CFA Level 1 (2009) - 5

Study Session 15

CrosseReference to CFA Institute Assigned Reading #63 - Understanding Yield Spreads

1.B An inverted or Jownward-sloping yield curve, under the pure expectations lheory, indicales that shon-term rales al'\:'eXIJCcred to decline in the future.

2.B Rational investors feel that long-term bonds have more risk exposure than short-tC1'm sl'curilies (i.e., long-term securities arc less liquid and subject to more price volaliliry). The orher statemeIHs are coneer.

3.C The market segmeIltation theory holds that certain types of financial institutions and investors [Hefer to confine (most of) their investment activity to certain maturity ranges of the f1xed-income market and that supply and demand forces within each segment ultimately determine the shape of the yield curve.

4.C Open market operations are canied on frequently. The Fed's selling ofTrcasurics in the open market rakes money out of the economy, reducing the amount of loanable funds and incrcasing inrcrest rates. The (1)I'0sitc occurs ,,",'hen the Fed buys Treasuries in lhe 0lwn nLlrku.

8.B. A contracting economy means lower corporate' e'arnings which increases the probability

of default on debt and increases yield spreads between corporate issues and Treasuries at a particular maturity.

10.A The tax rate that makes investors indifferent between two otherwise eqnivalent bonds is determined by solving for the tax rate in the equation: tax-exempt yield = (l - tax rate) x taxable yield. Rearranging this relationship, we have:

11.B As an economy expands, credit spreads decline as expected corporate earnings rise. This is because, with stronger earnings, corporations are less likely to default on their debt.

12.C The less liquidity a bond has, the higher its yield spread relative to Treasuries. This is because investors require a higher yield to compensate them for giving up liquidity, which results in a greater spread over Treasury issues, which arc very liquid.

13.B Since we usually speak of credit spreads as yield spreads to Treasuries, a change in the yield spread does not imply any change in the values of Treasuries.

Study Session 15

Cross-Reference to C\-<1\ Institute Assigned Reading #63 - Understanding Yield Spreads

5.175%, which is IL'ss than tilL' tax-L'xcmpt yiL'ld. kading [0 the samL' ckcisiun.

Thc following is a rcvicw of the Analysis of Fixed Income Invcstments principles dcsigned to address thc learning outcomc statcmcnts sct forth by CFA Institut"Q!). This topic is also covered in:

INTRODUCTION TO THE VALUATION OF

DEBT SECURITIES

Study Scssion 16

EXAM Focus

LOS 6/La: Explain the steps ill the bond vahntion process.

The general procedure for valuing fixed-income securities (or any security) is to take the present values of all the expected cash flows and add them up to get (he value of the seCUrIty.

There are three steps in the bond valuation process:

Step 1: Estimate the cash flows over the life of the security. For a bond, there are two types of cash flows: (l) the coupon payments and (2) the return of principal.

Step 2: Determine the appropriate discount rate based on the risk of (uncertainty about) the receipt of the estimated cash flows.

Step 3: Calculate the present value of the estimated cash flows by multiplying the bond's expected cash flows by the appropriate discount factors.

LOS 64.b: Identify the types of bonds for which estimating the expected cash flows is difficult, and explain the problems encountered when estimating the cash Hows ror t hesc honds.

Certainly, one problem in estimating future cash flows for bonds is predicting defaults and any potential credit problems that make the receipt of future cash flows uncertain. Aside from credit risk, however, we can identify three situations where estimating future cash flows poses additional difficulties.

1.The principal repayment stream is not known with certainty. This category includes bonds with embedded options (puts, calls, prepayment options, and accelerated sinking fund provisions), For these bonds, the future stream of principal paymen ts is uncertain and will depend to a large extent 011 the future path of interest rates. For example, lower rates will increase prepayments of mortgage passthrough securities, and principal will be repaid earlier.

Study Session 1G

Cross-Reference to CFA Institute Assigned Reading #64 - Introduction to the Valuation of Debt Securities

2.Thq coupon payments are not known with certainty. With floating-rate securities, future coupon payments depend on the path of interest rates. With some floatingrate securities, the coupon payments may depend on the price of a commodity or the rate of inflation over some future period.

3.The bond is convertible or exchangeable into another security. Without information about future stock prices and interest rates, we don'r know when the cash flows will come or how large they will be.

LOS ()/I.c: COm!JlIte the value or a hUl1d and the chalH'{'., in V;:!UL' t!J;u IS

;[ttributabk to a change in the discollnt rate.

For a 'rreasury bond, the appropriate rate used to value the promised cash flows is the risk-free rate. This may be a single rate, used to discount all of the cash flows, or a series of discount rates that correspond to the times until each cash flow arrives.

For non-Treasury securities, we must add a risk premium to the risk-free (Treasury) rate to determine the appropriate discount rate. This risk premium is one of the yidd spreJ.d measures covered in the previous review and is the added yield to compensate for greater credit risk, liquidity risk, call risk, prepayment risk, etc. When using J. single discount rate to VJ.lue bonds, the risk premium is added to the risk-free rate to get the appropriate discount rate for all of the expected cash flows.

yield on a risky bond = yield on a default-free bond + risk premium

Other things being equal, the riskier the security, the higher the yield differential (or risk premium) we need to add to the on-the-run Treasury yields.

Computing the Value of a Bond

Valuation with a single yield (discount rate). Recall that we valued an annuity using the time value of money keys on the calculator. For an option-free coupon bond, the coupon payments can be valued as an annuity. In order to take into aCCOUIH the payment of the par value at maturity, we will enter this final payment as the future value. This is the basic difference between valuing a coupon bond and valuing an annuity.

For simplicity, consider a security that will pay per year for ten years and make a single $1,000 payment at maturity (in ten years). If the appropriate discount rate is 8% for all the cash flows, the value is:

This is simply the sum of the present values of the future cash flows, $100 per year for ten years and $1,000 (the principal repayment) to be received at the end of the tenth year, at the same time as the final coupon payment.

Study Session 16

Cross-Reference to CFA Institute Assigned Reading #64 - Introduction to the Valuation of Debt Securities

The calculator solution is:

N" 10; PMT" 100; I~V" 1,000; I/Y" 8; CPT --, PV = -$1,134.20

where:

N = numbcr of years

PMT = the anllual coupon payment I/Y = the tlllIl/ud discount rate

rv "the par value or selling price at the end of an :lssumed holding period

Professor)' Note: j;,l<e note ofa couple ofpoilJts lu'l"e. The discount rate is entered as a whole number in pacent, 8, not 0.08. 'The ten COUPOII paymmts of$ 100 each are taken ({/re ofin the N = 10 CIItI), the prin('ljHII repayment is ill the FV

~ ,,1,000 ently. LflStfy, note that the PV is negative; it wilt be the opposite sign 10 ~~ the sign of PMT t1l1d FV The calculator is jwt "thinking" that ifyou receive the

The Change in Value When Interest Rates Change

Bond values and bond yields are inversely related. An increase in the discount r:lte will decrcase the present value of a bond's expected c:lsh flows; a decrease in the discount rate will increase the present value of a bond's expected cash flows. The ch:lnge in bond valuc in response to a change in the discount rate can be calculated as the difference between the present values of the cash flows at the two different discount rates.

Study Session 16 Cross-Reference [0 CFA Institute Assigned Reading #64 - Introduction to the Valuation of Debt Securities

Example: Changes in required yield

A bond 1t;\5 a par value of$1 ,000, a 6% semiannual coupon, and rhree years ro maruriry. Compure rhe bond values when rhe yield ro maruriry is 3%,6%, and 12%.

Answer:

Ar 1/ Y = 2; N = 3 X 2; FY = 1,000; PMT = 60; CPT ----t PY = -1,085.458

Ar II Y = ~; N = 3 X 2; FV = 1,000; PMT = 60; CPT ----t PY = -852.480

2 2

We have illusuared here a point covered earlier; if rhe yield to maturity equals the coupon rate, rhe bond value is equal to par. If the yield to maturity is higher (lower) than rhe coupon rate, the bond is trading at a discount (premium) to par.

We can now calculate the percentage change in price for changes in yield. If the required yield decreases from 6% to 3%, thevalue of the bond increases by

If the yield increases from 6% to 12%, the bond value decreases by

Price-yield profile. If you plot a bond's yield to its corresponding value, you'll get a graph like the one shown in Figure 1. Here we see that higher prices are associated with lower yields. This graph is called rhe price-.yield curve. Note that it is not a suaight line but is curved. For option-free bonds, the price-yield curve is convex toward the origin, meaning it looks like half of a smile.

Srudy Session 16

Cross-Reference to CFA Institute Assigned Reading #64 - Introduction to the Valuation of Debt Securities

Figure I: The Price-Yield Profile I

Bond

ValLie ('!,'llof ["II")

------------------ _ . _..._._-----

LOS 6!i.d: Explain how the price of a hond changes as the hond ;,ppro:lcbcs it.s maturity date, and compute the change in valuc that is attributablc to lhe passage of lime.

Prior ro maturity, a bond can be selling at a significant discount or premium to par value. However, regardless of its required yield, the price will converge to par value as maturity approaches. Consider the bond in the previous example ($1,000 par value, 3-year life, paying 6% semiannual coupons). The bond values corresponding to required yields of 3,6, and 12% as the bond approaches maturity are presented in Figure 2.

Figure 2: Bond Values and the Passage of Time

To compute the change in bond value due to the passage of time, just revalue the bond with the number of periods (remaining until maturity) reduced. Note that in the preceding example, the value of a 6% bond with three years until maturity and a yield to maturity of 3% is FV = 1,000; PMT = 30; N = 6; I/Y = 1.5; CPT -+ PV = $1,085.46. To see the effect of the passage of time (with the yield to maturity held

constant) just enter N = 5 CPT -+ PV to get the value one period (six months) from

Study Session 16 Cross-Reference to CFA Institute Assigned Reading #64 - Introduction to the Valuation of Debt Securities

now of $1 ,071.74, or N == 4 CPT .-> PV to get the value two periods (one year) from nowoF$I,057.82.

The change in value associated with the passage of time for the three bonds represented in Figure 2 is presented graphically in figure 3.

Figure 3: Premium, Par, and Discount Bonds

l)Ol1li

Vlll\(" ($)

Since a zero-coupon bond has only a single payment at maturity, the value of a zero is simply the present value of the par or face value. Given the yield to maturity, the calculation is:

maturity value

bond vaue=I ----,.--:..-----:;----

(1 +i rumbcr of ycars x 2

Note that this valuation model requires just three pieces of information:

1.The bond's maturity value, assumed to be $1,000.

2.The semiannual discount rate, i.

3.The life of the bond, N years.

Alternatively, using the TVM keys, we enter:

PMT == 0; FV == par; N == # years x 2; I/Y =YTM/2 = semiannual discount rate;

CPT - t PV

Although zero-coupon bonds do not pay coupons, it is customary to value zero-coupon bonds using semiannual discount rates. Note that N is now two times the number of years to maturity and that the semiannual discount rate is one-half the yield to maturity expressed as a BEY.

5tlldy Session 16

Cross-Reference to CFA Institute Assigned Reading #64 - Introduction to the Valuation of Debt Securities

Example: Valuing a ~ero-coupon bond

Compute the value ofa 10-year, $1,000 face value zero-coupon bono with a yield to maturity 01'8%.

Answer:

To hnd the value of this bond given its yield to maturity of 8% (a 4% semiannual rate), we can calculate:

Or, using a calculator, use the following inputs:

N = 10 X 2 = 20; FV = 1,000; l/Y = %= 4; PMT = 0; CPT ---+ PY = -$456.39

The difference between the current price of the bond ($456.39) and its par value

($1,000) is the amount of com pound interest that will be earned over the 1O-year life of the issue.

&~~-~ Professor's Note: Exam questions will likely specifY whether annual or semiannual

~~~discoullting should be used. Just be prepared to value a zero-coupon bond either way.

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LOS G/I.r: L>.pLill ihc ;;l"!)ilragcLcT ,':II\Lll;OIJ ;tppro~H:h alJd (he l1JarLCI pr()Cl:~" that forces the price of a bond toward its arhitT:1ge-fn;c valuc, and explain how a dcakr t;ilJ gnHTalc an :lrbitrage profit if:1 hond is 1l1ispriced.

Yield to maturity is a summary measure and is essentially an internal rate of return based on a bond's cash flows and its market price. In the traditional valuation approach, we get the yield to maturity of bonds with maturity and risk characteristics similar to those of the bond we wish to value. Then we use this rate to discount the cash flows of the bond to be valued.

Wi th the arbitrage-free valuation approach, we discoullt each Cflsh flow wing a discount

rate that is specific to the maturity ofeach cash flow. Again, these discount rates are called spot rates and can be thought of as the required rates of return on zero-coupon bonds maturing at various times in the future.

The arbitrage-free valuation approach simply says that the value of a Treasury bond based on (Treasury) spot rates must be equal to the value of the "parts" (i.e., the sum of the present values of all of the expected cash flows). If this is not the case, there must be an arbitrage opportunity. If a bond is selling for less than the sum of thc present valucs of its expected cash flows, an arbitrageur will buy the bond and sell the "picces."

Srudy Session 16

Cross-Reference to CFA Institute Assigned Reading #64 - Introduction ro rhe Valuation of Debt Securities

If ~he bond is selling for more rhan the sum of the values of the pieces (individual cash r~ows), one could buy the pieces, package them to "make" a bond, and then sell the bond "package" to earn an arbitrage profit.

The first step in checking for arbitrage-free valuation is to value a coupon bond using the appropriate spot rates. The second step is to compare this value to rhe market price of the bond. If the computed value is not equal to the market price, there is an arbitrage proflt to be earned by buying the lower-priced alternative (either the bond or the individual cash Haws) and selling thc higher-priced alternative. Of course, this assumes that there arc zero-coupon bonds available that correspond to the coupon bond's cash Haws.

Example: Arbitrage-free valuation

Consider a 6% Treasury note with 1.5 years to maturity. Spot rates (expressed as semiannual yields to maturity) are: 6 momhs = 5%, I year = 6%, 1.5 years = 7%" [f the note is selling for $992, compute the arbitrage proht, and explain how a dealer would perform the arbitrage.

Answer:

To value the note, note that the cash flows (per $1,000 par value) will be $30, $30, and $1,030 and that the semiannual discount rates are half the statcd yield to maturity.

Using the semiannual Spot rates, the present value of the expected cash flows is:

This value is less than the market price of the note, so we will buy the individual cash flows (zero-coupon bonds), combine them into a 1.5-year note "package," and sell the package for the market price of the note. This will result in an immediate and riskless profit 0[992.00 - 986.55 = $5.45 per bond.

Determining whether a bond is overor undervalued is a 2-step process. First, compute the value of the bond using either the Spot rates or yield to maturity, remembering that both are often given as twO times the semiannual discount rate(s). Second, compare this value to the market price given in the problem to see which is higher.

How a Dealer Can Generate an Arbitrage Profit

Recall that the Ti'easury STRIPS program allows dealers to divide Treasury bonds into their coupon payments (by date) and their maturity payments in order to create zerocoupon securities. The program also allows reconstitution of Treasury bonds/notes by putting the individual cash Bows back together to create Treasury securities. Ignoring any costs of performing these transformations, tbe ability to separate and reconstitute Treasury securities will insure thar the arbitrage-free valuation condition is met.