Important Finance Concepts
3. Calculate the present value of the expected cash flows found in Step 1 using the interest rate or interest rates determined in Step 2
In this book, we introduce many different valuation models. Here we discuss classic valuation models for stocks and bonds.
2 . 5 . 1 V a l u a t i o n M o d e l s f o r E q u i t i e s
Common stock can be thought of as a perpetual security—the owner of the shares has the right to receive a portion of cash flows from the company paid out as dividends. The value of one share should equal the present value of all future cash flows (dividends) the owner of the stock expects to receive from that share. Hence, to value one share, the investor must forecast future dividends. This approach to the valuation of common stock is referred to as the discounted cash flow approach, and the first dividend discount model dates back to John Williams (1938).
The theoretical (also called fair) value of the stock price is D1
(1+r ) + D2
(1+r )2 + D3
(1+r )3 + · · ·
where r is an appropriate discount rate. Note that this is an infinite sum.
Future dividends are not known with certainty, and whether a corpo-ration pays out dividends is decided by its board of directors. If a company does not pay dividends (e.g., it retains earnings), however, the same principle applies, as the retained earnings should be paid out as dividends eventually.
In that case, the fair value of the stock is defined to be the present value of the discounted cash flow stream.
Later in the book, we will discuss optimal portfolio allocation. One of the inputs to such models is the expected return of a security. It is intuitive how an estimate of this return can be produced from cash flow discount models for the stock price—the expected return on a stock would be the value of r in the previous formula that makes the value calculated with the formula equal to the observed market price of the stock. Since the formula for the price contains an infinite sum, additional assumptions must be made in order to be able to compute the expected return. For example, one can attempt to forecast the price PTof the stock at some future date T, in which case the formula becomes
P0= D1
(1+r )+ D2
(1+r )2 + D3
(1+r )3 + · · · + PT
(1+r )T.
From here, the value of the expected return can be found by trial and error.
Alternatively, one can make the assumption that future dividends will grow at a constant rate g. This model was developed by Gordon (1962) and is therefore often referred to as the Gordon model. We have
Dt+1=Dt·(1+g)=D1·(1+g)t−1.
Therefore, the infinite sum in the dividend discount model becomes
D1· 1 (1+r )·
1+(1+g)1
(1+r )1 +(1+g)2 (1+r )2 + · · ·
.
The infinite sum inside the parentheses can now be computed in closed form because it is a sum of an infinite geometric series of the kind
1+q+q2+q3+ · · · (where q in this case is (1+g)/(1+r)), which equals
1 1−q
Therefore, the Gordon model results in a stock price of D1
r−g
Given the current market price of the stock and using the preceding formula, a value for the expected return r can be derived.
There are numerous other valuation models, and a complete review is beyond the scope of this book. It is important to keep in mind, however, that all of the parameters that go into these models need to be estimated, and if the various quantities are defined by different accounting practices or forecasting models, they can lead to significantly different valuations.
2 . 5 . 2 V a l u a t i o n M o d e l s f o r F i x e d I n c o m e S e c u r i t i e s
The value of a bond is the discounted sum of its payments. In other words,
B0= C1
(1+r )+ C2
(1+r )2+ C3
(1+r )3+ · · · + CT
(1+r )T
where Ct denotes the cash flow (coupons or coupon plus principal) paid at date t. While the concept is simple, the details of its implementation, in-cluding how to determine an appropriate discount rate, can lead to different valuations. The minimum interest rate r the investor should require is the prevalent discount rate in the marketplace on a default-free cash flow. In the United States, U.S. Treasuries are considered the norm for default-free securities. This is one of the reasons the Treasury market is closely watched.
Consider a bond with a $100 principal, a coupon rate of 10% paid annually, and a term to maturity of 3 years. Assume a discount rate of 6%.
Today’s value of the bond is 10
(1+0.06)+ 10
(1+0.06)2 + 110
(1+0.06)3 =$110.69.10
If a 12% annual discount rate was used, the bond value today would have been
10
(1+0.12)+ 10
(1+0.12)2 + 110
(1+0.12)3 =$95.20.
Bond Price
Interest Rate E X H I B I T 2 . 2 Relationship between bond price and discount rate.
Note that the value of the bond is lower when the interest rate is higher.
This is an important property of bond prices: when interest rates go up, bond prices generally decline. The opposite is true as well—when interest rates go down, bond prices generally increase. (See Exhibit 2.2.) The relationship between bond prices and interest rates is actually nonlinear: the bond price is a convex function of interest rates.11You can observe this by computing the bond values for several different values of the discount rate. The shape of the relationship between bond prices and interest rates is important for risk management purposes. We will discuss it in more detail in the context of bond portfolio management in Chapter 10.
When one observes bond prices in the market, one can use the infor-mation about the bond and its market price to determine the interest rate that makes the present value of the stream of cash flows equal to the bond price. This implied interest rate is called yield to maturity (YTM), or simply yield. It is quoted on an annual basis. To find it, one would use an iterative trial-and-error procedure. In other words, different values for the yield will be tried until the computed present value of the bond cash flows equals the observed value.
Since the cash flows are every six months, the yield to maturity found by solving the equation for the bond price is a semiannual yield to maturity.
This yield can be annualized by either (1) doubling the semiannual yield or (2) compounding the yield. The market convention is to annualize the semi-annual yield by simply doubling its value. The yield to maturity computed on the basis of this market convention is called the bond-equivalent yield. It is also referred to as a yield on a bond-equivalent basis.
Yield
Time to Maturity (years) E X H I B I T 2 . 3 Upward-sloping yield curve.
Observed bond yields tend to be different depending on the time to maturity. The plot of bond yields versus the length of time corresponding to these yields is referred to as the yield curve. An example of a yield curve is given in Exhibit 2.3. The usual shape of the yield curve is upward-sloping;
however, the yield curve can have different shapes depending on market con-ditions. The reason why an upward-sloping curve is considered “normal” is because investors are assumed to demand more yield for long-term invest-ments than for short-term investinvest-ments. Long-term investinvest-ments are subject to more uncertainty. However, sometimes long-term bonds have lower yields than short-term bonds. In the latter case, we refer to the yield curve as an inverted yield curve. If the yields for all maturities were the same, of course, the yield curve would simply be a flat horizontal line.
Knowledge of the yield curve is helpful because it allows us to position a particular bond with regard to other bonds in its class. It allows us to compute the required yield for a bond. The required yield reflects the yield for financial instruments with comparable risk to a bond we are trying to price. Given the required yield, we can determine the price of the bond by using the required yield as the interest rate r in the pricing formula shown earlier in this section.
In the previous bond value calculation examples, we assumed that the discount rate is the same for all cash flows. In reality, the discount rates are typically different, depending on the length of time until the cash flow will be received. The U.S. Treasury rates, which are the relevant reference rates for default-free cash flows, are followed particularly closely. To see how the value of the bond would be computed when the discount rates for the
cash flows are different, consider again the example of the bond with $100 principal, a coupon rate of 10% paid annually, and a term to maturity of 3 years. Assume that the discount rates for year 1, 2, and 3 are of 6.00%, 6.80%, and 7.20%, respectively. Today’s value of the bond is
10
(1+0.06)+ 10
(1+0.0680)2 + 110
(1+0.0720)3 =$108.55.
2 . 6 I M P O R T A N T C O N C E P T S I N F I X E D I N C O M E
This section reviews important advanced concepts in fixed income.
2 . 6 . 1 S p o t R a t e s
The spot rate is the interest rate charged for money held from the present time until a prespecified time t. The convention is to quote spot rates as yearly rates, but the specific characteristics of the compounding (yearly, m time periods per year, or continuous) vary. The interest rates we used in the section on valuation were all spot rates.
Sometimes, spot rates are referred to as zero rates (short for zero-coupon rates) because they are in fact the return on a zero-coupon bond that is bought today and paid out a certain number of years in the future. Suppose the 5-year Treasury spot (zero) rate with continuous compounding is quoted as 6% per annum. This means that if the price of a 5-year zero-coupon bond today is B0, and the payout after five years is B5, then the following holds:
B5=B0·e0.06·5.
In general, let st denote the spot rate for time period t.12 Then, if Bt
denotes the price of a zero-coupon bond with maturity t, we have Bt =B0·est·t.
2 . 6 . 2 T h e T e r m S t r u c t u r e
The term structure defines the relationship between time and interest rates, with spot rates as the underlying interest rates. It is usually presented as a graph, as the yield curve in Exhibit 2.3. The term structure in interest rates is in fact a yield curve, but its meaning is a bit more academic, as it focuses on pure interest rates for default-free securities rather than yields. Knowledge of the term structure is crucial for pricing and trading purposes.
The obvious way to find the term structure of interest rates is to compute the interest rates implied in the prices of a series of zero-coupon default-free bonds. However, it is difficult to find zero-coupon bonds that span all ma-turities, especially long maturities.13This does not diminish the importance or usefulness of constructing the term structure. The term structure can be constructed from coupon-bearing bonds by a procedure called bootstrap-ping.14The idea is to use several coupon-bearing bonds, determine the spot rates implied by the bonds with the shortest maturity, use that knowledge to compute the spot rate implied by the bond with the next-shortest maturity, and proceed in this manner until the whole term structure is constructed.
The same method is used to construct other spot rate curves, such as the spot rate curve for LIBOR.15There will obviously be gaps in the term struc-ture because we cannot necessarily determine every point on the curve from market prices. Interpolation and polynomial approximations are used to complete the term structure given a few points.
Given a term structure of interest rates, discount factors for securities of different maturities can be determined. For example, if a coupon bond pays coupons six months from now and a year from now, we can read the interest rates corresponding to six months and one year from the term structure, and use those rates to discount the cash flows from the bond’s coupons when valuing the bond.
2 . 6 . 3 F o r w a r d R a t e s
Forward rates can be thought of as the current market consensus of future spot rates: they are interest rates for money to be invested between two dates in the future, but under terms agreed upon today. The concept of forward rates is very useful for pricing and investment purposes. Similarly to spot rates, forward rates are quoted as yearly rates.
Forward rates can be derived from the term structure of interest rates.
Examples of forward rates that can be extrapolated from the Treasury yield curve include
6-month forward rate six months from now.
6-month forward rate three years from now.
1-year forward rate one year from now.
3-year forward rate two years from now.
5-year forward rate three years from now.
We will use the following notation to denote forward rates:tfm, where the subscript t indicates a t-year interest rate, and the subscript m indicates that the t years begin m years from now. For example,0.5f3is the 6-month
forward rate three years from now, and3f2is the 3-year forward rate two years from now. When m=0, the forward rate is the same as the spot rate;
that is,tf0=st.
How do we compute forward rates? Consider, for example, an in-vestor with a 1-year investment horizon who is faced with the following two choices:
Buy a 2-year Treasury note.
Buy a 1-year Treasury bill and, when it matures in one year, buy another 1-year Treasury bill.
The investor would be indifferent between the two choices only if the return on the two is the same. If one of them offers a better return than the other, all investors will prefer that option, which will drive up its price, and hence reduce its return until it equals the return on the option that was less desirable at the beginning.16The value of an investment of $C with the first option after two years will be
C·(1+s2)2,
where s2is the 2-year spot rate. The return on a 2-year investment with the second option will be
C·(1+s1)·(1+1f1),
where s1 is the 1-year spot rate, and 1f1 is the implied rate in a 1-year Treasury bill purchased one year from now, that is, the 1-year forward one year from now. Since the returns on the two investments must be equal, we get
(1+s2)2=(1+s1)·(1+1f1), which allows us to compute the forward rate1f1as
1f1=(1+s2)2 (1+s1) −1.
For m time periods ahead, we have
1fm= (1+sm+1)m+1 (1+sm) −1.
In general, if we are given the term structure, and hence have information about spot rates st and sT for two times t and T, then we can estimate the forward rate between t and T,(T–t)ft, as
(T−t)ft= (1+sT) (1+st) −1.
2 . 6 . 4 C r e d i t S p r e a d s
When cash flows are not default-free, the Treasury rates cannot be used to discount them for valuation purposes. This is because, technically, investors should require a higher yield from default-risky than from default-free se-curities to compensate for the risk they are taking. For pricing default-risky securities, a term structure of credit spreads is often used, where a credit spread is defined as the difference between the yield on a default-free bond and a default-risky bond with the same cash flow characteristics. Dealer firms typically estimate a term structure for credit spreads for each credit rating and market sector. Generally, the credit spread increases with time to maturity. In addition, the shape of the term structure is not the same for all credit ratings. Typically, the lower the credit rating, the steeper the term structure of credit spreads. This is because the risk of default is higher when the default-risky security is a longer-term investment.
2 . 6 . 5 D u r a t i o n
The value of a bond investment is sensitive to changes in interest rates. This is because if the discount rates used to evaluate the different cash flows from the bond change, so will its price. All other things being equal, bonds with long maturities are more sensitive to changes in interest rates than bonds with short maturities. However, maturity alone is not sufficient for measuring the degree of interest rate sensitivity. Intuitively, the higher the bond’s coupon rate, the more dependent the bond’s total dollar return will be on the reinvestment of the coupon payments in order to produce the yield to maturity at the time of purchase. Hence, a change in interest rates during the life of the bond can have a substantial impact on the total return.
Duration is a measure of the sensitivity of a bond’s price with respect to changes in interest rates which takes into consideration the issues previously discussed. In mathematical terms, it is the derivative of the bond price with respect to interest rates, that is, it measures first-order sensitivity. As Ex-hibit 2.2 showed, the relationship between the bond price and interest rates is in fact nonlinear, so the duration would not explain the exact change in bond prices for a given change in interest rates. Convexity, which we will
describe in the next section, complements duration to provide a more accu-rate description of the sensitivity. It is important to note, however, that both duration and convexity describe the sensitivity of bond prices to interest rates when there is a parallel shift in the yield curve, that is, when the rates for all maturities move up or down simultaneously and by the same number of basis points. This clearly places a limitation on the usefulness of duration and convexity as measures of bond interest rate risk. Nevertheless, duration and convexity are very popular, fundamental tools in fixed income analysis.
Even though defining duration as the first derivative of the price/yield function is mathematically correct, it is not really used in practice because it is difficult to explain to clients what the relevance of such a mathematical concept is to measuring actual investment risk. Instead, duration is typically explained as the approximate price sensitivity of a bond to a 100-basis-point change in rates. Thus, a bond with a duration of 5 will change by approximately 5% for a 100-basis-point change in interest rates (that is, if the yield required for this bond changes by approximately 100 basis points).
For a 50-basis-point change in interest rates, the bond’s price will change by approximately 2.5%; for a 25 basis point change in interest rates, 1.25%, and so on.
Let us now define duration more rigorously. The exact formulation is Price if yields decline−Price if yields rise
2·(Initial price)·(Change in yield in decimal).
Let D denote duration, B0denote the initial price,y denote the change in yield, B− denote the price if yields decrease byy, and B+ denote the price if yields increase byy. We have
D= B−−B+ 2·B0·y.
It is important to understand that the two values in the numerator of the preceding equation, B+ and B−, are the estimated values obtained from a valuation model if interest rates change. Consequently, the duration measure is only as good as the valuation model employed to obtain these estimated values. The more difficult it is to estimate the value of a bond, the less confidence a portfolio manager may have in the estimated duration.
We will see in Chapter 10 that the duration of a portfolio is nothing more than a market-weighted average of the duration of the bonds comprising the portfolio. Hence, a portfolio’s duration is sensitive to the estimated duration of the individual bonds.
To illustrate the duration calculation, consider the following bond: a 6% coupon five-year bond trading at par value to yield 6%. The current price is $100. Suppose the yield is changed by 50 basis points. Thus,y= 0.005 and B0=$100. This is simple bond to value if interest rates or yield is changed. If the yield is decreased to 5.5%, the value of this bond would be
$102.1600. If the yield is increased to 6.5%, the value of this bond would be $97.8944. Therefore, B−=$102.1600 and B+=$97.8944. Substituting into the equation for duration, we obtain
$102.1600. If the yield is increased to 6.5%, the value of this bond would be $97.8944. Therefore, B−=$102.1600 and B+=$97.8944. Substituting into the equation for duration, we obtain