Question

What is the duration of a 20 year Treasury with a 2% coupon and a YTM of 2% and face value = $100,000? If rates fall 150bp, what is your estimate of the dollar change in value?

Answer 1

When the coupon rate and YTM is equal the price of the bond will be equal to its par value.

So bonds current value is 100000

Duration is a measure of a bond's sensitivity to interest rate changes. The higher the bond's duration, the greater its sensitivity to the change and vice versa.

It measures how long it takes, in years, for an investor to be repaid the bond’s price by the bond’s total cash flows

Macaulay duration = ∑[{tC/(1+y) ^t} + {nM/(1 + y) ^n}]/P

t = Period in which the coupon is received

C = Periodic coupon payment

y = the periodic YTM or required rate

n = number of periods

M = maturity value

P = market price of the bond

Pls refer below table

Year	Cash flow	PV factor @ ytm	PV of cash flow	Time weighted PV of cash inflow
a	b	c	b*c	a*b*c
1	2000	0.980392	1960.78	1960.784314
2	2000	0.961169	1922.34	3844.675125
3	2000	0.942322	1884.64	5653.934007
4	2000	0.923845	1847.69	7390.763408
5	2000	0.905731	1811.46	9057.308098
6	2000	0.887971	1775.94	10655.65659
7	2000	0.87056	1741.12	12187.8425
8	2000	0.85349	1706.98	13655.84594
9	2000	0.836755	1673.51	15061.59479
10	2000	0.820348	1640.7	16406.966
11	2000	0.804263	1608.53	17693.78686
12	2000	0.788493	1576.99	18923.83621
13	2000	0.773033	1546.07	20098.84565
14	2000	0.757875	1515.75	21220.50069
15	2000	0.743015	1486.03	22290.4419
16	2000	0.728446	1456.89	23310.26604
17	2000	0.714163	1428.33	24281.52712
18	2000	0.700159	1400.32	25205.7375
19	2000	0.686431	1372.86	26084.36887
20	102000	0.672971	68643.1	1372861.52
			Total	1667846.20

∑[{tC/(1+y) ^t} + {nM/(1 + y) ^n}] = 1667846.2011

Current price of the bond (P) = $100000

Let's put all the values in the formula to find the duration

Bond duration = 1667846.2011/ 100000

                               = 16.68

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If the YTM reduced to 0.5% price of the bond will be

Price of the bond could be calculated using below formula.

P = C* [{1 - (1 + YTM) ^ -n}/ (YTM)] + [F/ (1 + YTM) ^ -n]

Where,

                Face value = $100000

                Coupon rate = 2%

                YTM or Required rate = 0.5%

                Time to maturity (n) = 20 years

                Annual coupon C = $2000

Let's put all the values in the formula to find the bond current value

P = 2000* [{1 - (1 + 0.005) ^ -20}/ (0.005)] + [100000/ (1 + 0.005) ^20]

P = 2000* [{1 - (1.005) ^ -20}/ (0.005)] + [100000/ (1.005) ^20]

P = 2000* [{1 - 0.90506}/ 0.005] + [100000/ 1.1049]

P = 2000* [0.09494/ 0.005] + [90505.92814]

P = 2000* 18.988 + 90505.92814

P = 37976 + 90505.92814

P = 128481.92814

So price of the bond is $128481.93

$ change in price = $128481.93 – 100000 = 28481.93

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