CFA 2018 Question bank 02 The Arbitrage Free Valuation Framework

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Test ID: 7441665

The volatility assumption in a Monte Carlo simulation is important, because it determines the:

speed of prepayments. level of prepayments.

dispersion of future interest rates and the number of possible paths that may be followed.

Explanation

The volatility assumption in a Monte Carlo simulation is important because it determines the dispersion of future interest rates and the number of possible paths that may be followed.

Relativetothebinomialmodel, MonteCarlomethod ismostlikely:

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Question #4 of 38

Question ID: 463765

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Question #5 of 38

Question ID: 463774

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Question #6 of 38

Question ID: 463771

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The purposeofrelative valueanalysisisto determine: the return differential from riding the yieldcurve. whetherabond isfairly valued usingabenchmark yield. whetherastockisfairly valued using present value calculations.

Explanation

The purposeofrelative valueanalysisisto determinewhetherabond isfairly valued. Thebond'sspread oversome benchmarkis compared tothatofarequired spread to determinewhetherthebond isfairly valued. Therequired spread will bethatavailableon comparablesecurities.

Which of the following is a correctstatement concerning the backward induction technique used within the binomialinterest rate tree framework? From the maturity date ofabond:

the corresponding interest rates are weighted by the bond's duration to discount the value of the bond.

a deterministic interest rate pathisused to discount the value of the bond.

the correspondinginterest ratesand interest rate probabilitiesare used to discount the value of the bond.

Explanation

For abond that has N compounding periods, the current value of the bond is determined by computing the bond's possible valuesat period N and working "backwards" to the present. The value at any givennode is the probability-weighted average of the discounted valuesof the next period'snodal values.

With respect tointerest rate models, backward induction refers to determining:

convexity from duration.

one portionof the yield curve fromanother portion.

the current value ofabond based on possible final valuesof the bond.

Explanation

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Questions #7-1

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Question #7 of 38

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Question #8 of 38

QuestionID:463777

DawnAdams, CFA, alongwithherrecently hired staff, haveresponsibilitiesthatrequirethemtobefamiliarwithbackward

inductionmethodology asitisused withabinomial valuationmodel. Adams, however, is concerned thatsomeofherstaff, particularly thosenotenrolled intheCFA program, arealittleweakinthisarea. Toassesstheirunderstandingofthebinomial modeland itsuses, Adams presented herstaffwiththefirsttwo yearsofthebinomialinterestratetreeforan8% annually compounded bond (shownbelow). Theforward ratesand the corresponding valuesshowninthistreearebased onan assumed interestrate volatility of 20%.

AmemberofAdams" staffhasbeenasked torespond tothefollowing:

ComputeV , the valueofthebond atnode1L.

$101.05.

$95.99.

$103.58.

Explanation

V = (½)[(V + C) / (1 + r )] + [(V + C) / (1 + r )]

V = (½)[(99.455 + 8) / (1 + 0.05331)] + [(102.755 + 8) / (1 + 0.05331)] = $103.583

(Study Session14, LOS 47.i)

ComputeV , the valueofthebond atnode1U.

$91.72.

$99.01.

$99.13.

Explanation

V = (½)[(V + C) / (1 + r )] + [(V + C)/(1 + r )]

V = (½)[(98.565 + 8) / (1 + 0.079529)] + [(99.455 + 8) / (1 + 0.079529)] = $99.127

1L

1L 2LU 1L 2,LL 1L

1L

1U

1U 2,UU 1U 2,UL 1U

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Question #12 of 38

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Question #14 of 38

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Rearranging, the valueofthe put canbestated as: V = V − V

V was computed tobe$105.173inthe previous question, and V was determined tobe$104.755inthe question priortothat. Sothe valueoftheembedded putoptionforthebond underanalysisis:

$105.173 − 104.755 = $0.418 (Study Session14, LOS 47.e, i)

Whichofthefollowingstatementsregardingtheoptionadjusted spread (OAS)fora callablebond isleastaccurate? The OAS is the spread on a bond with an embedded option after the embedded

option cost has been removed.

The OAS isequaltothe Z-spread plustheoption cost.

The OAS fora corporatebond mustbe calculated usingabinomialinterestrate model.

Explanation

The OAS isequaltothe Z-spread minustheoption cost. Bothoftheother choicesaretruestatements. (Study Session14, LOS 47.g)

Why isthebackward induction methodology used to valueabond ratherthan aforward induction scheme?

The price of the bond is known at maturity. The convexity ofabond changesovertime.

Futureinterestrate changesare difficulttoforecast.

Explanation

Theobjectiveisto valueabond's current pricewhilethebond priceatmaturity isknown. Therefore, priceatmaturity is used asastarting

point, and weworkbackward tothe current value.

A3-year, 3% annual pay, $100 parbond is valued using pathwise valuation. Theinterestrate pathsare provided below: Path Year 1 Year 2 Year 3

1 2% 2.8050% 4.0787% 2 2% 2.8050% 3.0216% 3 2% 2.0780% 3.0216% 4 2% 2.0780% 2.2384% The valueofthebond in path 2 is closestto:

put putable nonputable

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Question #17 of 38

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Question #18 of 38

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Question #1

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QuestionID:463766

Thefollowingarethe yieldson variousbonds. TherelevantbenchmarkisthatofTreasury securities. Treasury Bond Yield 4.00%

Bond SectorYield 4.50% ComparableBond Yield 6.00% ABCBond Yield 6.50%

IstheABCbond undervalued orovervalued and why? Usingrelative valueanalysis, theABCbond is:

overvalued because its spread is greater than that of comparable bonds.

undervalued becauseitsspread isgreaterthanthatof comparablebonds.

undervalued becauseits yield isgreaterthanthatofTreasuries.

Explanation

The purposeofrelative valueanalysisisto determinewhetherabond isfairly valued. Thebond'sspread oversome benchmarkis compared tothatofarequired spread to determinewhetherthebond isfairly valued. Therequired spread will bethatavailableon comparablesecurities. Inthisexample, therelevantbenchmarkwasTreasury securities. Thespread for ABCbondsoverTreasurieswas 2.5%. Thespread for comparablebondsoverTreasurieswas 2.0%. Thehigherspread for ABCbondsmeansthatthey arerelatively undervalued (their priceislowbecausetheir yield ishigher).

Whichofthefollowing choicesisleast-likely a property ofabinomialinterestratetree?

Mean reversion of interest rates. Non-negativeinterestrates. Higher volatility athigherrates.

Explanation

Abinomialinterestratetreehastwo desirable properties:non-negativeinterestratesand higher volatility athigherrates. Binomialtrees donotforcemeanreversionofrates.

Thefollowingarethe yieldson variousbonds. Therelevantbenchmarkisthatofthebond sector. Treasury Bond Yield 3.00%

Bond SectorYield 3.25% ComparableBond Yield 5.75% ABCBond Yield 5.50%

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2 2% 2.8050% 3.0216% 3 2% 2.0780% 3.0216% 4 2% 2.0780% 2.2384% The valueofthebond in path3is closestto:

$101.85 $99.88 $100.02

Explanation

Answer:Path3 value =

Usingthefollowinginterestratetreeofsemiannualinterestrateswhatisthe valueofan option freesemiannualbond thathasone year remainingtomaturity and hasa6% coupon rate?

6.53%

6.30% 5.67%

97.53.

99.89. 98.52.

Explanation

Theoption-freebond pricetreeisasfollows:

100.00

A ==> 99.79 99.89 100.00

100.20

100.00

Asan example, the priceat nodeAisobtained asfollows:

Price = (prob × (P + coupon/2) + prob × (P + coupon/2))/(1 + rate) = (0.5 × (100 + 3) + 0.5 × (100 + 3))/(1 + 0.0653) = 99.79.

Thebond valuesattheother nodesareobtained in thesameway.

The calculation for node 0 ortime 0 is

0.5[(99.79 + 3)/(1+ 0.063) + (100.20 + 3)/(1 + 0.063) ] = 99.89

A up down 0.5 0.5

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QuestionID:472692

TimBrospackisgeneratingabinomialinterestratetreeassuminga volatility of15%. Current1-yearspotrateis5%. The 1-yearforward rateinthesecond yeariseitheralowestimateof5.250% orahighestimateof7.087%. Themiddle1-year forward ratein yearthreeisestimated at6.25%. Theuppernode1-yearforward ratein yearthreeisclosestto:

7.747% 6.445% 8.437%

Explanation

Uppernodeinterestrate = 6.25 × e = 8.437%

Whichofthefollowing choicesisleast-likelya property ofabinomialinterestratetree? Adjacent forward rates in a nodal period are one standarddeviation apart. Higher volatility athigherrates.

Non-negativeinterestrates.

Explanation

Abinomialinterestratetreehastwo desirable properties:non-negativeinterestratesand higher volatility athigherrates. Additionally, adjacentforward ratesinanodal period aretwostandard deviationsapart.

1 2% 2.8050% 4.0787% 2 2% 2.8050% 3.0216% 3 2% 2.0780% 3.0216%

4 2% 2.0780% 2.2384% The valueofthebond in path1is closestto:

$98.77 $100.18 $101.88

Explanation

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Question #3

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Path1 value =

A callablebond withan 8.2% annual coupon willmaturein two yearsat par value. The currentone-yearspotrateis7.9%. Forthesecond year, the yield-volatility modelforecaststhattheone-yearratewillbeeither6.8% or7.6%. The call priceis101. Usingabinomialinterest ratetree, whatisthe current price?

101.000.

100.279.

100.558.

Explanation

Thetreewillhavethree nodal periods: 0, 1, and 2. Thegoalistofind the valueat node 0. Weknowthe valueforallthe nodesin nodal

period 2:V=100. In nodal period 1, therewillbetwo possible prices: V =[(100+8.2)/1.076+(100+8.2)/1.076]/2 = 100.558

V =[(100+8.2)/1.068+(100+8.2)/1.068]/2= 101.311

SinceV isgreaterthan the call price, the call priceisentered intotheformulabelow:

V=[(100.558+8.2)/1.079)+(101+8.2)/1.079)]/2 = 101.000.

Fora putablebond, callablebond, or putable/callablebond, the nodal-decision processwithin thebackward induction methodology ofthe interestratetreeframeworkrequiresthatateach nodethe possible valueswill:

not be higher than the call price or lower than the put price.

includetheface valueofthebond.

be, in number, two plusthe numberofembedded options.

Explanation

Ateach node, therewillonly betwo values. Ateach node, theanalystmust determineiftheinitially calculated valueswillbebelowthe

put priceorabovethe call price. Ifa calculated valuefallsbelowthe put price:V = the put price. Likewise, ifa calculated valuefalls abovethe call price, then V = the call price. Thusthe putand call pricearelowerand upperlimits, respectively, ofthebond's valueata

node.

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1,L

0

i,U

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QuestionID:463767

Question #31 of 38

QuestionID:472597

Usingthefollowinginterestratetreeofsemiannualinterestrateswhatisthe valueofanoptionfreebond thathasone year remainingtomaturity and has5% couponratewithsemi-annual coupon payments.

Today 6 Months 7.30% 6.20%

5.90%

97.53.

98.98. 98.67.

Explanation

Theoption-freebond pricetreeisasfollows:

100.00

A → 98.89

98.67 100.00

99.56

100.00

Asan example, the priceat nodeAisobtained asfollows:

Price = (prob × (P + (coupon / 2)) + prob × (P + (coupon / 2)) / (1 + (rate / 2)) = (0.5 × (100 + 2.5) + 0.5 × (100 + 2.5) / (1 + (0.0730 / 2)) = 98.89. Thebond valuesattheother nodesareobtained in thesameway.

The calculation for node 0 ortime 0 is

0.5[(98.89 + 2.5) / (1+ 0.062 / 2) + (99.56 + 2.5) / (1 + 0.062 / 2)] =

0.5(98.3414 + 98.9913) = 98.6663

Thegovernmentbond spotrate curveisgivenbelow: Maturity (years) Spotrate

0.5 1.25%

1.0 1.30%

1.5 1.80%

2.0 2.00%

2.5 2.20%

3.0 2.25%

3.5 2.28%

4.0 2.30%

Computetheissue priceofa3-year, 3% semiannual coupongovernmentbond witha par valueof$100.

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Question #3

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Question #33 of 38

QuestionID:472690

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$102.15 $104.09 $102.20

Explanation

Value =

= $102.20

Abond witha10% annual coupon willmaturein two yearsat par value. The currentone-yearspotrateis8.5%. Forthesecond year, the

yield volatility modelforecaststhattheone-yearratewillbeeither8% or 9%. Usingabinomialinterestratetree, whatisthe current price?

102.659.

101.837.

103.572.

Explanation

Thetreewillhavethree nodal periods: 0, 1, and 2. Thegoalistofind the valueat node 0. Weknowthe valuein nodal period 2:V=100. In nodal period 1, therewillbetwo possible prices:

V =[(100+10)/1.09+(100+10)/1.09]/2= 100.917 V =[(100+10)/1.08+(100+10)/1.08]/2= 101.852

Thus

V=[(100.917+10)/1.085+(101.852+10)/1.085]/2= 102.659

1 2% 2.8050% 4.0787% 2 2% 2.8050% 3.0216% 3 2% 2.0780% 3.0216% 4 2% 2.0780% 2.2384% The valueofthebond in path4is closestto:

$100.02

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1,U

1,L

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Question #37 of 38

QuestionID:472605

Question #38 of 38

QuestionID:463772

Thenweusethespotratesto valuethe4-year, 5% annual pay bond:

Value =

Sam Roit, CFA, has collected thefollowinginformationonthe parrate curve, spotrates, and forward ratestogeneratea binomialinterestratetree consistentwiththis data.

Maturity ParRate SpotRate

1 5% 5.000%

2 6% 6.030%

3 7% 7.097%

Thebinomialtreegenerated isshownbelow (one yearforward rates)assuminga volatility levelof10%:

0 1 2

5% 7.7099% C

A 9.2625% B

Riotalsogenerated anothertreeusingthesamespotratesbutthistimeassuminga volatility levelof 20% asshownbelow:

0 1 2

5% 8.9480% 13.8180% 5.9980% 9.2625% 6.2088%

Theone-yearforward raterepresented by 'B' is closestto:

7.5835% 7.4223% 8.7732%

Explanation

Valuerepresented by 'B' = 9.2625 / e = 7.5835%

Abinomialmodelorany othermodelthat usesthebackward induction method cannotbe used to valueamortgage-backed security (MBS)because:

the cash flows for the MBS are dependent upon the path that interest rates follow. the prepaymentsoccurlinearly overthelifeofan interestratetrend (either up or down).

the cashflowsforan MBS only depend on the currentrate, notthe paththatrateshave followed.

Explanation

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