.
This implies that the long-term rates will be higher than the short-term rates.
But in the present value terms, the return from investing in a long-term
security will equal to the return from investing in a series of a short-term
security.
BONDS AND
THEIR
VALUATION
The Expectation Theory

The expectation theory assumes

capital markets are efficient

there are no transaction costs and

investors’ sole purpose is to maximize their returns

The long-term rates are geometric average of current and expected shortterm rates.

A significant implication of the expectation theory is that given their
investment horizon, investors will earn the same average expected returns
on all maturity combinations.

Hence, a firm will not be able to lower its interest cost in the long-run by the
maturity structure of its debt.
BONDS AND
THEIR
VALUATION
The Liquidity Premium Theory

Long-term bonds are more sensitive than the prices of the short-term bonds
to the changes in the market rates of interest.

Hence, investors prefer short-term bonds to the long-term bonds.

The investors will be compensated for this risk by offering higher returns on
long-term bonds.

This extra return, which is called liquidity premium, gives the yield curve
its upward bias.
BONDS AND
THEIR
VALUATION
The Liquidity Premium Theory

The liquidity premium theory means that rates on long-term bonds will be
higher than on the short-term bonds.

From a firm’s point of view, the liquidity premium theory suggests that as
the cost of short-term debt is less, the firm could minimize the cost of its
borrowings by continuously refinancing its short-term debt rather taking on
long-term debt.
BONDS AND
THEIR
VALUATION
The Segmented Markets Theory

The segmented markets theory assumes that the debt market is divided
into several segments based on the maturity of debt.

In each segment, the yield of debt depends on the demand and supply.

Investors’ preferences of each segment arise because they want to match
the maturities of assets and liabilities to reduce the susceptibility to
interest rate changes.
BONDS AND
THEIR
VALUATION
The Segmented Markets Theory

The segmented markets theory approach assumes investors do not shift
from one maturity to another in their borrowing—lending activities and
therefore, the shift in yields are caused by changes in the demand and
supply for bonds of different maturities.