Use the tools of financial mathematics to value debt and equity securities.
Apply the dividend growth model to value ordinary shares.
Explain the main differences between the valuation of ordinary shares based on dividends and earnings.

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Chapter 4Applying the Time Value of Money to Security ValuationLearning ObjectivesUse the tools of financial mathematics to value debt and equity securities.Apply the dividend growth model to value ordinary shares.Explain the main differences between the valuation of ordinary shares based on dividends and earnings.Learning Objectives (cont.)Explain the nature of interest rate risk.Understand the theories that are used to explain the term structure of interest rates.Apply the concept of duration to immunise a bond investment.Valuing a Financial AssetFinancial assets are valued under certainty by discounting the known future cash flows at the market interest rate and adding the resultant present values.In the case of shares, the future cash flows are dividends while, in the case of bonds, the future cash flows will be coupon / interest payments.Valuation of Shares Assuming CertaintyThe periodic cash flows from an investment in shares are dividends. Assuming the dividends continue indefinitely, the value of the share (P0) is: Valuing SharesThe previous 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.Valuing Shares (cont.)The previous valuation equation can be expressed in the following way to illustrate the point:Introducing UncertaintyThe presence of uncertainty results in investors requiring compensation in the form of a higher promised rate of return (ke): The Required Rate of Return on SharesThe required rate of return (ke) is determined 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.Constant DividendsIn share valuation, the constant dividend assumption is the simplest that can be made.Valuation applies the perpetuity formula: This formula can be used for shares that pay a constant dividend — i.e. a preference share where the dividend is maintained at a fixed rate.Growth in DividendsA more realistic assumption is that dividends per share will grow.If dividends are expected to grow at a constant rate, the following formula can be used to find the current price: Example: Variable Dividend GrowthValuation 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?Example: Variable Dividend Growth (cont.)This is best handled by 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).Substituting the data we have been given:Taking into account the variable dividend growth expected, the value of Rankine’s shares is $11.75.Example: Variable Dividend Growth (cont.)Share Valuation and the Price–Earnings RatioPrice–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:Linking Earnings and DividendsModification 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, riskShare ValuationKey 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.Valuation of Debt SecuritiesThe cash flows associated with a debt investment (debentures, bonds) are:Interest (coupon rate × par value)Face value at maturity Impact of Change in Market Rates (Yield) on Bond ValueThere 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.Interest Rate RiskThe 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, as the bond price will change as interest rates change.Interest Rate Risk (cont.)The cause of this change in bond price can be summarised as: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.)Determinants of Interest Rate RiskTerm to maturityThe 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 riskThe 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.Term Structure of Interest RatesDefinitionThe ‘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 hypothesisLiquidity premium hypothesisTerm 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.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 one-year bonds should yield the same result as investing in a two-year bond.In this case, the link between one-year and two-year yields is the expectation of what yields will be in year 2 for one-year bonds.Term Structure of Interest Rates (cont.)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.Term Structure of Interest Rates (cont.)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). Term Structure of Interest Rates (cont.)Inflation and the term structureWe 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.Term Structure of Interest Rates (cont.)Default Risk Structure of Interest RatesDefinition:While the size of the cash flows is known with debt securities, there is some possibility that these cash flows (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 rating agencies such as S&P and Moody’s.The higher the market’s assessment of the probability of default, the higher the required rate of return (or expected yield) on the debt.Other Factors Affecting Interest Rate StructuresMarketability of securitiesYield 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 exposedto greater risk.Duration and ImmunisationBonds 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).ImmunisationA strategy designed to achieve a target sum of money at a future point in time, regardless of interest rate changes.Duration and Immunisation (cont.)Zero-coupon bondsBonds 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.Duration and Immunisation (cont.)Bond durationA 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.Duration and Immunisation (cont.)DurationMeasure of the time period of an investment in a bond or debenture that incorporates cash flows that are made prior to maturity.Consider two five-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.Duration and Immunisation (cont.)Table A4.1: Present value of 5% and 15% coupon bondsDuration and Immunisation (cont.)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’.Duration and Immunisation (cont.)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 the bond’s price that the present value of that cash flow represents.Duration and Immunisation (cont.)Duration D is summarised in the formula: Duration and Immunisation (cont.)The previous equation for duration can be written in its more usual form: Duration and Immunisation (cont.)Duration and elasticityWhen there is a change in interest rates, all bond prices respond in the opposite direction, but they do not all respond 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.Duration and Immunisation (cont.)Interest elasticity of a bond’s price is proportional to its duration: Duration and Immunisation (cont.)Duration and bond price changesGiven 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: Duration and Immunisation (cont.)Duration and immunisationSuppose the yield curve is flat, but it makes 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.Duration and Immunisation (cont.)LimitationsOnly 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.SummaryFinancial 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.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.