Question

A 20-year, 6.500% annual payment bond settles on a coupon date. The bond's yield to maturity is 9.400%.

(a) What is the bond’s Macauley Duration (show your work, like you did in problem (16) above.)

(b) What is the bond’s approximate modified duration? Use yield changes of +/- 30 bps around the yield to maturity for your calculations.

(c) Calculate the approximate convexity for the bond.

(d) Calculate the change in the full bond price for a 40 bps change in yield.

Answer #1

a)

No of periods = 20 years

Coupon per period = (Coupon rate / No of coupon payments per year) * Face value

Coupon per period = (6.5% / 1) * $1000

Coupon per period = $65

Illustrating for Time period 1

Discount factor = 1 / (1 + YTM)^{Time period}

Discount factor = 1 / (1 + 9.4%)^{1}

Discount factor = 0.9141

Present value of Cashflow = Discount factor * Cashflow

Present value of Cashflow = 0.9141 * $65

Present value of Cashflow = $59.41

Weight = Present value of Cashflow / Total(Present value of Cashflow)

Weight = $59.41 / $742.65

Weight = 8%

Weighted average of Time = Weight * Time period

Weighted average of Time = 8.00% * 1

Weighted average of Time = 0.0800

Time period | Yield to Maturity | Discount Factor | Cashflow | Present value of Cashflow | Weight |
Weighted average of Time |

1 | 9.40% | 0.9141 | $65 | $59.41 | 8.00% | 0.0800 |

2 | 9.40% | 0.8355 | $65 | $54.31 | 7.31% | 0.1463 |

3 | 9.40% | 0.7637 | $65 | $49.64 | 6.68% | 0.2005 |

4 | 9.40% | 0.6981 | $65 | $45.38 | 6.11% | 0.2444 |

5 | 9.40% | 0.6381 | $65 | $41.48 | 5.59% | 0.2793 |

6 | 9.40% | 0.5833 | $65 | $37.91 | 5.11% | 0.3063 |

7 | 9.40% | 0.5332 | $65 | $34.66 | 4.67% | 0.3267 |

8 | 9.40% | 0.4874 | $65 | $31.68 | 4.27% | 0.3413 |

9 | 9.40% | 0.4455 | $65 | $28.96 | 3.90% | 0.3509 |

10 | 9.40% | 0.4072 | $65 | $26.47 | 3.56% | 0.3564 |

11 | 9.40% | 0.3722 | $65 | $24.19 | 3.26% | 0.3584 |

12 | 9.40% | 0.3402 | $65 | $22.12 | 2.98% | 0.3574 |

13 | 9.40% | 0.3110 | $65 | $20.22 | 2.72% | 0.3539 |

14 | 9.40% | 0.2843 | $65 | $18.48 | 2.49% | 0.3483 |

15 | 9.40% | 0.2599 | $65 | $16.89 | 2.27% | 0.3412 |

16 | 9.40% | 0.2375 | $65 | $15.44 | 2.08% | 0.3326 |

17 | 9.40% | 0.2171 | $65 | $14.11 | 1.90% | 0.3231 |

18 | 9.40% | 0.1985 | $65 | $12.90 | 1.74% | 0.3127 |

19 | 9.40% | 0.1814 | $65 | $11.79 | 1.59% | 0.3017 |

20 | 9.40% | 0.1658 | $1,065 | $176.61 | 23.78% | 4.7561 |

Total | $2,300 | $742.65 | 100.00% | 10.4173 |

**Macaulay Duration = 10.4173 years**

Using Texas Instruments BA 2 plus calculator

C01 = 65 Press ENTER (65 * 1)

Down Arrow

F01 = 1

Down Arrow

C02 = 130 Press ENTER (65 * 2)

Down Arrow

F02 = 1

Down Arrow

C03 = 195 Press ENTER

Down Arrow

F03 = 1

Down Arrow

C04 = 260 Press ENTER

Down Arrow

F04 = 1

Down Arrow

C05 = 325 Press ENTER

Down Arrow

F05 = 1

Down Arrow

C06 = 390 Press ENTER

Down Arrow

F06 = 1

Down Arrow

C07 = 455 Press ENTER

Down Arrow

F07 = 1

Down Arrow

C08 = 520 Press ENTER

Down Arrow

F08 = 1

Down Arrow

C09 = 585 Press ENTER

Down Arrow

F09 = 1

Down Arrow

C10 = 650 Press ENTER

Down Arrow

F10 = 1

Down Arrow

C11 = 715 Press ENTER

Down Arrow

F11 = 1

Down Arrow

C12 = 780 Press ENTER

Down Arrow

F12 = 1

Down Arrow

C13 = 845 Press ENTER

Down Arrow

F13 = 1

Down Arrow

C14 = 910 Press ENTER

Down Arrow

F14 = 1

Down Arrow

C15 = 975 Press ENTER

Down Arrow

F15 = 1

Down Arrow

C16 = 1040 Press ENTER

Down Arrow

F16 = 1

Down Arrow

C17 = 1105 Press ENTER

Down Arrow

F17 = 1

Down Arrow

C18 = 1170 Press ENTER

Down Arrow

F18 = 1

Down Arrow

C19 = 1235 Press ENTER

Down Arrow

F19 = 1

Down Arrow

C20 = 21300 Press ENTER

Down Arrow

F20 = 1

Down Arrow

PRESS NPV

I = 9.4

Down Arrow

NPV --> CPT NPV = 7736.4125

SET N = 20, PMT = 65, I/Y = 9.4, FV = 1000

CPT ---> PV = -742.6486

Macaulay Duration = NPV / PV

Macaulay Duration = 7736.4125 / 742.6486

**Macaulay Duration = 10.4173 years**

b)

Bond Price =
Coupon / (1 + YTM)^{period} + Face value / (1 +
YTM)^{period}

Bond Price = $65 / (1 + 9.4%)^{1} + $65 / (1 +
9.4%)^{2} + ...+ $65 / (1 + 9.4%)^{20} + $1,000 /
(1 + 9.4%)^{20}

Using PVIFA = ((1 - (1 + Interest rate)^{- no of
periods}) / interest rate) to value coupons

Bond Price = $65 * (1 - (1 + 9.4%)^{-20}) / (9.4%) +
$1,000 / (1 + 9.4%)^{20}

Bond Price = $742.649

Bond price at 30 bps increase in yield

Bond Price =
Coupon / (1 + YTM)^{period} + Face value / (1 +
YTM)^{period}

Bond Price = $65 / (1 + 9.7%)^{1} + $65 / (1 +
9.7%)^{2} + ...+ $65 / (1 + 9.7%)^{20} + $1,000 /
(1 + 9.7%)^{20}

Using PVIFA = ((1 - (1 + Interest rate)^{- no of
periods}) / interest rate) to value coupons

Bond Price = $65 * (1 - (1 + 9.7%)^{-20}) / (9.7%) +
$1,000 / (1 + 9.7%)^{20}

Bond Price at 30 bps increase in yield = $721.893

Bond price at 30 bps decrease in yield

Bond Price =
Coupon / (1 + YTM)^{period} + Face value / (1 +
YTM)^{period}

Bond Price = $65 / (1 + 9.1%)^{1} + $65 / (1 +
9.1%)^{2} + ...+ $65 / (1 + 9.1%)^{20} + $1,000 /
(1 + 9.1%)^{20}

Using PVIFA = ((1 - (1 + Interest rate)^{- no of
periods}) / interest rate) to value coupons

Bond Price = $65 * (1 - (1 + 9.1%)^{-20}) / (9.1%) +
$1,000 / (1 + 9.1%)^{20}

Bond Price at 30 bps decrease in yield = $764.339

Approximate Modified Duration = (Bond Price at 30 bps decrease in yield - Bond Price at 30 bps increase in yield ) / (2 * Bond price * Change in yield)

Approximate Modified Duration = ($764.339 - $721.893) / (2 * $742.649 * 0.3%)

**Approximate Modified Duration = 9.526**

c)

Approximate Convexity = (Bond Price at 30 bps decrease in yield
+ Bond Price at 30 bps increase in yield - 2 * Bond Price) / (Bond
price * (Change in yield)^{2})

Approximate Convexity = ($764.339 + $721.893 - 2 * $742.649) /
($742.649 * (0.3%)^{2})

**Approximate Convexity = 139.958**

d)

Change in the full bond price for a 40 bps increase in yield

Change in the full bond price = (- Modified Duration * Yield
change + 0.5 * Convexity * (Yield change)^{2}) * Bond
price

Change in the full bond price = (-9.526 * 0.4% + 0.5 * 139.958 *
(0.4%)^{2}) * $742.649

Change in the full bond price = -3.698% * $742.649

**Change in the full bond price = -$27.466**

New Bond price at 40bps increase in yield = Bond price + Change in the full bond price

New Bond price at 40bps increase in yield = $742.649 + (-$27.466)

**New Bond price at 40bps increase in yield =
$715.183**

Change in the full bond price for a 40 bps decrease in yield

Change in the full bond price = (- Modified Duration * Yield
change + 0.5 * Convexity * (Yield change)^{2}) * Bond
price

Change in the full bond price = (-9.526 * -0.4% + 0.5 * 139.958
* (-0.4%)^{2}) * $742.649

Change in the full bond price = 3.922% * $742.649

**Change in the full bond price = $29.129**

New Bond price at 40bps decrease in yield = Bond price + Change in the full bond price

New Bond price at 40bps decrease in yield = $742.649 + $29.129

**New Bond price at 40bps decrease** **in
yield = $771.778**

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