Question

1. A graph, plotting the relationship between a bond’s modified duration and its tenor for a zero, discount bond, premium bond and perpetuity.

2. In writing, and by showing numerical analysis, explain what drives the shape of the four curves you plotted in part 1

Answer 1

Duration is a measure of a bond's interest rate sensitivity. It provides an estimate how much a bond’s price is likely to rise or fall if interest rates change

ZCB
settlment date	maturity date	Years to maturity	coupon	yield	frequency (annual)	Macaulay Duration	Modified Duration	Slope of the curve= y/x= duration/ tenor
2/7/2019	2/7/2020	1	0%	5%	1	1.0000	0.9524	0.9524
2/7/2019	2/6/2021	2	0%	5%	1	1.9972	1.9048	0.9524
2/7/2019	2/6/2022	3	0%	5%	1	2.9972	2.8571	0.9524
2/7/2019	2/6/2023	4	0%	5%	1	3.9972	3.8095	0.9524
2/7/2019	2/6/2024	5	0%	5%	1	4.9972	4.7619	0.9524
2/7/2019	2/4/2029	10	0%	5%	1	9.9917	9.5238	0.9524
2/7/2019	2/3/2034	15	0%	5%	1	14.9889	14.2857	0.9524
2/7/2019	2/2/2039	20	0%	5%	1	19.9861	19.0476	0.9524
discount bond
settlment date	maturity date	Years to maturity	coupon	yield	frequency (annual)	Macaulay Duration	Modified Duration	Slope of the curve= y/x= duration/ tenor
2/7/2019	2/7/2020	1	4%	5%	1	1.0000	0.9524	0.9524
2/7/2019	2/6/2021	2	4%	5%	1	1.9584	1.8678	0.9339
2/7/2019	2/6/2022	3	4%	5%	1	2.8816	2.747	0.9157
2/7/2019	2/6/2023	4	4%	5%	1	3.7677	3.5909	0.8977
2/7/2019	2/6/2024	5	4%	5%	1	4.6175	4.4003	0.8801
2/7/2019	2/4/2029	10	4%	5%	1	8.3513	7.9615	0.7962
2/7/2019	2/3/2034	15	4%	5%	1	11.3278	10.799	0.7199
2/7/2019	2/2/2039	20	4%	5%	1	13.6668	13.0292	0.6515
premium bond
settlment date	maturity date	Years to maturity	coupon	yield	frequency (annual)	Macaulay Duration	Modified Duration	Slope of the curve= y/x= duration/ tenor
2/7/2019	2/7/2020	1	6%	5%	1	1.0000	0.9524	0.9524
2/7/2019	2/6/2021	2	6%	5%	1	1.9411	1.8513	0.9257
2/7/2019	2/6/2022	3	6%	5%	1	2.8330	2.7007	0.9002
2/7/2019	2/6/2023	4	6%	5%	1	3.6765	3.504	0.8760
2/7/2019	2/6/2024	5	6%	5%	1	4.4750	4.2645	0.8529
2/7/2019	2/4/2029	10	6%	5%	1	7.8838	7.5163	0.7516
2/7/2019	2/3/2034	15	6%	5%	1	10.5301	10.0392	0.6693
2/7/2019	2/2/2039	20	6%	5%	1	12.6080	12.0208	0.6010
perpetuity bond
settlment date	maturity date	Years to maturity	coupon	yield	frequency (annual)	Macaulay Duration	Modified Duration
2/7/2019	-	1	5%	5%	1	21.0000	20
				6.00%			20
				7.00%			20
				8.00%			20
				9.00%			20
				10.00%			20
				11.00%			20
				12.00%			20
				13.00%			20
settlment date	maturity date	Years to maturity	coupon	yield	frequency (annual)	Macaulay Duration	Modified Duration
2/7/2019	-	1	5%	5%	1	21.0000	20
				6.00%			20
				7.00%			20
				8.00%			20
				9.00%			20
				10.00%			20
				11.00%			20
				12.00%			20
				13.00%			20

Explanation- for zero coupon bond, the relationship between the tenor and modified duration is linear with slope of the curve being almost equal to one and modified duration being slightly lower than the tenor. This is because, the changes in interest rate do not impact the price of a zcb as there are no coupon payments.

Explanation- for discount bond, the relationship between the tenor and modified duration is linear with slope of the curve being less than one. The slope of the curve goes decreasing at an increasing pace as the tenor increases. This is because the impact of difference between yield and coupon rate gets magnified as the tenor increases.
Explanation- for premium bond, the relationship between the tenor and modified duration is linear with slope of the curve being less than one. The slope of the curve goes decreasing at an increasing pace as the tenor increases, at a rate greater than that for a discount bond. This is because the impact of difference between yield and coupon rate gets magnified as the tenor increases, more than that for a discount bond.
Explanation- for perpetual bond, there is not relationship between the tenor and duration of the bond as the tenor is technically infinite. Hence, the curve is a straight horizontal line. A more appropriate relationship is between the duration and the yield of the bond. the relationship between the 2 is inverse i.e. modified duration= 1/rate

Note- pay attention to the horizontal scale. The tenor is not changing at a linear scale after 5 years. This is done to emphasis on a rate of change of duration with tenor