Fixed Income - Calculation

Question 1

What is the yield to call on a bond that has an 8% coupon paid annually, $1,000 face value, 10 years to maturity and is first callable in 6 years? The current market price is $1,100. The call price is the face value plus 1-year’s interest.

A) 6.00%.
B) 7.14%.
C) 7.02%.

Answer 1

N = 6; PV = -1,100.00; PMT = 80; FV = 1,080; Compute I/Y = 7.02%.

Question 2

An investor holds $100,000 (par value) worth of TIPS currently trading at par. The coupon rate of 4% is paid semiannually, and the annual inflation rate is 2.5%. What coupon payment will the investor receive at the end of the first six months?

A) $2,025.
B) $2,050.
C) $2,000.

Answer 2

A - par value rise with inflation

payment = $100K * (1+2.5%/2)*2% = 2,2025

Question 3

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) $1,047.68.
C) $1,000.00.

Answer 3

B

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.

Question 4

An investor purchased a 10-year zero-coupon bond with a yield to maturity of 10% and a par value of $1,000. What would her rate of return be at the end of the year if she sells the bond? Assume the yield to maturity on the bond is 9% at the time it is sold and annual compounding periods are used.

A) 15.00%.
B) 16.00%.
C) 19.42%.

Answer 4

C
Purchase price: I = 10; N = 10; PMT = 0; FV = 1,000; CPT → PV = 385.54

Selling price: I = 9; N = 9; PMT = 0; FV = 1,000; CPT → PV = 460.43

% Return = (460.43 − 385.54) / 385.54 × 100 = 19.42%

Question 5

Georgia Corporation has $1,000 par value bonds with 10 years remaining maturity. The bonds carry a 7.5% coupon that is paid semi-annually. If the current yield to maturity on similar bonds is 8.2%, what is the current value of the bonds?

A) $952.85.
B) $1,123.89.
C) $569.52.

Answer 5

A
The coupon payment each six months is ($1,000)(0.075 / 2) = $37.50. To value the bond, enter FV = $1,000; PMT = $37.50; N = 10 × 2 = 20; I/Y = 8.2 / 2 = 4.1%; CPT → PV = -952.85.

Question 6

An investor wants to take advantage of the 5-year spot rate, currently at a level of 4.0%. Unfortunately, the investor just invested all of his funds in a 2-year bond with a yield of 3.2%. The investor contacts his broker, who tells him that in two years he can purchase a 3-year bond and end up with the same return currently offered on the 5-year bond. What 3-year forward rate beginning two years from now will allow the investor to earn a return equivalent to the 5-year spot rate?

A) 3.5%.
B) 4.5%.
C) 5.6%.

Answer 6

B
(1.04^5 / 1.032^2)1/3 - 1 = 4.5%.

Using the average method, (4%5 - 3.2%2)/3=4.53%

Question 7

What is the present value, stated as a percentage of par, of a three-year security that pays a fixed annual coupon of 6% using a discount rate of 7%?

A) 100.00.
B) 92.48.
C) 97.38.

Answer 7

C
This value is computed as follows:
Present Value = 6/1.07 + 6/1.07^2 + 106/1.07^3 = 97.38

(or i just used financial calculator)

Question 8

Using the following spot rates for pricing the bond, what is the present value of a three-year security that pays a fixed annual coupon of 6%?
Year 1: 5.0%
Year 2: 5.5%
Year 3: 6.0%

A) 102.46.
B) 100.10.
C) 95.07.

Answer 8

B
This value is computed as follows:
Present Value = 6/1.05 + 6/1.055^2 + 106/1.06^3 = 100.10

The value 95.07 results if the coupon payment at maturity of the bond is neglected.

• This is the same as using spot rate to valuing bond price.
• if using NPV method, PMT=$0, since there is no payment in the middle (n keeping with the notion that each cash flow is a separate bon)

Question 9

The six-month spot rate is 4.0% and the 1 year spot rate is 4.5%, both stated on a semiannual bond basis. The implied six-month rate six months from now, stated on a semiannual bond basis, is closest to:

A) 4%.
B) 6%.
C) 5%.

Answer 9

C
6m6m/2 = [(1 + S2/2)^2 / (1 + S1/2)] - 1 = [(1.0225)^2/(1.02)1] - 1
[1.0455 / 1.02] - 1 = 0.025

6m6m = 0.025 × 2 = 0.05

*using the average method, (4% + x)/2 = 4.5% = > x = 5%

Question 10

What value would an investor place on a 20-year, $1,000 face value, 10% annual coupon bond, if the investor required a 9% rate of return?

A) $1,091.
B) $879.
C) $920.

Answer 10

A

N = 20; I/Y = 9; PMT = 100 (0.10 × 1,000); FV = 1,000; CPT → PV = 1,091.

Question 11

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) $968.
B) $912.
C) $1,080.

Answer 11

B
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.

Question 12

Consider a 5-year, semiannual, 10% coupon bond with a maturity value of 1,000 selling for $1,081.11. The first call date is 3 years from now and the call price is $1,030. What is the yield-to-call?

A) 3.91%.
B) 7.28%.
C) 7.82%.

Answer 12

C
N = 6; PMT = 50; FV = 1,030; PV = -1,081.11; CPT → I = 3.91054

3.91054 × 2 = 7.82

Question 13

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) $1,000.
B) $983.
C) $1,010.

Answer 13

B
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.

Question 14

A 20-year, $1,000 face value, 10% semi-annual coupon bond is selling for $875. The bond’s yield to maturity is:

A) 11.43%.
B) 11.62%.
C) 5.81%.

Answer 14

B
N = 40 (2 × 20 years); PMT = 50 (0.10 × 1,000) / 2; PV = -875; FV = 1,000; CPT → I/Y = 5.811 × 2 (for annual rate) = 11.62%.

Question 15

Find the yield to maturity of a 6% coupon bond, priced at $1,115.00. The bond has 10 years to maturity and pays semi-annual coupon payments.

A) 5.87%.
B) 8.07%.
C) 4.56%.

Answer 15

C
N = 10 × 2 = 20; PV = -1,115.00; PMT = 60/2 = 30; FV = 1,000.
Compute I = 2.28 (semiannual) × 2 = 4.56%

Question 16

What is the yield to maturity (YTM) on a semiannual-bond basis of a 20-year, U.S. zero-coupon bond selling for $300?

A) 7.20%.
B) 6.11%.
C) 3.06%.

Answer 16

B

N = 40; PV = −300; FV = 1,000; CPT → I = 3.055 × 2 = 6.11.

Question 17

Today an investor purchases a $1,000 face value, 10%, 20-year, semi-annual bond at a discount for $900. He wants to sell the bond in 6 years when he estimates the yields will be 9%. What is the estimate of the future price?

A) $946.
B) $1,079.
C) $1,152.

Answer 17

B
In 6 years, there will be 14 years (20 − 6), or 14 × 2 = 28 semi-annual periods remaining of the bond’s life So, N = (20 − 6)(2) = 28; PMT = (1,000 × 0.10) / 2 = 50; I/Y = 9/2 = 4.5; FV = 1,000; CPT → PV = 1,079.

Note: Calculate the PV (we are interested in the PV 6 years from now), not the FV.

Question 18

A $1,000 par value, 10% annual coupon bond with 15 years to maturity is priced at $951. The bond’s yield to maturity is:

A) equal to its current yield.
B) greater than its current yield.
C) less than its current yield.

Answer 18

B
The bond’s YTM is:
N = 15; PMT = 100; PV = −951; FV = 1,000; CPT I/Y = 10.67%

Current Yield = annual coupon payment / bond price
CY = 100 / $951 = 0.1051 or 10.51%

Question 19

Given a required yield to maturity of 6%, what is the intrinsic value of a semi-annual pay coupon bond with an 8% coupon and 15 years remaining until maturity?

A) $1,202.
B) $1,095.
C) $1,196.

Answer 19

C
This problem can be solved most easily using your financial calculator. Using semiannual payments, I = 6/2 = 3%; PMT = 80/2 = $40; N = 15 × 2 = 30; FV = $1,000; CPT → PV = $1,196.

Question 20

An investor plans to buy a 10-year, $1,000 par value, 8% semiannual coupon bond. If the yield to maturity of the bond is 9%, the bond’s value is:

A) $934.96.
B) $935.82.
C) $1,067.95.

Answer 20

A

N = 20, I = 9/2 = 4.5, PMT = 80/2 = 40, FV = 1,000, compute PV = $934.96

Question 21

Consider a bond that pays an annual coupon of 5% and that has three years remaining until maturity. Assume the term structure of interest rates is flat at 6%. If the term structure of interest rates does not change over the next twelve-month interval, the bond’s price change (as a percentage of par) will be closest to:

A) 0.84.
B) -0.84.
C) 0.00.

Answer 21

A
The bond price change is computed as follows:
Bond Price Change = New Price − Old Price = (5/1.06 + 105/1.06^2) − (5/1.06 + 5/1.06^2 + 105/1.06^3) = 98.17 − 97.33 = 0.84.

or use financial calculator N=2 vs N=3

Question 22

A 20-year bond with a par value of $1,000 and an annual coupon rate of 6% currently trades at $850. It has a yield to maturity of:

A) 7.9%.
B) 7.5%.
C) 6.8%.

Answer 22

B

N = 20; FV = 1,000; PMT = 60; PV = -850; CPT → I = 7.5

Question 23

A $1,000 bond with an annual coupon rate of 10% has 10 years to maturity and is currently priced at $800. What is the bond’s approximate yield-to-maturity?

A) 11.7%.
B) 13.8%.
C) 12.6%.

Answer 23

B
FV = 1,000, PMT = 100, N = 10, PV = -800

Compute I = 13.8

Question 24

An analyst wants to estimate the yield to maturity on a non-traded 4-year, annual pay bond rated A. Among actively traded bonds with the same rating, 3-year bonds are yielding 3.2% and 6-year bonds are yielding 5.0%. Using matrix pricing the analyst should estimate a YTM for the non-traded bond that is closest to:

A) 3.8%.
B) 3.6%.
C) 4.1%.

Answer 24

A
Interpolating: 3.2% + [(4 - 3) / (6 - 3)] × (5.0% - 3.2%) = 3.8%

(6-3, not 6-4….)

Question 25

Assume a city issues a $5 million bond to build a new arena. The bond pays 8% semiannual interest and will mature in 10 years. Current interest rates are 9%. What is the present value of this bond and what will the bond’s value be in seven years from today?

     Present Value	Value in 7 Years from Today A)          4,674,802	                      4,931,276 B)          5,339,758	                      4,871,053 C)          4,674,802	                      4,871,053

Answer 25

C
Present Value:
Since the current interest rate is above the coupon rate the bond will be issued at a discount. FV = $5,000,000; N = 20; PMT = (0.04)(5 million) = $200,000; I/Y = 4.5; CPT → PV = -$4,674,802

Value in 7 Years:
Since the current interest rate is above the coupon rate the bond will be issued at a discount. FV = $5,000,000; N = 6; PMT = (0.04)(5 million) = $200,000; I/Y = 4.5; CPT → PV = -$4,871,053

Question 26

An 11% coupon bond with annual payments and 10 years to maturity is callable in 3 years at a call price of $1,100. If the bond is selling today for 975, the yield to call is:

A) 9.25%.
B) 10.26%.
C) 14.97%.

Answer 26

C
PMT = 110, N = 3, FV = 1,100, PV = 975
Compute I = 14.97

Question 27

An investor gathered the following information about two 7% annual-pay, option-free bonds:

Bond R has 4 years to maturity and is priced to yield 6%
Bond S has 7 years to maturity and is priced to yield 6%
Both bonds have a par value of $1,000.
Given a 50 basis point parallel upward shift in interest rates, what is the value of the two-bond portfolio?

A) $2,086.
B) $2,030.
C) $2,044.

Answer 27

C
Given the shift in interest rates, Bond R has a new value of $1,017 (N = 4; PMT = 70; FV = 1,000; I/Y = 6.50%; CPT → PV = 1,017). Bond S’s new value is $1,027 (N = 7; PMT = 70; FV = 1,000; I/Y = 6.50%; CPT → PV = 1,027). After the increase in interest rates, the new value of the two-bond portfolio is $2,044 (1,017 + 1,027).

*first time used the old rate…. it should be 6.5%, not 6%