4-1 copyright 2009 mcgraw-hill australia pty ltd ppts t/a business finance 10e by peirson slides...

4-1

Copyright 2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by PeirsonSlides prepared by Farida Akhtar and Barry Oliver, Australian National University

Chapter 4

Applying the Time Value of Money to Security

Valuation

4-2


Learning Objectives• Use financial mathematical tools to value

securities.

• Value ordinary shares — using dividend growth model.

• Explain the main differences between ordinary shares valuation based on dividends and earnings.

• Explain the nature of interest rate risk.

• Theories used to explain interest rates term structure.

• Apply duration concept to immunise a bond investment.

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Valuing a Financial Asset

• Financial assets are valued under certainty by discounting the known future cash flows at the market interest rate and adding the resultant present values of the future cash flows.

• For example: In the case of shares, future cash flows are dividends while, in the case of bonds, future cash flows will be coupon/interest payments.

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Valuation of Shares with Certainty• The periodic cash flows from an investment in shares

are dividends. Assuming the dividends continue indefinitely, the value of the share (P0) is:

• This equation does not ignore the capital gain component of returns, as the price a share is sold for at time n should represent the discounted value of dividends beyond time n.

( )∑∞

= +=

10

1tt

t

i

DP

period per time rateinterest

periodin shareper dividend

==i

tDWhere

t

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Valuing Shares (cont.)

( ) ( )nn

n

tt

t

i

P

i

DP

++

+= ∑

= 1110

• The previous valuation equation can be expressed in the following way to illustrate the point:

period per time rateinterest

period of end at the value terminalexpected

period holding sinvestor’

period of end at the dividend

pricemarket current

:where

0

i

nP

n

tD

P

n

t

4-6


Introducing Uncertainty

• The presence of uncertainty results in investors requiring compensation in the form of a higher promised rate of return (ke):

( )( )∑

∞

= +=

10

1 tt

e

t

k

DEP

( )shares on thereturn of rate required

period of end at the dividend expected

:where

=

=

e

t

k

tDE

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Shares Required Rate of Return• The required rate of return (ke) is determined

by using the concept of the opportunity cost of capital.

• For a risky security, the opportunity cost is at least the return on the risk-free security (i).

• The amount by which ke exceeds i is referred to as the security’s risk premium.

• Several theories, such as the capital asset pricing model and the arbitrage pricing theory, have been developed to help determine this risk premium.

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Constant Dividend and Growth in Dividends• Constant Dividend

– In share valuation, the constant dividend assumption is the simplest that can be made. Valuation applies the perpetuity formula:

– Use this formula for shares that pay a constant dividend.

• Growth in Dividend

– Let’s assume that dividends per share will grow. If dividends grow at a constant rate, then use following formula to find the current price: ( )

( )gk

gDP

e −

+=

100

ek

DP 0

0 =

4-9


Example: Variable Dividend Growth• Valuation can be further generalised, accounting

for changes in dividend growth rates.

Example:

– Rankine Ltd has just paid a $0.90 annual dividend.

– The required return on Rankine’s shares is 15%.

– Current growth rate (g) of 10% per annum applies for the next 3 years.

– After year 3, growth rate (g) falls to 6% indefinitely.

– What is the value of Rankine shares today?

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Example: Variable Dividend Growth (cont.)• Combining the present value of a variable cash flow

stream with the constant dividend growth formula.

• The first three terms account for the higher 10% growth (first three dividends), while the fourth term accounts for the all dividends thereafter (lower 6% growth).

( )( )

( )( )

( )( )

( )( ) ( )( )gk

ggD

k

k

gD

k

gD

k

gDP

ee

eee

−

++

++

+

++

+

++

+

+=

1' 1

1

1

1

' 1

1

' 1

1

' 1

30

3

3

30

2

200

0

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• Substituting the data we have been given:

• Taking into account the variable dividend growth expected, the value of a Rankine share is $11.75.

( ) ( )( )

( )( )

( )( )

( )75.11$

06.015.0

06.110.190.0$

15.1

1

15.1

10.190.0$

15.1

10.190.0$

15.1

10.190.0$

3

3

3

3

2

2

0

=

−+

++=P

Example: Variable Dividend Growth (cont.)

4-12


Share Valuation and the Price–Earnings (P/E) Ratio• Price–earnings ratio is often used to value shares

(Value = P/E × EPS).

• Earnings and dividends are related, as a company’s after-tax earnings (profit) must be either retained or paid out as dividends:

( ) tt EbD 1 −=

shareper earnings period

retained earnings of proportion

:where

tE

b

t ==

4-13


Linking Earnings and Dividends• Modification of dividend model using EPS:

– With a constant growth rate, E1 = E0 (1 + g).

– Thus, the P/E ratio can be expressed as:

• Factors influencing P/E ratio:

– Growth opportunities.

– Risk.

( ) ( )gkEPSbP e −−= 10 1

( )( ) ( )gkgbEP e −+−= 1 100

4-14


Share Valuation

• Key differences between dividend and EPS approaches.

• Valuation based on dividends:

– Dividends are discounted to a present value to provide a share valuation.

• Valuation based on earnings (EPS):

– Earnings are capitalised into a share value using a price–earnings ratio.

4-15


Valuation of Debt Securities• The cash flows associated with a debt investment

(debentures, bonds) are:– Interest (coupon rate × par value)– Face value at maturity

( ) ( )nn

n

tt

t

i

P

i

IP

1 110

++

+= ∑

=

(yield) rateinterest market

maturity toperiods ofnumber

maturityat repayment) (principal valueface

at timepayment interest

:where

====

inP

tI

n

t

4-16


Impact of Change in Market Rates (Yield) on Bond Value

• There is an inverse relationship between interest rates and bond value.

• Thus, as interest rates rise (relative to the coupon rate), the value of the bond falls.

• If interest rates fall (relative to the coupon rate), the value of the bond increases.

4-17


Interest Rate Risk• The chance that interest rates will change in the

future, thereby changing the value of an asset.

• Even for a risk-free bond, in respect of the cash payments being certain, risk exists because the bond price will change as interest rates change. Reasons for bond price changes:

– Price effects — the valuation of the stream of future cash flows is carried out using a new market interest rate (i), leading to a different price.

– Reinvestment effects — the coupons flowing from the bond can be invested at the new market interest rate (i) rather than the old one. (Favourable if the interest rate has risen.)

4-18


Determinants of Interest Rate Risk• Term to maturity

– The longer the term to maturity, the greater the effect of the new interest rate through compounding.

– Longer bonds are usually more price-sensitive to interest rates.

• Default risk

– The chance that the bond issuer will fail to make a coupon or principal payment.

– As default risk rises, the value of a bond will fall.

4-19


Term Structure of Interest Rates• Definition

– The term structure is the relationship between the term to maturity and interest rate for securities in the same risk class.

• The term structure is illustrated by the yield curve, which plots bond yield against term to maturity.

• Determinants of term structure:

– Market expectations hypothesis.

– Liquidity premium hypothesis.

4-20


Term Structure of Interest Rates (cont.)

• Yield curve can be downward or upward sloping, or flat.

• Each case represents different information about interest rates and expectations of future interest rate movements.

4-21


• Expectations theory:

– Interest rates are set such that investors in bonds or other debt securities can expect, on average, to achieve the same return over any future period, regardless of the security in which they invest.

• For example, investing in a sequence of two 1-year bonds should yield the same result as investing in a 2-year bond.

• In this case, the link between 1-year and 2-year yields is the expectation of what yields will be in year 2 for 1-year bonds.


4-22


• Liquidity premium (risk premium) theory:

– Although future interest rates are determined by investors’ expectations, investors require some reward (liquidity premium) to assume the increased risk of investing long term.

• The key issue is that interest rates could rise in the short term. If investors hold long-term bonds, they miss the opportunity to invest at the higher rate (a lack of liquidity).

• Thus, investing in long-term bonds requires some compensation for this risk — liquidity premium.


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• Empirical evidence on the term structure:

– Some support for term structure premium but not beyond 8–9 months — Fama (1984).

– Some evidence in support of term structure in Australian data — Tease (1988), Robinson (1998), and Young and Fowler (1990).

– However, more rigorous statistical studies found little evidence to support expectations theory — Ales (1995) and Heaney (1994).


4-24


• Inflation and the term structure

– We would expect lenders to require the nominal interest rate to compensate them for expected inflation.

– The higher the expected inflation rate, the higher the observed nominal interest rate.

– Inflation expectations:

raise expectations about future interest rates,

raise liquidity risk, and

have an impact on the term structure.


4-25


Default Risk Structure of Interest Rates• Definition:

– While debt securities cash flows size is known, there is some possibility that these payments may be defaulted on by the bond issuer.

• The financial health of a company is assessed to determine the chance of such a default.

• These assessments are summarised by credit ratings, provided by S&P & Moody’s rating agencies.

• The higher the market’s assessment of the probability of default, the higher the required rate of return (or expected yield) on the debt.

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Other Factors Affecting Interest Rate Structures

• Marketability of securities

– Yield differentials on securities may also result from differences in marketability.

– An investor will buy a security of low marketability only if the yield is greater than that on a security of high marketability.

– Higher expected rate of return on equity (shares) than on debt because ordinary shareholders are exposed to greater risk.

4-27


Duration and Immunisation

• Bonds are subject to interest rate risk — a change in interest rates will change the value of a bond or bond portfolio.

• It is possible to structure a bond portfolio so that changes in interest rates have a minimal effect on the value of the bond (portfolio).

• Immunisation

– A strategy designed to achieve a target sum of money at a future point in time, regardless of interest rate changes.

4-28


Duration and Immunisation (cont.)

• Zero-coupon bonds

– Bonds that pay only one cash flow — the payment at maturity.

– An investor will know with certainty the price/value of the bond at maturity because it will be worth its face value, and will be unaffected by changes in interest rates, because there are no coupons.

– However, it is more usual to have bonds with non-zero coupons.

4-29



• Bond duration

– A technique certain to immunise an investment in coupon-paying bonds against all possible changes in interest rates has never been achieved.

– However, there is a technique that will immunise a bond investment in a relatively simple environment in which the yield curve is flat, but may make a parallel shift up or down.

– This technique is based on the concept of bond duration.

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• Duration

– Measure of the time period of an investment in a bond or debenture that incorporates cash flows that are made prior to maturity.

– Consider two 5-year bonds, both of which have a face value of $1000, pay interest annually, and are currently priced to yield 10% p.a. They differ, however, in that one has a coupon rate of 5% p.a. and the other a rate of 15% p.a.

4-31



Table A4.1: Present value of 5% and 15% coupon bonds

5% COUPON 15% COUPONYEAR CASH FLOW PRESENT VALUE CASH FLOW PRESENT VALUE

($) ($) ($) ($)

1 50 45.45 150 136.362 50 41.32 150 123.973 50 37.57 150 112.704 50 34.15 150 102.455 50 31.05 150 93.14

1000 620.92 1000 620.92Total 810.46 1189.54

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• For the low-coupon bond, the face-value payment ($1000) represents about 77% of its price.

• For the high-coupon bond, the face-value payment ($1000) represents only about 52% of its price.

• The low-coupon bond brings returns to the investor later in its life, relative to the high-coupon bond. In this sense, the low-coupon bond is ‘longer’.

• This timing feature can be incorporated into a duration measure by weighting the number of periods that will elapse before a cash flow is received by the fraction of bond’s price that the PV of that cash flow represents.

4-33


Duration and Immunisation (cont.)• Duration D is summarised in the formula:

• See Table A4.2 for duration calculation example.• The previous equation for duration can be written in

its more usual form:

( )( )

. flowcash of PV ,1

(coupon) , at time flowcash

:where

ttt

t

t

Ci

CCPV

tC

+=

=( )

tP

CPVD

n

t

t∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1 0

( )

( )∑

∑

=

=

+

+×

=n

tt

t

n

tt

t

i

C

i

tC

D

1

1

1

1

4-34



• Duration and elasticity

– When interest rates change, all bond prices respond in the opposite direction, but not to the same extent.

– Different bonds have different interest elasticities.

– Bond duration has a tight link with interest rate elasticity and the price response to interest rate changes.

– Interest elasticity of a bond’s price is proportional to its duration. See example on p. 101.

Di

iE ⎟

⎠

⎞⎜⎝

⎛+

−=1

4-35



• Duration and bond price changes

– Given that duration can be related to interest elasticity, it follows that it is possible to use duration to work out the approximate percentage price change that will occur for a given change in interest rate.

– For ‘small’, discrete changes in interest rates and bond prices, we have the following approximation: See example on p.102.

iDiP

PΔ⎟

⎠

⎞⎜⎝

⎛+

−≈Δ

11

0

0

4-36


Duration and Immunisation (cont.)• Duration and immunisation

– Suppose the yield curve is flat, but it may make a parallel shift up or down.

– If an investor is holding a bond whose duration matches the remaining investment period, the investment is immunised against the shift.

• Limitations

– Only immunised for a single yield shift.

– Requires costly and cumbersome rebalancing of the investment after each shift to remain immunised.

– Only a flat yield curve subject to parallel shifts has been considered.

4-37


Summary• Financial assets such as shares and bonds are

streams of cash flows that can be valued by summing the present value of these cash flows.

• Ordinary shares provide a dividend stream that can be valued in various ways depending on expected growth.

• Debt securities provide interest payments and a repayment of principal.

• The price of debt securities varies inversely with the interest rate.

4-38


Summary (cont.)• Term structure of interest rates

– Expectations and liquidity premium theories along with inflation and market segmentation.

• Risk of default is important in valuing debt securities.

• Bond duration

– Related to interest rate elasticity of bond prices.

– Can be used to immunise bond holdings, ensuring availability of a fixed amount of funds at a given future date, despite interest rate risk.

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