Question

You find a bond with 24 years until maturity that has a coupon rate of 9.0 percent and a yield to maturity of 6.1 percent. Suppose the yield to maturity on the bond increases by 0.25 percent. a. What is the new price of the bond using duration and using the bond pricing formula? (Do not round intermediate calculations Round your answers to 2 decimal places.) Book Print Merences Estimated price Actual price b. Now suppose the original yleld to maturity is increased by 1 percent. What is the new price of the bond? (Do not round intermediate calculations. Round your answers to 2 decimal places.) Estimated price Actual price

Answer 1

To answer both the parts of the question, we need to first calculate the current bond price and modified duration first.

To calculate the price of any financial instrument, one needs to identify the cash flows arising out of the instrument and discount them using the yield (discount rate) to determine the present value.

For example, in case of a Bond, with an annual coupon C, face value F, time to maturity n years, yield r, the present value of this bond will be:

The bond will pay a coupon C after 1 year, another coupon C after 2^nd year, and so on till the end of n^th year. At the end of n^th year the bond will also payback the face value.

discounting these coupons and the face value by the yield r we can find the present value (PV) or the bond price as follows.

Bond price = C/(1+r) + C/(1+r)² + C/(1+r)³ + C/(1+r)⁴ + ........ + C/(1+r)ⁿ + F/(1+r)ⁿ

This can be simplified as   

Bond price = C * [(1-(1+r)^-n ) / r] + F/(1+r)ⁿ

Given values in the problem:

n = 24 years

r = 6.1%

F = 1000 (assumed)

C = 9% = 9% of 1000 = 90

Substituting the given values in this formula,

Bond price = 90 * [(1-(1+0.061)^-24 ) / 0.061] + 1000/(1+0.061)²⁴ = 1360.62

Duration it is the measure of how long on an average the holder of the bond has to wait before he receives his payments on the bond.

We need to calculate Modified duration which is the sensitivity of bond's price to change in yield.

Modified duration = Macaulay Duration / (1+ r/n) where n is the number of periods in a year. For annual rate n =1, semi-annual n = 2 and so on.

we need to calculate Macaulay Duration first. What is Macaulay duration?

Macaulay‟s duration is the weighted average of the times when the payments are made. And the weights are a ratio of the coupon paid at time t to the present bond price. This can be expressed as:

Macaulay Duration =

1*C/(1+r) +2* C/(1+r)2 + 3 * C/(1+r)3 +4*C/(1+r)+ + ........ +n* C/(1+r)" +n* F/(1+r)" Current BondPrice

Please note the difference in formula with respect to the bond price formula is that we have multiplied time (t) with each cash flow as is 2*C..., or 4*C..... etc. and divided the cash flow by current bond price to determine the weight.

The value is calculated in excel and the screenshot is copied below:

time period (t)	cash flow (C )	PV of cash flow (PV) = C/(1+r)^t	PV of time weighted cashflow (PV * t)
1	90	84.83	84.83
2	90	79.95	159.90
3	90	75.35	226.06
4	90	71.02	284.08
5	90	66.94	334.68
6	90	63.09	378.53
7	90	59.46	416.23
8	90	56.04	448.34
9	90	52.82	475.39
10	90	49.78	497.84
11	90	46.92	516.14
12	90	44.22	530.69
13	90	41.68	541.86
14	90	39.29	549.99
15	90	37.03	555.40
16	90	34.90	558.36
17	90	32.89	559.15
18	90	31.00	558.01
19	90	29.22	555.14
20	90	27.54	550.76
21	90	25.95	545.05
22	90	24.46	538.18
23	90	23.06	530.29
24	90	21.73	521.54
24	1000	241.45	5794.85
	Total	1360.62	16711.28

The current bond price = $1360.62

The numerator of Macaulay Duration formula = 16711.28

Macaulay Duration = 16711.18 / 1360.62 = 12.28209

Now calculate Modified duration.

Modified duration = 12.28209 / (1+0.061) = 11.576

Modified duration is the sensitivity of change in bond price to change is yield. The formula for this is:

AB/B= -(ModifiedDuration) * AY

Now let us answer the two questions:

a.

Calculate new bond price using duration:

change in yield is given as +0.25%

substitute the values in the modified duration formula:

change in bond price = B * -(Modified Duration ) * change in yield

= 1360.62 * -11.576 * +0.0025 = -39.3763

New Bond price - current bond price = change in bond price

New Bond price = change in bond price + current bond price

= --39.3763 + 1360.62 = 1321.24

Bond price using duration = 1321.24

Calculate new bond price using bond pricing formula:

Bond price = C * [(1-(1+r)^-n ) / r] + F/(1+r)ⁿ

Bond price = 90 * [(1-(1+0.0635)^-24 ) / 0.0635] + 1000/(1+0.0635)²⁴ = 1322.09

Bond price using bond pricing formula = 1322.09

Part b:

Change in yield to maturity = + 1.0 %

change in bond price = B * -(Modified Duration ) * change in yield

= 1360.62 * -11.576 * +0.0100 = -157.50537

New Bond price - current bond price = change in bond price

New Bond price = change in bond price + current bond price

= -157.50537+ 1360.62 = 1203.12

Bond price using duration = 1203.12

Calculate new bond price using bond pricing formula:

Bond price = C * [(1-(1+r)^-n ) / r] + F/(1+r)ⁿ

Bond price = 90 * [(1-(1+0.0710)^-24 ) / 0.0710] + 1000/(1+0.0710)²⁴ = 1216.02

Bond price using bond pricing formula = 1216.02