t4.basic bond pricing (tuckman)

L1.T4. Valuation & Risk Models,

Basic bond pricing (spot and forward rates, yield) (based on Chapters 1 & 2 of Tuckman):

FRM 2011 Practice Questions

By David Harper, CFA FRM CIPM

www.bionicturtle.com

FRM 2011 VALUATION: BASIC BOND PRICING 1 www.bionicturtle.com

Table of Contents

Question 10: Time value of money .................................................................... 2

Question 11: Treasury STRIPS .......................................................................... 5

Question 12: Spot rates ................................................................................. 8

Question 13: Forward rates .......................................................................... 10

Question 14: Yield to maturity (YTM) ............................................................... 12


Question 10: Time value of money

AIMs: Describe and contrast individual and market expressions of the time

value of money. Define discount factor and use a discount function to

compute present and future values. Define the “law of one price”,

support it using an arbitrage argument, and describe how it can be

applied to bond pricing.

10.1. Three different U.S. Treasury notes pay semi-annual coupons and mature in exactly one year; i.e., each pays the next coupon in six months and matures six months subsequently. The price of Bond A with a coupon rate of 2.0% per annum is $99.02 and the price of Bond C with a coupon rate of 7.0% per annum is $103.91. If Bond B has a coupon rate of 4.0% per annum, what is the price of Bond B? (this is a variation on an actual previous exam question)

a) $99.12 b) $100.56 c) $100.98 d) $101.12

10.2. Assume that a U.S. Treasury bill will pay $1,000 in one year and the security is default free (there is absolutely no credit risk). The price of this bill today is given by P(0). Which of the following statements, according to Tuckman, is most true about individual versus market expressions of the theory of the time value of money?

a) Rational, well-informed individuals are each willing to pay a DIFFERENT price, P(0); and therefore the market should exhibit various (DIFFERENT) fair prices for the security

b) Rational, well-informed individuals should arrive at the SAME willingness-to-pay price, P(0); and therefore the market should reflect this (SAME) price upon which all participants agree

c) Rational, well-informed individuals are each willing to pay a DIFFERENT price, P(0); but the market should reflect only one (SAME) fair price

d) Rational, well-informed individuals should arrive at the SAME willingness-to-pay price, P(0); but the market should reflect various (DIFFERENT) prices

10.3. The first U.S. Treasury bond has a price of $99.98, matures in six months, and pays a semi-annual coupon at a rate of 3.0% per annum. The second U.S. Treasury bond has a price of $101.11, matures in one year, and pays a semi-annual coupon at a rate of 4.0% per annum. What are, respectively, the six-month and one-year discount factors?

a) d(0.5) = 0.9790, d(1.0) = 0.9830 b) d(0.5) = 0.9850, d(1.0) = 0.9720 c) d(0.5) = 1.0020, d(1.0) = 0.9830 d) d(0.5) = 0.9650, d(1.0) = 1.0340


10.4. Assume the following discount function, which is a set of discount factors: d(0.5) = 0.990, d(1.0) = 0.970, d(1.5) = 0.960, d(2.0) = 0.950. A U.S. Treasury bond pays a semi-annual coupon at a rate of 5.0% per annum and matures with a face value of $1,000 in eighteen months (T = 1.5 years). What is the price of the bond?

a) $985.00 b) $1,002.00 c) $1,015.00 d) $1,033.00


Answers: 10.1. C. $100.98 This is an application of the law of one price. A portfolio of 60% of Bond A plus 40% of Bond C pays the same coupon as Bond B: (60% * $2) + (40% * $7) = $4. By the law of one price, the price of Bond B should be approximately (60% * $99.02) + (40% * $103.91) = $100.976. At this price, a portfolio of 100% of Bond B will cost the same as a portfolio of 60% Bond A + 40% Bond C and return the same future cash flows (specifically, $4 coupon in six months and $104 in one year, per $100 invested). Note, you can also infer a 3.0% yield on Bond's A and C and use the 3.0% yield to compute a bond price of $100.978; i.e., n = 2, I/Y = 1.5, PMT = 2, FV = 100, and CPT PV = $100.978. (although technically the law of one price does not imply yields are the same, but rather than the six-month and one-year spot/zero rates are the same, and as we can't infer the six-month zero rate, we can't really bootstrap the one-year zero rate). 10.2. C. Rational, well-informed individuals are each willing to pay a DIFFERENT price, P(0); but the market should reflect only one (SAME) fair price Please keep in mind this very narrowly refers to a risk-free asset: this is the condition that allows us to say this is only about the "time value of money." The "law of one price" is a theoretical function of the market, not different participants or individuals: its violation creates arbitrage opportunities. Please also keep in mind this is a theory about fundamentals; it does not preclude technical factors from interfering. Individuals are expected to express DIFFERENT preferences toward the time value of money. Tuckman's point is that the market price reflects a SINGLE CONSENSUS: "While the three people in the examples are willing to pay different amounts for $1,000 next year, there exists only one market price for this $1,000. If that price turns out to be $950 then the first person will pay $950 today to fund a $1,000 party in a year. The second person would be indifferent between buying the $950 stereo system today and putting away $950 to purchase the $1,000 stereo system next year. Finally, the third person would refuse to pay $950 for $1,000 in a year because the business can transform $940 today into $1,000 over the year. In fact, it is the collection of these individual decisions that determines the market price for $1,000 next year in the first place." 10.3. B. d(0.5) = 0.9850, d(1.0) = 0.9720 $99.98=d(0.5)*$101.50, so that d(0.5) = 99.98/101.50 = 0.9850 $101.11 = d(0.5)*$2.00 + d(1.0)*$102.00, so that d(1.0) = [101.11 - (0.9850 * 2.0)]/102.00 = 0.9720 10.4. D. $1,033.00 Price = $25*0.990 + $25*0.970 + $1,025*0.960 = $1,033


Question 11: Treasury STRIPS

AIMs: Discuss the components of a U.S. Treasury coupon bond, and

compare and contrast the structure to Treasury STRIPS, including the

difference between P-STRIPS and C-STRIPS. Compute the price of a fixed

income security with certain cash-flows and compare its value to fixed-

income securities with different, but certain, cash flow characteristics.

Identify arbitrage opportunities for fixed income securities with certain

cash flows.

11.1. A $10 million Treasury bond (note) with a 10-year maturity pays semi-annual coupons at a coupon rate of 4.0% per annum. If the bond is fully "stripped" such that STRIPS are created, each of the following is TRUE except:

a) The stripping creates 21 zero-coupon bonds b) Each of the C-STRIPS and the P-STRIP implies an exact spot (a.k.a., zero) interest rate c) The duration of the P-STRIP is greater than the duration of the original Treasury bond d) The C-STRIPS each have durations near to zero

11.2. A U.S. Treasury note with 1.5 years to maturity has a market price of $101.75 and pays a semi-annual coupon with a coupon rate of 5.50%. The market's discount function is the following set of discount factors: d(0.5) = 0.970, d(1.0) = 0.950, and d(1.5) = 0.920. Is the bond trading cheap, rich, or fair?

a) Trading cheap b) Trading fair c) Trading rich d) Cheap at six months, fair at one year, and rich at 1.5 years.

11.3. Which of the following would be the most likely reason for a C- or P-STRIP to "trade rich" or "trade cheap?"

a) Arbitrageurs b) Technical (non-fundamental) factors; e.g., liquidity, supply/demand c) A shift in the spot rates which changes discount rate(s) abruptly d) Individual investors have different views (preferences) with respect to the time value of

money


11.4. A fixed income manager has determined that an eighteen-month (1.5 year maturity) Treasury note with a market price at $102 that pays a 4.0% semi-annual coupon is overvalued. She conducts an arbitrage by shorting the bond and buying the replicating portfolio, as it trades cheap, that consists of positions in the following three bonds:

1) Bond B(1): A six-month (0.5 year to maturity) bond with a market price of $99.80 that pays a 1.0% semi-annual coupon (i.e., 1.0% per annum coupon rate pays 0.5% every six months)

2) Bond B(2): A one-year (1.0 year to maturity) bond with a market price of $99.40 that pays a 2.0% semi-annual coupon

3) Bond B(3): A 1.5-year bond with a market price of $98.00 that pays a 3.0% semi-annual coupon

What are face amounts, respectively, of the replicating portfolio? (note: more tedious than a typical exam question)

a) B(1) = 0.00, B(2) = 0.10, B(3) = 103.54 b) B(1) = 0.49, B(2) = 0.49, B(3) = 100.49 c) B(1) = 1.60, B(2) = 2.85, B(3) = 4.67 d) B(1) = 1.43, B(2) = 99.65, B(3) = 103.66


Answers: 11.1. D. The C-STRIPS are not floating-rate notes, they are zero-coupon bonds corresponding to the respective coupons. So, in this case, the twenty C-STRIPS have Macaulay durations of: 0.5, 1.0, 1.5, ..., 20. In regard to (A), (B), and (C), each is TRUE about the STRIPS. In regard to (A), the bonds twenty coupons (10 years * 2 coupons/year) create 20 C-STRIPS plus the principal repayment creates a single P-STRIP. 11.2. C. Trading rich The model ("predicted") price is the discounted present value = $2.75 * d(0.5) + $2.75 * d(2) + $102.75 * d(3) = 99.81 As the market price ($101.75) is greater than than the predicted price ($99.81), the bond is trading rich.

If market price > model (predicted) price, bond is "trading rich" If market price < model (predicted) price, bond is "trading cheap"

In regard to (D), this is meaningless. 11.3. B. Technical (non-fundamental) factors; e.g., liquidity, supply/demand The essence of Tuckman's explanation for the difference between market and model (predicted) prices is technical factors; i.e., factors not included in the fundamental pricing model. In regard to (A), arbitrage tends the reduce the trading premium/discount. In regard to (D), this is a valid but inferior answer because Tuckman already incorporates varying individual views and cites the discount rates as a consensus view; i.e., that individual preferences vary is already "built-into" the theory of the law of one price. (of course, different preferences inform supply/demand, so this is indirectly correct). 11.4. B. B(1) = 0.4853, B(2) = 0.4877, B(3) = 100.49 We don't need the market prices to generate the replicating portfolio. Starting with the final cash flow at 1.5, where F(x) = face amount: As F(3) * (1+3%/2) = $102.00, F(3) = 102/1.015 = 100.4926 As F(2) * (1+2%/2) + F(3)*3%/2 = $2.00, F(2) = (2 - 100.4926* 3%/2)/(1+2%/2) = 0.4877; As F(1) * (1+1%/2) + F(2)*(2%/2) + F(3)*3%/2 = $2.00, F(1) = (2 - 100.4926* 3%/2 - 0.4877*2%/2)/(1+1%/2) = 0.4853 The replicating portfolio must generate cash flows of: $2 at 0.5, $2 at 1.0, and $102 at 1.5 (same as the overvalued bond). To confirm/check just the 0.5 year cash flows: B(1): 0.4853*(1+1%/2) = $0.4877; i.e., coupon plus par B(2): 0.4877*2%/2 = $0.00488 B(3): 100.49 * 3%/2 = $1.5074 And $0.4877 + $0.00488 + $1.507 = $2.00.


Question 12: Spot rates

AIMs: Calculate and describe the impact of different compounding

frequencies on a bond’s value. Calculate holding period returns under

different compounding assumptions. Derive spot rates from discount

factors. Calculate the value of a bond using spot rates.

12.1. The spot rate curve is flat for all maturities at 4.0% per annum. A $100 face value bond with a three-year maturity pays an annual coupon of 6.0%. If we first price the bond under annual compounding (i.e., 4.0% per annum discounted annually), but then re-price the bond under continuous compounding (i.e., 4.0% per annum discounted continuously), what is the change in bond price the results solely from the change to continuous discounting?

a) No change (the yield is unchanged) b) Increase of $0.17 (i.e., from annual to continuous) c) Decrease of $0.19 d) Decrease of $0.23

12.2. An initial investment of $100 made ten (10) years ago has grown to $415 today. What is the (realized) holding period return under an assumption, respectively, of semiannual (s.a.) and annual (CAGR) compounding?

a) 14.23% (s.a.) and 14.75% (CAGR) b) 14.75% (s.a.) and 15.29% (CAGR) c) 15.29% (s.a.) and 15.72% (CAGR) d) 15.72% (s.a.) and 16.36% (CAGR)

12.3. The following discount function contains semi-annual discount factors out to two years: d(0.5) = 0.9970, d(1.0) = 0.9911, d(1.5) = 0.9809, d(2.0) = 0.9706. What is the implied eighteen-month (1.5 year) spot rate (aka, 1.5 year zero rate)?

a) 0.600% b) 1.176% c) 1.290% d) 1.505%

12.4. If the spot rate term structure is flat, what is true of the discount function (i.e., the set of discount factors) as function of maturity?

a) Flat b) Increasing with maturity c) Decreasing with maturity d) Insufficient information: we need the yield (YTM) to answer


12.5. The spot rate term structure is upward-sloping: 1.0% at 0.5 years, 2.0% at 1.0 years, 3.0% at 1.5 years, and 4.0% at 2.0 years. What is the price of two-year $100 face value bond that pays a semi-annual coupon with a 6.0% per annum coupon rate?

a) $99.74 b) $101.67 c) $102.27 d) $103.95

Answers:

12.1. D. Decrease of $0.23 Under annual compounding, bond price = $6/(1.04) + $6/(1.04)^2 + $106/(1.04^3) = 105.550 Under continuous compounding, bond price = $6*exp(-4%*1) + $6*exp(-4%*2) + $106*exp(-4%*3) = $105.317 Price change = 105.317 - 105.550 = -$0.233 12.2. B. 14.75% (s.a.) and 15.29% (CAGR) s.a. "holding period" return = 2*[(415/100)^(1/20)-1] = 14.75% CAGR "holding period" return = 415/100^(1/10)-1 = 15.29% 12.3. C. 1.290% As r(t) = 2*[(1/d(t))^(1/2t) - 1], r(1.5) = 2*[(1/0.9809)^(1/3) - 1] = 1.2898% 12.4. C. Decreasing with maturity Greater maturity requires more discounting. For example if the spot rate term structure is flat at 5%, then semi-annual discount function is: d(0.5) = 0.9756, d(1.0) = 0.9518, d(1.5) = 0.9286 ... In regard to (D), please note that a flat spot/zero term structure is the special case where the yield must match; e.g., flat spot rates at 5% imply yield must also be 5%. 12.5. D. $103.95 See spreadsheet here http://db.tt/CT6e5amX PV (0.5 year cash flow) = $3/(1+1%/2)^(0.5*2) = $2.99 PV (1.0 year cash flow) = $3/(1+2%/2)^(1.0*2) = $2.94 PV (1.5 year cash flow) = $3/(1+3%/2)^(1.5*2) = $2.87 PV (2.0 year cash flow) = $103/(1+4%/2)^(2.0*2) = $95.16 Sum of PV of cash flows = $103.951

http://db.tt/CT6e5amX


Question 13: Forward rates

AIM: Define and interpret the forward rate, and compute forward rates

given spot rates.

13.1. The price of a three-year zero coupon government bond is $94.23 and the price of a similar five-year bond is $82.99. Under annual compounding, what is the two-year implied forward rate from year three to year five, F(3,5)?

a) 3.67% b) 4.55% c) 5.83% d) 6.56%

13.2. The price of a six-month zero-coupon bond (bill) is $99.90 and the price of a one-year zero-coupon bond is $98.56. What is the implied six-month forward rate, under semi-annual compounding?

a) 1.30% b) 2.95% c) 2.73% d) 3.08%

13.3. The six-month and one-year discount factors are, respectively, d(0.5) = 0.9920 and d(1.0) = 0.9760. What is the implied six-month forward rate, under semi-annual compounding?

a) 2.34% b) 3.28% c) 3.95% d) 4.01%

13.4. The term structure of spot rates is: 0.60% at 1 year; 0.90% at 2 years; 1.00% at 3 years; 2.20% at 4 years; 3.10% at 5 years. What is the two-year forward swap rate starting in three years, F(3,5), under respectively, semi-annual (s.a.) and annual compounding?

a) 4.89% (s.a.) and 5.07% (annual) b) 5.25% (s.a.) and 5.22% (annual) c) 6.29% (s.a.) and 6.33% (annual) d) 7.03% (s.a.) and 7.14% (annual)

13.5. The eighteen-month forward curve is upward sloping with the following sequence of six-month forward rates: F(0,0.5) = S(0.5) = 1.0%, F(0.5, 1.0) = 3.0%, F(1.0,1.5) = 5.0%. What is the price of a $100 face value bond that matures in 1.5 years with a semi-annual coupon that pays a coupon rate of 6.0%?

a) $99.02 b) $100.56 c) $103.89 d) $104.43


Answers:

13.1. D. 6.56% Note that prices are a function of spot rates: P(3) = F/[1+s(3)]^3 and P(5) = F/[1+s(5)]^5 no-arbitrage expectation is : [1+s(3)]^3 * [1+f(3,5)]^2 = [1+s(5)]^5, such that [1+f(3,5)]^2 = [1+s(5)]^5 / [1+s(3)]^3, and taking square root of both sides: 1+f(3,5) = SQRT([1+s(5)]^5 / [1+s(3)]^3), such that f(3,5) = SQRT([1+s(5)]^5 / [1+s(3)]^3) - 1, and substituting price in: f(3,5) = SQRT[P(3)/P(5)] - 1. In this case, f(3,5) = SQRT[94.23/82.99] - 1 = 6.56% 13.2. C. 2.73% (99.90/98.56-1)*2 = 2.728% 13.3. B. 3.28% (0.9920/0.9760-1)*2 = 3.279% 13.4. C. 6.29% (s.a.) and 6.33% (annual) Semiannual F(3,5) = ([(1+3.1%/2)^(5*2)/(1+1%/2)^(3*2)]^(1/4)-1)*2 = 6.291% Annual F(3,5) = SQRT(1.031^5/1.01^3)-1 = 6.332% 13.5. D. $104.43 PV (1st coupon) = $3.0/(1+1%/2) = $2.99 PV (2nd coupon) = $3.0/[(1+1%/2)*(1+3%/2)] = $2.94 PV (final cash flow) = $103/[(1+1%/2)*(1+3%/2)*(1+5%/2] = $98.51. Sum of cash flows = $104.43 (exactly) or 104.44


Question 14: Yield to maturity (YTM)

AIMs: Define, interpret, and apply a bond’s yield-to-maturity (YTM) to

bond pricing. Compute a bond's YTM given a bond structure and price.

Establish the relationship between spot rates and YTM. Understand the

relationship between coupon rate, YTM, and bond prices. Define and

describe: Discount bond; Premium bond; Coupon effect; Pull-to-par.

14.1. A three (3)-year bond with a current price of $105.90 pays a semi-annual coupon with a coupon rate of 5.0% per annum. What is the bond's yield-to-maturity (YTM) on a bond-equivalent basis?

a) 1.97% b) 2.25% c) 2.93% d) 3.56%

14.2. An eight (8)-year bond with a current price of $975.00 pays an annual coupon of 6.0%. What is the bond's yield-to-maturity (YTM)?

a) 5.88% b) 6.41% c) 6.89% d) 7.14%

14.3. Each of the following is necessarily TRUE about a bond's yield-to-maturity (YTM) EXCEPT:

a) A bond that sells at a premium to par has a yield (YTM) that is less than its coupon rate b) A bond that sells at a discount to par has a yield (YTM) that is greater than its coupon

rate c) The yield (YTM) of a zero-coupon bond equals the spot (zero) rate of the bond's maturity d) If the same term structure of spot rates applies to two bonds with identical maturities,

the bond with the higher yield (YTM) is a superior investment 14.4. Assume the two-year term structure of spot rates is upward-sloping as follows: 1.0% at 0.5 years, 2.0% at 1.0 years, 3.0% at 1.5 years, 4.0% at 2.0 years. Consider the following two statements:

I. The yield (YTM) of a two-year bond must be less than 4.0% II. Given a two-year bond, an increase in the coupon rate implies an increase in the yield

(YTM) Which of the above statements is (are) TRUE?

a) Neither b) I. only c) II. only d) Both I. and II.


14.5. Which of the following bonds offers the highest yield (YTM)?

a) 7-year bond with a 3% coupon trading at par b) 20-year bond with a 4% coupon trading at a 15% premium to par c) 10-year bond with 4% coupon trading at a 15% discount to par d) 15-year bond with a 4% coupon 15% trading at a 15% discount to par

14.6. A ten (10)-year bond pays a semi-annual coupon with a coupon rate of 7.0% and the bond's yield (YTM) is 6.0%. If the yield remains unchanged, what happens to the price of the bond in six months?

a) Lower price b) Same price c) Higher price d) Need more information (the initial bond price)

14.7. Each of the following is necessarily true about a bond's yield (YTM) EXCEPT:

a. If the term structure of spot rates is flat at X%, a bond's yield must be also be X% b) b. Regardless of the slope of the term structure (e.g., upward- or downward-sloping) and

number of spot rates, the yield (YTM) is a single value c) c. The yield on a coupon-paying bond is sensitive to (i.e., will change in response to) a

change in spot rates at specific maturities d) d. For a given bond with a fixed coupon rate, an increase in the bond's maturity implies a

decrease in the bond's yield (YTM) 14.8. You just purchased a ten (10)-year bond at a discount to par. The bond pays a quarterly coupon with a coupon rate of 4.0% per annum; i.e., 1% each quarter. The bond's yield is 8.0% per annum. Your broker says to you, "The 8% yield-to-maturity represents the return you will realize--that is, your realized return--on this bond." Which qualifier or caveat is BEST attached to this assertion?

a) This statement is already true: realized return will equal yield (YTM) b) The statement is only true if there is no shift in the term structure of spot rates c) This statement is true if you (the bondholder) hold the bond to maturity: realized return

will equal yield (YTM) if the bond is held to maturity d) This statement is true only if both the bond is held to maturity and the coupons (interim

cash flows) are reinvested at the same yield (YTM)


Answers: 14.1. C. 2.93% N = 6, PV = -105.91, PMT = 2.5, FV = 100, CPT I/Y = 1.46392 * 2 = 2.92784%; i.e., bond-equivalent refers to semi-annual compound frequency. 14.2. B. 6.41% N = 8, PV = -975, PMT = 60, FV = 1000, CPT I/Y = 6.40913% 14.3. D. False, this ignores the coupon effect. Fairly prices bonds will have various yields. In regard to (A), (B), and (C), each is TRUE. 14.4. B. I. only. As the yield-to-maturity is a summary of all the spot rates that enter into the bond pricing equation, the yield must be less than the highest spot rate (and greater than the lowest spot rate) In regard to II. This is false: if the term structure of spot rates is increasing, as the coupon increases, generally the yield will decrease. 14.5. C. 10-year bond with 4% coupon trading at a 15% discount to par In regard to (A), this bond trades at par so its yield must equal its coupon of 3% In regard to (B), this bond trades at a premium, so it's yield must be less than its coupon of 4%. The bonds in both (C) and (D) must have yields greater than their 4% coupon; however, the 10-year bond returns the 15% discount faster, so its yield is higher. 14.6. A. Lower price: As yield < coupon rate, the bond currently trades at a premium to par. With unchanged yield, the price will get "pulled to par" and therefore must decrease. We have enough information to price the bonds, but please note we do NOT need to price the bond if we are familiar with pulling to par concept! at 10 years, price = -PV(6%/2, 10*2, $7/2, $100) = $107.44; at 9.5 years, price = -PV(6%/2, 9.5*2, $7/2, $100) = $107.16. 14.7. D. False. This is only true for a discounted bond; in the case of a premium-priced bond, the yield increases with maturity. 14.8. D. We don't expect the ex-post realized return to equal the ex-ante yield (YTM); it is only the case if both the bond is held to maturity and the coupons are reinvested at the same rate. In regard (B), a static zero rate terms structure will not give a realized return equal to yield if the bond is sold before maturity. (Any non-flat term structure implies an interim mark-to-market yield that differs from the ex ante yield.)

t4.basic bond pricing (tuckman)

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