karoliukas

Corinne Mueller is explaining how to derive the theoretical Treasury spot rate curve from the prices of Treasury coupon bonds. She states the following:
Statement 1: To calculate a theoretical Treasury spot rate curve from the yields on coupon bonds, we must know at least two actual Treasury spot rates.
Statement 2: To compute the theoretical 3-year Treasury spot rate, first determine the spot rates for each of the bond’s coupon periods from 0.5 to 2.5 years. Discount each coupon payment to its present value using the theoretical spot rate for each period. The theoretical 3-year spot rate is the discount rate on the final coupon and principal payment that sets the sum of the present values of all the bond’s cash flows equal to its price.
With respect to Mueller’s statements:

A) only one is correct.

B) both are correct.

C) both are incorrect.

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Statement 1 is incorrect. If we know one actual spot rate, we can calculate the theoretical spot rate for the next longer period. With these two spot rates we can calculate the next theoretical spot rate, and so on up the coupon curve. Statement 2 is a correct description of the methodology for computing a theoretical Treasury spot rate.

karoliukas

An investor gathers the following information about a 2-year, annual-pay bond:

• Par value of $1,000
• Coupon of 4%
• 1-year spot interest rate is 2%
• 2-year spot interest rate is 5%

Using the above spot rates, the current price of the bond is closest to:

A) $983.

B) $1,000.

C) $1,010.

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The value of the bond is simply the present value of discounted future cash flows, using the appropriate spot rate as the discount rate for each cash flow. The coupon payment of the bond is $40 (0.04 × 1,000). The bond price = 40/(1.02) + 1,040/(1.05)2 = $982.53.

karoliukas

Which of the following statements regarding zero-coupon bonds and spot interest rates is least accurate?

A) Price appreciation creates all of the zero-coupon bond's return.

B) Zero-coupon bonds have no coupons.

C) Spot interest rates will never vary across the term structure.

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Zero-coupon bonds are quite special. Because zero-coupon bonds have no coupons (all of the bond’s return comes from price appreciation), investors have no uncertainty about the rate at which coupons will be invested. Spot rates are defined as interest rates used to discount a single cash flow to be received in the future.

karoliukas

An investor gathers the following information about a 3-year, annual-pay bond:

• Par value of $1,000
• Coupon of 8%
• Current price of $1,100
• 1-year spot interest rate is 5%
• 2-year spot interest rate is 6%

Using the above information, the 3-year spot rate is closest to:

A) 4.37%.

B) 4.27%.

C) 8.20%.

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The value of the bond is simply the present value of discounted future cash flows, using the appropriate spot rate as the discount rate for each cash flow. The coupon payment of the bond is $80 (0.08 × 1,000) and the face value is $1,000. Hence, bond price of 1,100= 80/(1.05)+ 80/(1.06)2 + 1,080/(1 + 3-year spot rate)3. Using the yx key on our calculator, we can solve for the 3-year spot rate of 4.27%.

karoliukas

The 3-year spot rate is 10%, and the 4-year spot rate is 10.5%. What will the 1-year rate be 3 years from now?

A) 12.0%.

B) 10.0%.

C) 11.0%.

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[(1 + Z4)4 / (1 + Z3)3] − 1 = 12.01% = 12%.

karoliukas

Given the following spot rate curve:

Spot Rate
1-yr zero = 9.50%
2-yr zero = 8.25%
3-yr zero = 8.00%
4-yr zero = 7.75%
5-yr zero = 7.75%

What will be the market price of a five-year, 9% annual coupon rate bond?

A) $1,067.78.

B) $1,000.00.

C) $1,047.68.

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90 / (1 + 0.095) + 90 / (1 + 0.0825)2 + 90 / (1 + 0.08)3 + 90 / (1 + 0.0775)4 + 1,090 / (1 + 0.0775)5 = $1,047.68.

karoliukas

Using the following spot rates, what is the price of a three-year bond with annual coupon payments of 5%?

• One-year rate: 4.78%
• Two-year rate: 5.56%
• Three-year rate: 5.98%

A) $98.87.

B) $93.27.

C) $97.47.

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The bond price is computed as follows:
Bond price = (5 / 1.0478) + (5 / 1.05562) + (105 / 1.05983) = $97.47

karoliukas

An analyst observes that the current 6-month T-Bill rate is 8% (4% semi-annually) and the one-year T-Bill rate is 9% (4.5% semi-annually). There is an existing 1.5-year, 9% semi-annual coupon bond selling for $990. What is the annualized 1.5-year spot rate?

A) 8.8%.

B) 9.5%.

C) 9.8%.

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45 / (1.04) + 45 / (1.045)2 + 1045 / (1 + Z3)3 = 990
(1045 / 905.53 )0.3333 − 1 = Z3 = 4.89%
Annualized = 9.8%.

karoliukas

Assume that a callable bond's call period starts two years from now with a call price of $102.50. Also assume that the bond pays an annual coupon of 6% and the term structure is flat at 5.5%. Which of the following is the price of the bond assuming that it is called on the first call date?

A) $100.00.

B) $102.50.

C) $103.17.

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The bond price is computed as follows:
Bond price = 6/1.055 + (102.50 + 6)/1.0552 = $103.17

karoliukas

Can spot interest rates be used to value a callable bond?

A) Yes.

B) No.

C) It depends on the slope of the term structure.

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Any complex debt instruments (like callable bonds, putable bonds, and mortgage-backed securities) can be viewed as the sum of the present value of its individual cash flows where each of those cash flows are discounted at the appropriate zero-coupon bond spot rate. It should be noted that while the appropriate spot interest rate can be used to discount each cash flow, determining the actual pattern of cash flows is uncertain due to the possibility of the bond being called away.