Question

Given a 5 year bond with an annual step-up coupon (1% initial coupon, 1% step-up per year), trading at 105% of face value (100), what is the YTM? Would this bond have a duration that is higher or lower than an equivalent bond with a flat 5% coupon?

Answer 1

Year	CF	PVF@1%	Disc CF	PVF@2%	Disc CF
1	1	0.9901	$     0.99	0.9804	$     0.98
2	2	0.9803	$     1.96	0.9612	$     1.92
3	3	0.9706	$     2.91	0.9423	$     2.83
4	4	0.9610	$     3.84	0.9238	$     3.70
5	5	0.9515	$     4.76	0.9057	$     4.53
5	100	0.9515	$   95.15	0.9057	$   90.57
PV of cash inflows		$ 109.61		$ 104.53
Price of the bond (Given)		$ 105.00		$ 105.00
NPV		$     4.61		$    -0.47

YTM :
YTM is the rate at which PV of Cash inflows are equal to Bond price when the bond is held till maturity.

YTM = Rate at which least +ve NPV + [ NPV at that Rate / Change in NPV due to 1% inc in disc rate ] * 1%

= 1 % + (4.61 / 5.08) * 1%

= 1 % + 0.91 * 1%

= 1 % + 0.91 %

= 1.91 %

YTM of flat 5% coupon bond:

Year	CF	PVF@3%	Disc CF	PVF@4%	Disc CF
1	5	0.9709	$     4.85	0.9615	$     4.81
2	5	0.9426	$     4.71	0.9246	$     4.62
3	5	0.9151	$     4.58	0.8890	$     4.44
4	5	0.8885	$     4.44	0.8548	$     4.27
5	5	0.8626	$     4.31	0.8219	$     4.11
5	100	0.8626	$   86.26	0.8219	$   82.19
PV of cash inflows		$ 109.16		$ 104.45
Price of the bond (Given)		$ 105.00		$ 105.00
NPV		$     4.16		$    -0.55

YTM :
YTM is the rate at which PV of Cash inflows are equal to Bond price when the bond is held till maturity.

YTM = Rate at which least +ve NPV + [ NPV at that Rate / Change in NPV due to 1% inc in disc rate ] * 1%

= 3 % + (4.16 / 4.71 )* 1%

= 3 % + 0.88 %

= 3.88 %

Duration of Bond,  with Initial coupon 1 % ans Step up 1 % per year

Year	CF	[email protected]%	Disc CF	Weights	Weights * Year
1	$     1.00	0.9813	$     0.98	0.0093	0.0093
2	$     2.00	0.9629	$     1.93	0.0183	0.0367
3	$     3.00	0.9448	$     2.83	0.0270	0.0810
4	$     4.00	0.9271	$     3.71	0.0353	0.1413
5	$     5.00	0.9097	$     4.55	0.0433	0.2167
5	$ 100.00	0.9097	$   90.97	0.8666	4.3332
Duration of the Bond	4.8182

Duration of flat 5% coupon bond

Year	CF	[email protected]%	Disc CF	Weights	Weights * Year
1	$     5.00	0.9626	$     4.81	0.0458	0.0458
2	$     5.00	0.9267	$     4.63	0.0441	0.0883
3	$     5.00	0.8921	$     4.46	0.0425	0.1274
4	$     5.00	0.8588	$     4.29	0.0409	0.1636
5	$     5.00	0.8267	$     4.13	0.0394	0.1968
5	$ 100.00	0.8267	$   82.67	0.7873	3.9365
Duration of the Bond	4.5584

Duration of 1% initial coupon, 1% step-up per year is higher than Duration of flat 5% coupon bond

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