Question

4. Draw a time line showing the cash flows for a bond that has a four year maturity, semiannual coupon payments, a coupon rate of 5 percent, and a par value of $1,000.

5. Using the information in question 4, value the bond under the following interest rate assumptions: a. Market Rate = 3% b. Market Rate = 5% c. Market Rate = 7%

6. Assuming the market rate is 6.5 percent, what is the value of a bond that pays an annual coupon payment, a coupon rate of 8 percent, a par value of $1,000, and a maturity of 10 years. a. Is the bond in question 6 selling for a discount, premium, or par?

7. Find the yield to maturity (YTM) for a bond with the following characteristics:

Maturity = 15 years, Coupon Rate = 10%, Coupon Payments made Annually, Par = $1,000. a. Bond Selling Price = $1,125 b. Bond Selling Price = $1,000 c. Bond Selling Price = $975

Answer 1

Solution 4:

Coupon interest payment done half annually (i) = $1,000 * 5% / 2

= $25

Number of half years (N) = 4 * 2 = 8 half years

Investment of $1000 will be done at start of period i.e. when N = 0. Bond will pay $25 coupon payment at the end of each half year and $1000 face value at the time of maturity.

Time line of bond cash inflows will be as below:

0	1	2	3	4	5	6	7	8
-1000	25	25	25	25	25	25	25	25
								1000

Value of bond is equal to present value of its cash inflows till maturity.

Bond cash inflows will be discounted at the market rate to arrive at value of bon.

Formula for annuity PV factor:

Formula for PV:

Bond market price exceeds its maturity amount when bond coupon rate exceeds prevailing market rate of interest and vice versa.

Solution 5 (a)

Bond valuation when market rate is 3%

Semiannual market rate = 3% / 2 = 1.50% semi annually

Annuity PV factor of 1.50% for 8 periods = 7.485925

PV of coupon payments = $25 * 7.485925

= $187.15

Present value of maturity amount = $1000 / (1.015^8)

= $1000/1.126493

= $887.71

Value of bond = $187.15 + $887.71

= $1074.86

Bond is selling at premium

Solution 5 (b)

When market rate of same as coupon rate, bond’s valuation will be same as to its maturity value.

Same is also proved in below calculation:

Bond valuation when market rate is 5%

Semiannual market rate = 5% / 2 = 2.50% semi annually

Annuity PV factor of 2.50% for 8 periods = 7.17013

PV of coupon payments = $25 * 7.17013

= $179.25

Present value of maturity amount = $1000 / (1.025^8)

= $1000/1.218403

= $820.75

Value of bond = $179.25 + $820.75

= $1000

Bond is selling at par.

Solution 5 (c)

Bond valuation when market rate is 7%

Semiannual market rate = 7% / 2 = 3.50% semi annually

Annuity PV factor of 3.50% for 8 periods = 6.87395

PV of coupon payments = $25 * 6.87395

= $171.85

Present value of maturity amount = $1000 / (1.035^8)

= $1000/1.316809

= $759.41

Value of bond = $171.85 + $759.41

= $931.26

Bond is selling at a discount.

Solution 6

Number of years = 10

Market rate = 6.50%

Coupon rate = 8% paid annually

Coupon amount = $80

Annuity PV factor of 6.50% for 10 periods = 7.18883

PV of coupon payments = $80 * 7.18883

= $575.10

Present value of maturity amount = $1000 / (1.065^10)

= $1000/1.877137

= $532.73

Value of bond = $575.10 + $532.73

= $1107.83

Bond is selling at a premium.

Solution 7

In absence of a financial calculator, Estimate YTM can be calculated using below formula:

For exact YTM calculation, Rate calculator in Excel can be used:

Formula used: =RATE(B9,B11,B10,B12,)

Where,

Bond period is filled in B9

Coupon payment is filled in B11

Current value is filled in B10

Par value is filled in B12

Part a

N	15
PV	$ (1,125.00)
PMT	$100
FV	$1,000
Compute I	8.50% (YTM)

Part b

N	15
PV	$ (1,000.00)
PMT	$100
FV	$1,000
Compute I	10.00% (YTM)

YTM of bond is equal to coupon rate when selling price is equal to par value.

Part c

N	15
PV	$     (975.00)
PMT	$100
FV	$1,000
Compute I	10.33% (YTM)