Chapter 4

 

Applying the time value of money to security valuation

 

Solutions to questions

 
1.    Given certainty, the future returns on all investments will be known, and hence, the same rate of return will be required on all financial assets. The required rate of return is the market rate of return that balances the total supply of, and demand for, loanable funds. The required rate of return will reflect the time value of money (which in turn depends on the consumption preferences of individuals).
2.    It is true that the value is often sensitive to the forecast growth rate. For example, if the current dividend is 40 cents per share and the required rate of return is 8 per cent per annum, then the valuation using the dividend growth model (Equation 4.8) is $6.80 if the forecast growth rate is 2 per cent per annum, but is $8.24 if the forecast growth rate is 3 per cent per annum. This feature implies that the analyst should not place blind faith in the model; it should be used with care. Note, however, that this feature does not necessarily mean that the model is defective. The nature of compounding implies that small differences in growth rates can imply (correctly) large differences in present values.
3.    In the chapter we show that the value of a company’s shares equals the present (discounted) value of the future dividends paid on those shares. This is consistent with valuation based on cash flows because, in the case of shares, dividends are the net cash flows to shareholders. Since dividends are not the same as earnings, the value of a share cannot also be equal to the present value (PV) of earnings. In principle, the value of a share should be equal to the PV of earnings less the PV of the funds invested to generate those earnings. In other words, the value would be equal to PV (earnings)—PV (retained earnings) which must, of course, be the same as PV (dividends). While a share cannot be valued by simply discounting earnings to a present value, a valuation based on earnings can be carried out by capitalising earnings using a price-earnings ratio.
       In summary, share valuation can be based on either dividends or earnings, but discounting, using a required rate of return, should be applied only to dividends.
4.    Once a bond has been issued (sold), it offers investors a fixed schedule of dates and cash flows. If the bond has no default risk, this schedule will be followed regardless of any future developments. Therefore, if the required rate of return (required yield) increases, the present value of every promised cash flow will be lower. Therefore, its price, which is the sum of these present values, must also be lower. It is true that, all other things being equal, investors will find a higher rate of return more attractive than a lower rate of return. Therefore, all other things being equal, a bond offering a higher coupon interest rate will be more attractive—and hence have a higher price—than a bond offering a lower coupon interest rate. But, once the bonds are issued and the schedules of dates and cash flows is fixed, if the required rate of return increases, the price of both bonds will decrease.
5.    Government debt securities are subject to interest rate risk.
6.    There are two types of risk that are relevant: interest rate risk and default risk. Interest rate risk relates to the change in the price of a debt security as a result of a change in interest rates. All fixed-interest debt securities are subject to this risk. Default risk relates to the risk that the interest payments and principal repayments will not be met by the borrower. Default risk differs across debt securities and it is one factor that explains differential yields on debt securities. In addition, debt securities with equivalent risks of default, but different terms of maturity, will offer different yields. Theories to explain the term structure of interest rates are outlined in the chapter.
7.    The term ‘structure of interest rates’ expresses the relationship between interest rate and term to maturity, at a particular point in time, for a particular risk-class of securities. Theories to explain the term structure of interest rates are discussed in the chapter.
8.    An upward-sloping yield curve means that short-term interest rates are lower than long-term interest rates. However, provided that the capital market is efficient, this does not imply that companies would necessarily be better off raising new debt by issuing short-term debt securities. If the expectations theory is correct, a rising term structure usually implies that short-term interest rates are expected to increase in the future. Hence, a borrower issuing short-term debt is expected to experience rising costs in the future.
9.    The liquidity premium theory suggests there is a premium due to uncertainty about the future level of interest rates. This causes an upward bias in the yield. This means that compared with the yield curves that would be observed if only expectations mattered, an upward-sloping yield curve will become steeper, a downward-sloping yield curve will become less steep and a flat yield curve will become slightly upward-sloping. A downward-sloping yield curve would be more steeply downward-sloping in the absence of a liquidity premium.

Solutions to problems

 
 
1.    The value of this promise today is:
 
No, we don’t know what the value will be in a year’s time. The value at that time will depend on the 3-year interest rate at that time. We do not know, today, what that rate will be.
2.    (a)     Given D₁ = $0.64, g = 0.10 and k_e = 0.20, then:
             
       (b)     P₁ is given by:
                
       P1 = D2/[ke-g] = $0.64x1.10/[0.2-0.1] =$7.04
 
3.    (a)   P₀   =  
                     =  
                     =   $10.00
 
       (b)   P₀   =  
                     =  
                     =   $8.40
       (c)   Let g be the growth rate in the first 3 years. Then: