Financial Analysts Journal
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The Problem with Redemption Yields
Stephen M. Schaefer
To cite this article: Stephen M. Schaefer (1977) The Problem with Redemption Yields, Financial
Analysts Journal, 33:4, 59-67, DOI: 10.2469/faj.v33.n4.59
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by StephenM. Schaefer
The
Problem
with
Redemption
Yields
4 A coupon bond resembles a portfolioof pure
discount bonds in that it makes payments at a
numberof futuredates. Inan efficient market,the
price of a coupon bond should equal the price of the
corresponding portfolioof pure discount bonds.
Such a marketdiscounts payments made by
differentbonds at the same point in time at the same
spot rate - the rate of exchange between money
today and money at a single date in the futurewhereas it discounts payments made by the same
bond at differentpoints in time at differentspot
rates.
These propertiesof spot rates contrast sharply
with the propertiesof the redemptionyield - the
single internalrate of returnequating the
discounted value of all futurepaymentsto a bond's
price. Redemptionyield is a derived figure;we need
to know the bond's price before we compute it. The
redemptionyield, therefore, cannot help us estimate
the bond's value.
When comparingtwo or more bonds, it is far more
useful to estimate the relevantsequence of spot
rates than to calculate the bonds' redemption
yields. >
anddeviationsfromsucha curvewill not implyanythingaboutover- or underpricing.
Thirdly,volatilities calculatedon a yield basisare not comparable,
andthereforetell us nothingabouta bond'sopportunityor risk.
The redemptionyieldon a bond is, of course,just
the internalrateof returnover the life of the bond.
Manyof the issuesraisedin thispapercorrespondto
well knownobjectionsto the use of the internalrate
of returnas an investmentcriterion.
Yield as a Measure of Value
In economics,a spot interestrateusuallymeasures
the rate of exchange between money today and
moneyat a single date in the future.On a couponbearingbond,wherethereareinterveningpayments,
the spotrateis distinctfromthe redemptionyield.If
the one-yearspotrateis RI, the presentvalueof one
dollarpaid in a year'stime is:
dl_
(1 +1RI)
Similarly,if the two-yearspotrateis R2,the present
value of one dollarpaid in two years'time is:
d2
(1 +R2)2
YIELD is such a basic
HE REDEMPITION
item in the bond manager'stoolboxthat it is Spot rateswill dependon the consensusof individsometimesuseful to rememberits serious uals'time preferences,expectationsof futureyields
limitations:Firstly,yield is an imprecisemeasureof and degreesof risk-aversion.
investmentvalue;secondly,since yields dependon
We can thinkof d1 as the marketpriceof a pure
the coupon,yields of bondswith differentcoupons discountbond promisingto pay one dollar in one
will not generallylie alonga smoothcurve.Fittinga year'stime and d2 as the value of a similarbond
curvethroughpointsthatdo not andshould not lie promisingto pay one dollarin two years'time.' A
alonga curveis unlikelyto be a profitableexercise, couponbondresemblesa portfolio of purediscount
bonds in that it makes paymentsat a numberof
Stephen Schaefer, Prudential Research Fellow in Indates.An efficientmarketofferslittle reward
future
vestment at the London Graduate School of Business
to
packaging
securities,hencethe priceof a coupon
the
at
Professor
Assistant
Visiting
is
currently
Studies,
University of Chicago Graduate School of Business. bond should be similarto the price of the correHe is more than usually grateful to Professor Richard spondingportfolioof purediscountbonds.For exBrealey for his assistance in the preparation of this ar- ample,a bond witha 10 per cent couponanda face
ticle, though the author is naturally responsible for
any errors in the end result.
1. Footnotesappearat end of article.
FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1977 E1 59
The CFA Institute
is collaborating with JSTOR to digitize, preserve, and extend access to
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yield on their holdings,the yield on the portfolio
doesnot in factequalthe averageof theyieldson the
individualholdings,andindeedcannotbe calculated
10
110
fromtheseyields.For example,supposewe invested
(1 + R1) (1 + R2)2
$100 in a three per cent five-yearbond priced at
Similarly,a bond with a six per cent coupon ma- 91.25 to yieldfivepercentand$100 in a 12 percent
turing in three years would have a presentvalue five-yearbondpricedat 107.72to yield 10 percent.
The averageredemptionyieldis 7.5 percent,butthe
equaling:
redemptionyield on the portfoliois 7.42 per cent.
9
6
+
6
+
106
Its inabilityto tell us anythingaboutthe characteris(1 +RI) (1 +R2)2 (1 +R3)3
tics of the portfolioseriouslylimitsthe usefulnessof
The above formulasunderscoretwo important the redemptionyield.
In discussingthe redemptionyield, manyinvespoints:(1) Paymentsmadeby differentbondsat the
samepoint in time are discountedat the samerate tors place greatemphasison the importanceof the
and(2) paymentsmadeby the samebondat different rates at which futurecoupons may be reinvested.
points in time are typicallydiscountedat different They point out that only if coupons can be reinrates.Preciselythe oppositeis true of the redemp- vestedat a rateequalto the redemptionyieldwillthe
tion yield,whichis the singleinternalrateof return realizedreturnequal the redemptionyield. While
equatingthe discountedvalueof all futurepayments correct,this factprovesquiteirrelevantto the probto the bond'scurrentprice.Thus,if the priceof the lemof assessingthe relativevaluesof risklessbonds.
two-year10 per cent bond is P, we calculatethe re- The importantparametersare,rather,the spot rates
demptionyieldby findingthevalueof y thatsatisfies implicitin today'sprices;becausetheypermitcalculation of presentvalues, spot rates -not future
the equation:
valuesbasedon some assumedreinvestmentraterepresentthe importantquantitiesfor the invest10
110
p
+
mentdecision.
(1+ y)2
(1+ y)
The Yield Curve
This impliesthat paymentsmadeby the samebond
Since
we
can
so littlefroma directcompariin
time
deduce
are
all
at
the
at differentpoints
discounted
one dollar derived son of redemptionyields,it is perhapsnot surprising
same rate, y. Correspondingly,
froma 10 percent bond is not assumedto be worth that more elaboratecalculationsbased on these
the same as one dollarderivedfroma six per cent measuresyieldlittlefurtherinformation.
As we have
alreadyindicated,the redemptionyielddependsnot
bond.
A furthermajordisadvantageof the redemption only on the spotratesandtermto maturity,but also
yield lies in the factthatit is a derivedfigure.In our on the size of the coupon.Ingeneral,twobondswith
economicmodel,individualpreferencesand expec- the same maturitybut differentcouponswill have
tations combine to determine spot rates. Dis- differentredemptionyields. Tryingto fit a smooth
countingthe paymentsat the appropriatespot rates curvethrougha set of pointsthat do not lie on the
then determinesthe price of any coupon bond. samecurveis thereforepointless,andit is evenmore
Finally,the priceandcouponpaymentstogetherim- pointlessto tryto readomensin the deviationsfrom
ply a redemptionyield. In otherwords,we need to the curve.
A moreusefulexerciseis to estimatethe sequence
knowthe pricebeforewe can computethe redemption yield. Knowingthe redemptionyield for one of spot rates,since we mayexpectthese to lie on a
bond, with its particularpatternof payments,will smoothcurve.Thereare severalalternativewaysto
not tell us anythingaboutthe appropriate
yieldfor a do this.3The mostappropriate
methodin anyparticsecond bond with a differentpatternof payments. ularcasewill dependprimarilyon the characteristics
Since we need to know a bond'svalueto compute of the data. The calculatedspot rates (or "zerothe redemptionyield, the redemptionyield cannot couponyield curve")maybe substantiallydifferent
be of anyuse in estimatingthe bond'svalue.We can- from the redemption yields for corresponding
maturities.Figure 1 shows an estimatedspot rate
not reachheavenby haulingon our bootstraps.
The complexityof the yieldconceptbringswithit curveand correspondingredemptionyields for the
a furtherseriousshortcoming.Institutionsholdport- BritishGovernmentSecuritiesMarketin September
folios, ratherthanindividualbonds;it is the charac- 1974. Notice that, while yields on 25-year bonds
teristicsof the portfoliothatareof primaryinterest. were approximately15 per cent, the 25-year spot
Although many institutionscalculate the average rateis estimatedat over 25 per cent.
value of $100 maturingin two yearswould have a
presentvalueof 1Od1+ 110d2, whichequals:
60 0 FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1977
The Relationship between Redemption
Yields and Spot Rates
principal repayment).For example, a three-year
bondwitha 10 percentcouponmaybe regardedas
It was suggestedearlier that a coupon-bearing a portfoliocontaining10 one dollarperyearthreebond could be regardedas a portfolioof puredis- year annuitiesand 100 three-yearpure discount
countbonds.In the sameway,so longas the coupon bonds,each payingone dollarat maturity.
streamis uniform,we may consider an n-period
Now, the price of an n-year annuity depends
bond as a portfoliocontainingan n-periodannuity solelyon the spotratesR1,R2, . . ., R,. Thusthe re(the couponstream)and a purediscountbond (the demptionyield on an annuity,the annuity yield,
Figure 1: Redemption Yields and Estimated TermStructure for September 1974
(U.K.Government Securities Market)
25
20
E
00 0(9
15
0
0
O0
0
~
0
~
_
15
5
5
10
20
15
25
0
-YIELDS
*
-
SPOT
30
Maturity(Years)
FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1977 E) 61
also dependson R1, R2, . . ., Rn.For any spot rate will elaborateon this point later.
curvethereexists,therefore,a correspondingannuMeanwhile,considerthe effectof couponsize on
ityyieldcurve.The redemptionyieldon an n-period the redemptionyieldwhenmaturityis heldconstant.
coupon-bearingbond is an averageof the n-period Justas a discountbond maybe thoughtof as a zeroannuityyieldandthe n-periodspotrate,in the sense coupon bond, so an annuitymay be consideredan
thatthe yield on the bond alwayslies betweenthese infinite-couponbond.We mightimagine,therefore,
two numbers.4Figure2 showsa spot ratecurveand thatas the size of the couponincreases,the redempthe correspondingannuityyield curve;the redemp- tion yield tends away from the spot rate curve
tion yields on all coupon-bearingbonds lie in the towardsthe annuityyield curve.5This is indeedthe
shadedareabetweenthe curves.Becausethe annuity case: Whenthe spot rate is higherthan the correyieldis itselfan averageof the spotrates,the annuity spondingannuityyield, redemptionyields decline
yield curvewill tend to lie below a risingspot rate with increasingcoupon and vice versa.This result
curveandabovea fallingspotratecurve.It is possi- maybe conciselystatedas:
ble for the annuityyield curve and the spot rate
dy,
curveto intersect-as they do at point X in Figure
OasRn An
dc
2-and in this case all coupon-bearingbondswith
that maturityhave the same redemptionyield. We
Figure 2: Spot Rate Curve, Annuity Yield and Redemption Yields
Redemption Yields
on Coupon-Bearing Bonds
12
10
Spot RateCurve
CL
8
6
0
10
20
Maturity(Years)
62 O FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1977
30
40
yieldcurvesfor low couponbonds.
where y,nis the redemptionyield on an n-period constant-coupon
bondwithcouponc, A,,is the n-periodannuityyield These result from the differencein characteristics
and,as before,Rnis the n-periodspotrate.Redemp- between short- and long-maturitylow coupon
tionyieldson bondswitha givencouponwill lie on a bonds. The former behave much like discount
smooth curve; Figure 3 illustratesseveral such bonds,and theirredemptionyields are close to the
bonds,
spot rates.The latter,like all long-maturity
curves.
The resultgiven above has to do only with the are similarto annuities;when spot ratesare monorelativelocation of constant-couponyield curves; tonic increasing,the yield on these bonds may be
whatabouttheirshape?In particularwe mightcon- substantiallybelowthe termstructureandthe assosider whether,for a given maturity,all constant- ciatedyield curvesmaythus be humped.
If the spotrateis higherthanthe annuityyieldand
couponyield curveshaveslopeswiththe samesign.
In otherwords,even if we restrictour attentionto the spotratecurvehas a smallpositiveslope,a given
yieldcurvecan be upwardsloping,
constant-couponyield curves, can we talk about constant-coupon
yield curvesrisingor fallingat a particularmaturity at maturityn, when:
withoutspecifyingthe coupon size? Perhaps,surC/F> Rn-1/n,
prisingly,we cannot.For example,a rising(monotonic) spot rate curve can produce hump-shaped whereC/F is the ratioof couponto face value and
Figure 3: Constant-Coupon Yield Curves
12 2
X/
\.
10
10
8
D
D
c= 10%
Spot Rate Curve
-
W:
/
Yield Curve
~~~~Annuity
8
6
0
10
20
30
40
Maturity(Years)
FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1977 0 63
R, is the n-period spot rate. Yield curves for tive, this is simplya resultof the arithmetic.On the
coupons lower than this value may be downward other hand,there is no reasonwhy the underlying
sloping at this point; all yield curves for higher spot rate curveshouldeventuallybecomehorizoncoupons- including,of course, the annuityyield tal.
For practicalpurposesthe detailsof these results
curve-will be upwardslopingat this maturity.
Figure4 showsa risingmonotonicspotratecurve, areperhapsof only slightimportance.Theydo illusannuityyieldcurve,andconstant- trate,however,how difficultit is, in view of their
the corresponding
couponyieldcurvesforcouponsof one percent,five very complexnature,to drawany sensibleconclupercent and 10 percent.The one percentcurvehas sions from the direct comparisonof redemption
a pronouncedhump,the five per cent curvea slight yields.Spotrates,on the otherhand,do permitlogihumpand the 10 per cent curve,like the spot rate cally consistentcomparisonsof bond prices.
curve and the annuityyield curve, is monotonic.
yieldcurvesex- Intersecting Yield Curves and the
Noticealsothatall constant-coupon
cept the spot rate curvetend towardsthe annuity Par Yield Curve
We pointedout earlierthat, when the spot rate
yield curveas maturityincreases.They are also asymptoticallyhorizontalno matterwhat shape the curveand the annuityyield curveintersect,the rebondsequal
spotratecurveadopts.So longas spotratesareposi- demptionyieldson all coupon-bearing
Figure 4: Showing Humped Constant-Coupon Yield Curves
Derived from Monotonic Spot Rate Curve
16
Spot Rate Curve
-
14
%
I~~~~~~~~~~~~~~~
12
10
10
~
~
~ ~
~
~
~
~
~
~
~
~ ~ ~~c
0
8
AnnuityYield Curve
6
4
0
10
20
Maturity(Years)
64 E FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1977
30
40
TABLE 1: A Term Structure That Produces Intersecting Yield Curves for n = 3
Annuity
Discount
Annuity
Spot
Yield
Rate
Factor
Price
Period
A1
d1
R1
j
ai
0.9383
0.9383
0.0658
1
0.0658
1.7354
0.7971
0.1000
2
0.1200
2.4868
0.1000
3
0.7513
0.1000
the spot rateand the annuityyield at theirpoint of ever;one cannotcalculatepar yields directlyfrom
intersection.Thisphenomenonoccurswhenthe spot redemptionyields.Figure5 showsthe estimatedpar
yield curvefor the datadisplayedin Figure1.
ratecurvesatisfiesthe followingcondition:
Volatility
We haveshownthat redemptionyieldson bonds
an
where,as before,Rnis the n-periodspot rate,dnis withdifferentcouponsarenot properlycomparable.
The same point appliesto measuresbased on rethe presentvalueof one dollarpaid at time n, or
demptionyields,of whichvolatilityis the mostcommon example.Volatility is usuallydefined as the
d
change.in price(or proportionalchangein price)for
(1+
a given changein yield. It is used both to measure
anda, is the presentvaluenof an n-periodannuity,or risk and to select bonds, given an assessmentof
futureinterestrates.Thus if bond A has a higher
an=
Idi
volatilitythan bond B, it would be arguedthat a
J=1
given fall in yield would producea largerproporTable 1 givesan exampleof a termstructurethat tional price incrementin A than in B. We mustresatisfiesthis conditionfor n = 3. Notice that the member,however,thatredemptionyieldsarealtered
right-handside of the equationequals:
by changesin spot rates,andthata givenchangein
1-0.7513
spotrateswillgenerallychangeby differentamounts
0.10
l-d3=
the redemptionyields of two bonds with different
2.4868
a3
coupons.Thus,eventhoughA hasa highervolatility
whichis equalto R3, the three-yearspot rate.If we thanB, it is quitepossiblethatspotratesmaychange
now calculatethe priceof a bondwithan arbitrarily in such a waythat B's proportionalpriceincrement
chosen coupon, we shall find that the impliedre- will exceed A's. The relationshipbetween price
demptionyieldis always10 percent.Forexample,if change,volatilityand changein redemptionyield is
we makethe coupon10 percent,the priceof a bond a tautology,andtells us nothingabouthow pricesof
with face value $100 is givenby:
differentbondswill changeas the underlyinginterest
rateschange.
p
10d1+ 10d2 + 110d3 = 10a3 + 100d3.
the valuesgivenin Table 1, we obtaina
Substituting
Conclusion
priceof exactly$100. In otherwords,a three-year
for
bondwitha couponof 10 percentwouldsell at par; The explanation the wide use of redemption
consequentlythe redemptionyield mustbe 10 per yield is obvious:Giventhe termsof a bond and its
price,the redemptionyieldcan be calculatedunamcent.6
The right-handside of the equationhas another biguouslyevenif onlyone bondis available.ByconIt is the couponthathas trast,in the absenceof purediscountbonds,we can
moregeneralinterpretation:
to be offeredto makean n-periodbond sell at par. only estimatespotrates- andthenonly if we havea
The "paryield curve,"a plot of thesevaluesagainst sampleof issues.Similarly,the redemptionyield apmaturity,will be a smoothcurveand representsan pearsto offerthe advantagesof economy,for a bond
alternative,logicallyconsistentwayof describingthe hasonly one redemptionyield,evenif eachpayment
termstructure.Indeedit mayappealto practitioners is discountedat a differentrate.Butthesecharactermorethanthe spot ratecurve,since paryields will isticsare weaknesses,not strengths.Becausea yield
generallylie closer to redemptionyields than spot is uniqueto a security,ratherthan to a payment,
ratesdo. The calculationof paryieldsrequiresesti: comparisonbetween bonds is wholly impossible.
mationof spot ratesas an intermediatestep, how- The redemption yield is calculable and unR
1 - dn
FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1977 0
65
Figure 5: Par Yield Curve and Redemption Yields for September 1974
(U.K.Government Securities Market)
25
20
E
0)
c:
a)
0~~~
&
00
10
?
0???
0gc
5
0-YIELDS
*-PAR
I
l
l
l
lI
5
10
15
20
25
30
Maturity(Years)
Footnotes
ambiguousonly becauseits calculationis basedon
the priceor value.As a summarydescriptionof the 1. d1,d2,. . . aresometimescalled"discountfactors,"and
waythingsare,it hasno placein anyanalysisof how
R1, R2 ... the term structure.
thingsshouldor will be.
2. Whilethis is the standarddiscountingformula,it is
worthwhilebearing in mind the conditions under
Bond man-agement
will make real progressonly
which it holds. Bonds will be pricedaccordingto a
when investorsstart to think in terms of present
commonsetof spotratesprovidedthata bondmatures
valuesratherthanfuturevalues,andtheycando this
in each futureperiodand that shortsellingis allowed.
only if they think in termsof the underlyingspot
(This
guaranteesa completemarket.)However,it will
rates,ratherthanthe yield to maturity.a
66 E FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1977
not hold if, for example (a) short selling is not allowed
and (b) coupon payments and principal repayments
are taxed at different rates for some investors and (c)
there is some heterogeneity in investor tax rates.
3. See W.T. Carleton and IA. Cooper, "Estimation and
Uses of the Term Structure of Interest Rates" (paper
delivered at Annual Meeting of European Finance Association, London, September, 1975); J.H. McCulloch, "Measuring the Term Structure of Interest
Rates," Journal of Business (January 1971),
pp. 19-31; S.M. Schaefer, "On Measuring the Term
Structureof Interest Rates" (London Graduate School
of Business Studies, Institute of Finance and Accounting Working Paper IFA-2-74); and S.D. Hodges
and S.M. Schaefer, "A Model for Bond Portfolio Improvement," forthcoming in Journal of Financial
and Quantitative Analysis.
4. The proof of this and the other propositions in the text
are not given here for the sake of brevity but are included in a supplement available from the author.
5. See A. Buse, "Expectations, Prices, Coupons and
Yields," Journal of Finance (September 1970),
pp. 809-818; J.L. Carr, P.J. Halpern and J.S. McCallum, "Correctingthe Yield Curve: a Re-Interpretation of the Duration Problem" Journal of Finance
(September 1974), pp. 1287-1294; C. Khang, "Expectation, Prices, Coupons and Yields: Comment," Journal of Finance (September 1975), pp. 1137-1140;
B.G. Malkiel, The Term Structure of Interest Rates
(Englewood Cliffs, NJ.: Prentice-Hall, 1966); and
Schaefer, "On Measuringthe Term Structureof Interest Rates." Their results are special cases of this result.
The results of these authors concern the sign of dy/dc
when the term structure is rising or falling. A rising or
falling term structure is sufficient but not necessary for
establishingthe sign of dy/dc, whereas this condition is
both necessary and sufficient. In the U.K., where yields
have generally increased with maturity, the relationship between redemption yields and coupon level has
usually been the reverseof that predicted by the result.
The explanation is connected with the tax system and,
more particularly,with the fact that the conditions for
the simple discounting formula to apply do not, most
probably, obtain. See Footnote 2.
6. Table 1 provides a counterexample to Weingartner's
proposition that "two bonds having the same term-tomaturity and the same frequency of coupons, but having different uniform coupons, cannot have the same
yield-to-maturity unless the market yield curve is absolutely flat." See H.H. Weingartner,"The Generalised Rate of Return," Journal of Financial and
Quantitative Analysis (September 1966), p. 14.
7. The Bank of England has, for some years, been
publishing estimates of par yields in the Bank of England Quarterly Bulletin. See J.P. Burman and W.R.
White, "Yield Curves for Gilt Edged Stocks," Bank of
England Quarterly Bulletin (December 1972),
pp. 467-486 and J.P. Burman, "Yield Curves for Gilt
Edged Stocks,"
BEQB (September
1973),
pp. 315-326 and (June 1976), pp. 212-215.
~~~~~~~~~~~~~CENTRAL
AND SOUTH WEST
P
CORPORATION
Benelic!ialcPP0W......
CO MMONSTOCK
DIVIDEND
192ndCONSECUTIVE
QUARTERLY
CASH
COMMON
STOCK
DIVIDEND
The Board of Directors has declared per share cash dividends payable June 30, 1977 to stockholders of record at
the close of business June 3, 1977.
Common Stock-Quarterly-$.40
.
.
5 /o Cumulative Preferred Stock-Semi-annual-$1.25
$4.50 Dividend Cumulative Preferred SlockSemi-annual-$2.25
k,'
...
The Board of Directors of Central
and South West Corporation at its
meeting held on April 21, 1977,
declared a regular quarterly dividend ofthirty-one and one-half cents
(311/2?) per share on the Corpora-
gis
payable May 31, 1977, to stockholders of record April 29, 1977.
LeroyJ. Scheuerman
Secretary and Treasurer
Wilmington Delaware 19899
g
Payable July 31, 1977 to stockholders of record at the close
of business July 8, 1977.
$5.50 Dividend Cumulative Convertible
Preferred Stock-Quarterly-$1.375
WesternAutoSupply
Spiegel, Inc.
System Operating Companies
Edwin M. Stokes
Vice-President and
Beneficial Finance System
Secretary
HEhiHE
May 20, 1977
gSouthwestern
CentralPowerand LightCompany
PublicServiceCompany
of Oklahoma
ElectricPowerCompany
West Texas Utilities Company
FINANCIAL ANALYSTS JOURNAL / jULY-AUGUST 1977 0 67