Reading 65: Introduction to the Valuation of Debt Securities- 2011, valuation, generate, process, center

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Session 16: Fixed Income: Analysis and Valuation
Reading 65: Introduction to the Valuation of Debt Securities

LOS f: Explain the arbitrage-free valuation approach and the market process that forces the price of a bond toward its arbitrage-free value, and explain how a dealer can generate an arbitrage profit if a bond is mispriced.

 

 

You are considering the purchase of a three-year annual coupon bond with a par value of $1,000 and a coupon rate of 5.5%. You have determined that the spot rate for year 1 is 5.2%, the spot rate for year two is 5.5%, and the spot rate for year three is 5.7%. What would you be willing to pay for the bond now?

A) $995.06.

B) $937.66.

C) $1,000.00.

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You need the find the present value of each cash flow using the spot rate that coincides with each cash flow.
The present value of cash flow 1 is: FV = $55; PMT = 0; I/Y = 5.2%; N = 1; CPT → PV = -$52.28.
The present value of cash flow 2 is: FV = $55; PMT = 0; I/Y = 5.5%; N = 2; CPT → PV = –$49.42.
The present value of cash flow 3 is: FV = $1,055; PMT = 0; I/Y = 5.7%; N = 3; CPT → PV = –$893.36.
The most you pay for the bond is the sum of: $52.28 + $49.42 + $893.36 = $995.06.

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A three-year bond with a 10% annual coupon has cash flows of $100 at year 1, $100 at year 2, and pays the final coupon and the principal for a cash flow of $1,100 at year 3. The spot rate for year 1 is 5%, the spot rate for year 2 is 6%, and the spot rate for year 3 is 6.5%. What is the arbitrage-free value of the bond?

A) $1,050.62.

B) $975.84.

C) $1,094.87.

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Spot interest rates can be used to price coupon bonds by taking each individual cash flow and discounting it at the appropriate spot rate for that year’s payment. To find the arbitrage-free value:

Bond value = [$100 / (1.05)] + [$100 / (1.06)²] + [$1,100 / (1.065)³] = $95.24 + $89.00 + $910.63 = $1,094.87

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Which of the following packages of securities is equivalent to a three-year 8% coupon bond with semi-annual coupon payments and a par value of 100? A three-year zero-coupon bond:

A) with a par of 100 and six zero-coupon bonds with a par value of 4 and maturities equal to the time to each coupon payment of the coupon bond.

B) with a par of 100 and six zero-coupon bonds with a par value of 8 and maturities equal to the time to each coupon payment of the coupon bond.

C) with a par value of 150 and six 8% coupon bonds with a maturity equal to the time to each coupon payment of the above bond.

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This combination of zero-coupon bonds has exactly the same cash flows as the above coupon bond and therefore it is equivalent to it.

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Which of the following statements concerning the arbitrage-free valuation of non-Treasury securities is CORRECT? The credit spread is:

A) a function of default risk and the term to maturity.

B) only a function of the bond's term to maturity.

C) only a function of the bond's default risk.

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For valuing non-Treasury securities, a credit spread is added to each treasury spot yields. The credit spread is a function of default risk and the term to maturity.

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The arbitrage-free bond valuation approach can best be described as the:

A) use of a series of spot interest rates that reflect the current term structure.

B) use of a single discount factor.

C) geometric average of the spot interest rates.

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The use of multiple discount rates (i.e., a series of spot rates that reflect the current term structure) will result in more accurate bond pricing and in so doing, will eliminate any meaningful arbitrage opportunities. That is why the use of a series of spot rates to discount bond cash flows is considered to be an arbitrage-free valuation procedure.

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A 3-year option-free bond (par value of $1,000) has an annual coupon of 9%. An investor determines that the spot rate of year 1 is 6%, the year 2 spot rate is 12%, and the year 3 spot rate is 13%. Using the arbitrage-free valuation approach, the bond price is closest to:

A) $1,080.

B) $912.

C) $968.

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We can calculate the price of the bond by discounting each of the annual payments by the appropriate spot rate and finding the sum of the present values. Price = [90 / (1.06)] + [90 / (1.12)²] + [1,090 / (1.13)³] = 912. Or, in keeping with the notion that each cash flow is a separate bond, sum the following transactions on your financial calculator:

N = 1; I/Y = 6.0; PMT = 0; FV = 90; CPT → PV = 84.91
N = 2; I/Y = 12.0; PMT = 0; FV = 90; CPT → PV = 71.75
N = 3; I/Y = 13.0; PMT = 0; FV = 1,090; CPT → PV = 755.42

Price = 84.91 + 71.75 + 755.42 = $912.08.

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A 2-year option-free bond (par value of $1,000) has an annual coupon of 6%. An investor determines that the spot rate of year 1 is 5% and the year 2 spot rate is 8%. Using the arbitrage-free valuation approach, the bond price is closest to:

A) $992.

B) $966.

C) $1,039.

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The arbitrage free valuation approach is the process of valuing a fixed income instrument as a portfolio of zero coupon bonds. We can calculate the price of the bond by discounting each of the annual payments by the appropriate spot rate and finding the sum of the present values. Bond price = [60 / (1.05)] + [1,060 / (1.08)²] = $966. Or, in keeping with the notion that each cash flow is a separate bond, sum the following transactions on your financial calculator:

N = 1; I/Y = 5.0; PMT = 0; FV = 60; CPT → PV = 57.14
N = 2; I/Y = 8.0; PMT = 0; FV = 1,060; CPT → PV = 908.78
Price = 57.14 + 908.78 = $966.

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A 2-year option-free bond (par value of $10,000) has an annual coupon of 15%. An investor determines that the spot rate of year 1 is 16% and the year 2 spot rate is 17%. Using the arbitrage-free valuation approach, the bond price is closest to:

A) $9,694.

B) $8,401.

C) $11,122.

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We can calculate the price of the bond by discounting each of the annual payments by the appropriate spot rate and finding the sum of the present values. Price = [1,500/(1.16)] + [11,500/(1.17)²] = $9,694. Or, in keeping with the notion that each cash flow is a separate bond, sum the following transactions on your financial calculator:

N=1, I/Y=16.0, PMT=0, FV=1,500, CPT PV=1,293
N=2, I/Y=17.0, PMT=0, FV=11,500, CPT PV=8,401

Price = 1,293 + 8,401 = $9,694.

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Current spot rates are as follows:

1-Year: 6.5%
2-Year: 7.0%
3-Year: 9.2%

Which of the following is CORRECT

A) For a 3-year annual pay coupon bond, all cash flows can be discounted at 9.2% to find the bond's arbitrage-free value.

B) The yield to maturity for 3-year annual pay coupon bond can be found by taking the geometric average of the 3 spot rates.

C) For a 3-year annual pay coupon bond, the first coupon can be discounted at 6.5%, the second coupon can be discounted at 7.0%, and the third coupon plus maturity value can be discounted at 9.2% to find the bond's arbitrage-free value.

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Spot interest rates can be used to price coupon bonds by taking each individual cash flow and discounting it at the appropriate spot rate for that year’s payment. Note that the yield to maturity is the bond’s internal rate of return that equates all cash flows to the bond’s price. Current spot rates have nothing to do with the bond’s yield to maturity.

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Which of the following statements concerning arbitrage-free bond prices is NOT correct?

A) Credit spreads are affected by time to maturity.

B) The determination of spot rates is usually done using risk-free securities.

C) It is not possible to strip coupons from U.S. Treasuries and resell them.

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It is possible to both strip coupons from U.S. Treasuries and resell them, as well as to aggregate stripped coupons and reconstitute them into U.S. Treasury coupon bonds. Therefore, arbitrage arguments ensure that U.S. Treasury securities sell at or very near their arbitrage free values. For valuing non-Treasury securities, a credit spread is added to each treasury spot yields. The credit spread is a function of default risk and the term to maturity.