CFA Level 1 (2009) - 5

Option-adjusted

SLUdy Session 16

Cross-Reference to CFA Institute Assigned Reading #65' - Yield Measures, Spot Rates, and Forward Rates

LOS 65.e

I

The theoretical Treasury spot rate curve is derived by calculating the Spot rate for each successive period N based on the spot rate for period N - 1 and the market price of a bond wilh N coupon payments.

To compule lhe value of a bond using spot rates, discOlllll each separate cash flow llsing lhe spot rate corresponding to the number of periods until the cash flow is to be received.

LOS 65.1'

Three commonly used yield spread measures: Nominal spread: bond YTM - Treasury YTM.

Zero-volatility spread (Z-spread or static spread): lhe equal Jmounr of additional yield that must be added to each Treasury spot rate to get spot rates that will produce a present value for a bond equal to its marker price.

spretld (OAS): spread to the SPOl yield curve after adjusting for the effects of embedded options. OAS rd~eets the spread for credit risk and liquidity risk primarily.

There is no difference between the nominal and Z-spread when the yield curve is flat. The steeper the spot yield curve and the earlier bond principal is paid (amortizing securities), the greater the difference in the two spread measures.

LOS 65.g

The option cost for a bond with an embedded option is Z-spread - GAS.

For callable bonds, Z-spread > OAS and option cost> O.

For putable bonds, Z-spread < GAS and option cost < O.

LOS 65.h

Forward rates are current lending/borrowing rates for shan-term loans to be made in future periods.

A spot rate for a maturity of N periods is the geometric mean of forward rates over the

N periods. The same relation can be used to solve for a forward rate given spot rates for two different periods.

To value a bond using forward rates, discounr the cash flows at times 1 through N by the product of one plus each forward rate for periods 1 to N, and sum them.

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©2008 Kaplan Schweser

Study Session 16 Cross-Reference to CFA Institute Assigned Reading #65 - Yield Measures, Spot Rates, and Forward R.,tes

Use the following data to answer Questions 1 through 4.

An analyst observes a Widget & Co. 7.125%, 4-year, semiannual-pay bond trading at 102.347% of par (where par = $1 ,ClOO). The bond is cdlable at 101 in two years, and putable ar 100 in two years.

1.What is the bond's current yield?

A.6.962%.

B.7.328%.

C.7.~26%.

2.What is the bond's yield to maturity?

A.3.225%.

B.5.864%.

C.6.450% .

.3. What is the bond's yield lO call?

A.3.167%.

B.5.664%.

C.6.334%.

4.What is the bond's yield to pur?

A.4.225%.

B.5.864%.

C.6.450%.

5.Based on semiannual compounding, what would the YTM be on a 15-ycar, zero-coupon, $1,000 par value bond that's currently trading at $331.40?

A.3.750%.

B.5.151%.

C.7.500%.

6.An analyst observes a bond with an annuaL coupon that's being priced to yield 6.350%. What is this issue's bond equivalent yield?

A.3.175%.

B.3.126%.

C.6.252%.

7.An analyst determines that the cash flow yield of GNMA Pool 3856 is 0.382% per month. What is the bond equivalent yield?

A.4.628%.

B.9.363%.

C. 9.582%.

©2008 Kaplan Schweser

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Study Session 16

Cross-Reference to CFA Institute Assigned Reading #65 - Yield Measures, Spot Rates, and Forward Rates

8.1f the YTM equals the actual compound return an investor realizes on an

investment in a cbupoll bond purchased at a premitU11 to par, it is leflSt likely that:

A. cash Haws will be paid as promised.

B. the bond will nor be sold at a capital loss.

C. cash flows will be reinvested at the YTM rate.

9.	The 4-year spar ratl' is 9.'i'5%, and the 3-year spar rate is 9.85°/b. What is thc
	I-year forward rate ducc years from roday?
	A.	8.258 l )'1l.
	B.	9.850%.
	c:.	I 1.059%.

10.An invesror purchasl's a bond thar is plIlabJe at the option of rhe holder. The option has value. He has calculated the 7-spread as 223 basis pOiIHS. The option-adjusted spread will be:

A.eLlual to 223 basis pOiIHS.

B.ll'sS than 223 basis poilHs.

C. greater than 223 basis poilHs.

Use the following data to answer Questions 11 and 12.

Given:

•CUITClH I-year rate", 5.'5%.

•/1 '" 7.63 IYo.

•/2 '" 12.18%.

•/, '" 15.5%.

II.	The value of a 4-ycar, 10% annual-pay, $1,000 par value bond would be closest
	to:

A.$995.89.

B.$1,009.16.

C.$1,085.62.

12.Using annual compounding, the value of a 3-year, zero-coupOIl, $1,000 par value bond would be:

A.$785.

B.$852.

C.$948.

13.A bond's nominal spread, zero-volatility spread, and option-adjusted spread will all be equal for a coupon bond if:

A.the yield curve is flat.

B.the bond is option free.

e. the yield curve is flat and the bond has no embedded options.

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©2008 Kaplan Schweser

Study Session 16 Cross-Reference to CFA Institute Assigned Reading #65 - Yield Measures, Spot Rates, and Forward Rates

f';,:' ;,: ,~~ : ~".'-'" ." •,.... '. I .. ,." " ~:~ ~. "	'- . , '•. 1 ...	'	~ I	. , l' ~.
~~~~~~'·:~:<~J?~,P~¥~~~:C.:fi';e~~~~>.; .._~'~:.' '....			,,::: ,,_....	~>., ':;;~"

I.A

2.C

3.C

4.B

	.	71.25
current yield ~		= 0.06%2, or 6.%2°/(,
		J ,023.17
1 023 /'7	R	35.625		1,000
	""	(II YTM/2l'	1 - > YTM . 6,450%
, . , j	= ~		.	0+ YTM/2)H
	, --I
N ~ 8; FV = 1,000; PMT = 35625; PV = -1,023A7 -> CPT IN = 3.225						x 2 ~ (JA5(Yt,
1,023.47 = t		35.625	+-	J,0 I a	'*YTC = 6.334%
	(•. 1 (I + YTC I 2)'			(J +YTCld
N = 4; FV = 1,010; PMT = 35.625; PV = -1,023.17; CPT --; I/Y = 3.167						x 2 =
6.334%
1,023.47 ~ ,,'!......		35.625	+	1,000 ,	=> YTP = 5.864%
	7-1	(I + YTP/21'	(J +YTPI2)'
N = 4; FV = 1,000; PMT = 035.625; PV = -1,023.47; CPT -> \lY = 2.'J32						x 2 ~
5.864%

'i.	C	II 1,000 JiD	.-IL2=7.5% or,
		I331.40	j
		Solving with a financial calculator:
		N = 30; FY =	1,000; PMT = 0; PY = -331.40; CPT -> IN = 3.750 x 2 = 7.500%
6.	C	bond eljuivaIcm yield = [(I		+ EAY) 1/2 -	I] x 2 = [(1.0635)1/2 -- I] x 2 = 6.252%
7.	A	bond equivalent yield = [(I		+ CFY)6 -	I] x = [(1.00382)6 -- 1] x 2 = 4.628%

8.B For a bond purchased at a premium to par value, a decrease in the premium over time (a capital loss) is already factored into the calculation ofYTM.

10. C For embedded purs (e.g., purable bonds): oprion cost < 0, => GAS> Z-spread.

©2008 Kaplan Schweser

Page 125

Srudy Session 16

Cross-Reference to Cf-A Instilllte Assigned Reading #65 - Yield Measures, Spot Rates, and Forward Rates

II. B Spot rates: 51 = 5.5%.

S2 = [(1.055)(1.0763)]112 - 1 = 6.5MIc,

S, = [(I.055)(1.076.1)(1.1218)jl/l- I = 8.:)')%

\= [(J.05,))(I.0763)(1.1218)(1.155)]lh - I ~ \013'j{,

Bond valuc:
N = 1; rv =	100; 1/Y = 5.5; CPT-> PV=	-94.79
N = 2: rV = 100; IIY = G.56; CPT --; PV =		-88.07
N = 3; FV =	100; IIY = 8.39; CPT -~ PV=	-78.53
N = 4: FV = 1,100: I/Y = IO.n CPT -> flV=		-747.77
	TOlal:	$1,009.16

12. A Find rhe spot rale f(lr .3-year lenJing:

	Value of the bond: N = 3; rV = 1,000; l/Y = 8..'19; CPT·I'V = -78').29
	or
	$1,000	= $785.05
	(I.055)(1.0763)(1.1218)
1'I.C	If the yield curve is flat,	the nominal ,prt"ad ;lI1d the Z-spread arc equal. If the bond is
	oprion-(ree, the Z-spread and OAS arc equal.
14. C	A Treasury bond is rhe besr answer. The Treasury spot yield curve will correctly price ;1Il
	on-the-run Treasury bond at its arbitrage-free price, so thc Z-spread is zero.

15.A Thc Z-spread will be greater than the nominal spread when the spar yield curve is upward sloping.

" 4 _

,ANSWERS -. QOMPREHENSIVE PRO~LEMS		.:;
f- ' . - _ - . • _ _.. ' ..	_"~ ~ ~ •• > l ' . ,~_ - _ ~ ",' ... " .. ! ~.	. r •• " '''' ~ ....' ~ - : •

I.The thrce sources of rerum are coupon interest payments, recovery of principal/capiraI gain or Joss, and reinvesrment income.

Coupon interest payments: 0.07 / 2 x $100,000 x 6 = $21,000

Recovery ofprincipal/capital gain or loss: Calcularc the sale price of the bond: N = (1O - 3) x 2 = 14; I1Y = 6.9 / 2 = 3.45; PMT = 0.07 / 2 x 100,000" 3,500; FV = 100,000; CPT

.--; PV = -100,548

Capital gain = 100,548 - 92,800 = $7,748

ReinveJtment income: We can ,olve this by treating the coupon payments as a 6-period annuity, calculating rhe future value based on the semiannual imerest rate, and subtracting the coupon payments. The difference must be the interest earned by reinvesting the coupon payments.

N = .3 x 2 " 6; I1Y = 5/2 = 2.5; PV = 0; PMT '" -3,500; CPT -, FV = $22,357

Page 126

©2008 Kaplan Schweser

Study Session 16 Cross-Reference to CfA Institute Assigned Reading #65 - Yield Measures, Spot Rates, and Forward Rates

Reinvcstmcnt incomc = 22,357 - (6 x 3,500) ~ $1,357

2.BEY = 2 x semianllllal discount ratc

	semiannual discount rate = (1.07) 1/2	- I = 0.31\1J = 3.41J%
	BEY = 2 x 3.41\ 'x,= iJ.RR'X,
3.	0.07)2	= 0.0712 = 7.12%
	annual-pay YTM = [1 -I -2- - 1
1\.	A. Since the T-bills are zero coupon instruments, thcir YTMs arc the 6-molHh and

I-year spot rates. -II) solve lor the			I.S-year spot rate we set the bond's market price
eLJuallO the present value of its (discoullled) cash flows:
	')	...,		102	;-
	._-~--	+- --_":'_--_.\-----
100·	I +~'i~~	[1 + 0.~.)2r	[1.1	Si) ]\

s12	1.1	=	102
1+-			----=1.0615
r, 2. 100--1.9724-1.9375
S			1/

1+~=].OiJI5/5=1.0201

2

51)= 0.0201 x 2 = 0.0402 = 4.02%

R.Compute the YTM on the corporate bond:

N = 1.5 x 2 = 3; PV = -102.395; PMT = 7 I 2 = 3.5; FV = 100; CPT ---> I/Y =

2.6588 x 2 = 5.32%

nominal spread = YTM n	, - YTM.!,	= 5.32% - 4.0% = 1.32%>, or 132 bl)
()OI1U		H.';L~IJrY

Sol ve for the zero-volatility spread by sctting the present value of thc bond's cash flows equal to the bond's price, discounting each cash flow by the Treasury spot rate plus a fixed Z-sprcad.

Substituting each of the choices inlO this equation gives the following bond valucs:

Z-spread	Bond value
127 bp	102.4821
130 bp	102.4387
133 bp	102.3953

©2008 Kaplan Schweser

Page 127

Study Session 16

Cross-Reference to CFA Institute Assigned Reading #65 - Yield Measures, Spot Rates, and Forward Rates

Since the price of the bond is 102.395, a 7.-spread of 133 bp is the correct one.

I

Note that, assuming one of the three zero-volatility spreads given is correct, you could calculate the bond value using the middle spread (130) basis points, get a bond value (102.4387) that is roo high, and know that tile higher zero-volatility spread is the only one that could generate a present value equal to the bond's marker prtce,

Also note that according to the LOS, you arc not responsible for this calculation. Working through this example, however, shonld ensure that you understand the concept of a zero-volatility spread well.

5,	A.	S	]:\	[	S	)2 [ f10	J
		(1+~		=	I+~	1-1-°'2

0,5 flO = 0.03103 X 2 = 0.0621 = 6.21 %

B. 1f l here, refers to the I-year rate, one year from today, expressed as a BEY.

S2)4	r		SI)2(	11'(1)2
(1+-	=	1+-		1+-
2	l		2	2

, I fl)2 l-r--

[ 2

IfI		[H S22 f
			-1
2		(Hif
		1	0.054)4
If)		(	+ --2- -1 = 0.0320
2		(1-1-~'~~~r
I fl	= 2 X		0.0320 = 6.40%

Note that the approximation 2 x 5.4 - 4.4 = 6.4 works very well here and is quite a

bit less work.

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©2008 Kaplan Schweser

---7

Study Session 16 Cross-Reference to CFA Institute Assigned Reading #65 - Yield Measures, Spot Rates, and Forward Rates

C. DiscoulH each of the bond's cash flows (as a percent of par) by du.'appropriatc spot ra tc: I

			2.25	2_2')	2.25			102.2')	c 98.36
		. , -- j			+ ------ j
			1.02	1.0445	1.0769			\.1125
6.	A.	510]2	[	505J(	05fo.5]	(		0.035][	0.038J	6
	[1+ - =		1+ - 1+ -- =				1+ -- 1+ -- =1.0318
	51.0	2		2	2			2	2
		= \.03(,11	112	1=0.0182
	-
	2

S10 = 0.0182 x 2 = 0.0364 = 3.64%

~ = 1.0576 111 -I = 0.0188

2
5 15 = 0.0188 x 2 = 0.0376 = 3_76%
520]4	[	S05J[ ..	0.';fo.5](	0.51"J.()Jl'	051"1.5J'
[1-1--	= 1+-- J+ -- 1+--- 1+ --
. 2		2	2	2	2
		0.o35J(	0.038J(	0.040)(	0.044)	:=\.O	8	09
	= [1+ -- 1+ -- 1+ -- 1+ --
		2	2	2	2
52.0 = 1.0809 114		_ I = 0.0196
2

52 .0 = 0.0196 x 2 = 0.0392 = 3.92%

R.

7.A. Bond (I) has no reinvestment income and will realize its current YTM at maturity unless it defanlts. For the coupon bonds to realizc their current YTM, their coupon income would have to be reinvested at the YfM.

Bond (II): (1.04)10 (100,000) - 100,000 - 10(4,000) = $8,024.43

Bond (III): First, we must calculate the current bond value. N = 5 x 2 = 10; l/Y = 8 / 2 = 4; FY = 100,000; PMT = 4,500; CPT PY = -104,055.45

(1.04)10 (l 04,055.45) - 100,000 - 10(4,500) = $9,027.49

©2008 Kaplan 5chwescr

Page 129

Study Session 16

Cross-Reference to CFA Institute Assigned Reading #65 - Yield Measures, Spot Rates, and FOlward Rates

B.Reinvestment income is most important to thc investor with the 9'Yr) coupon bopd, followed by the 8'Yo coupon bond dnd the zcro-coupon bond. In gcncral, reinveSllllcnt risk inCITdSeS wilh the coupon rale 011 a bond.

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©2008 Kaplan Schweser

The following is a review of the Analysis of Fixl'd Inmme Investments principles designed to address the

learning outcome statements set forth by CFA lll'tilllleQ<J. This IOpil' is also covered in:

INTRODUCTION TO THE MEASUREMENT

OF INTEREST RATE RISK

Study Session 16

EXAM Focus

This topic review is about the relation of yield changes and bond price changes, primarily based on the concepts of duration and convexity. There is really nothing in this study session that can be safely ignored; the calculation of duration, the usc of duration, and the limitations of duration as a mcasure of hond price risk are al1 important. Vou should work to understand what convcxity

is	and its relation to the interest	rate	risk
of	fixed-income securities. There	are	two

important formulas: the formula for effeerive duration and the formula for estimating the price effeer of a yield change hased on hoth duration and convexity. Finally, you should get comfortablc with how and why the convexity of a bond is ant:cted by the presence of cmbedded options.

LOS 66.a: Distinguish between the full valuation approach (the scenario analysis approach) and the duration/convexity approach for measuring interest rate risl<, and explain the advantage of using the full valuation approach.

The full valuation or scenario analysis approach to measuring interest rate risk is based on applying the valuation techniques we have learned for a given change in the yield curve (i.e., for a given interest rt1/£' sr£'nario). For a single option-free bond, this could

be simply, "if the YTM increases by SO bp or ] 00 bp, what is the impact on the value of the bond?" More complicated scenarios can be used as well, such as the effeer on the hond value of a steepening of the yield curve (long-term rates increase more than shan-term rates). If our valuation model is good, the exercise is straightforward: plug

in the rates described in the interest rate scenario(s), and sec what happens to the values of the bonds. For more complex bonds, such as callable bonds, a pricing model that incorporates yield volatility as well as specific yield curve change scenarios is required

to usc the ful1 valuation approach. If the valuation models used arc suHiciently good, this is the theoretically preferred approach. Applied to a portfolio of bonds, one bond at a time, we can get a very good idea of how different interest rate change scenarios will affect the value of the portfolio. Using this approach with extreme changes in interest rates is cal1ed stress testing a bond portfolio.

The duration/convexity approach provides an approximation of the actual interest rate sensitivity of a bond or bond portfolio. Its main advantage is its simplicity compared to the full valuation approach. The full valuation approach can get quite complex and time consuming for a portfolio of more than a few bonds, especially if some of the bonds

have more complex structures, such as call provisions. As we will see shortly, limiting our scenarios to parallel yield curve shifts and "settling" for an estimate of il1(erest rate risk allows us to use the summary measures, duration, and convexity. This greatly simplifies the process of estimating the value impact of overall changes in yield.

©2008 Kaplan Schweser

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