Question

RCL issues a 7-year fixed coupon bond paying coupons semi annually at a coupon rate of 7.50%. The bonds current price as a percentage of par is $92.0185 and its future value is $1000.

a. What is the bonds YTM ?

b. What is the bonds Duration?

Answer 1

Part A:

YTM :

YTM is the rate at which PV of Cash inflows are equal to Bond price when the bond is held till maturity. Yield to maturity (YTM) is the total return anticipated on a bond if the bond is held until it matures. Yield to maturity is considered a long-term bond yield but is expressed as an annual rate

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

Period	Cash Flow	PVF/PVAF @ 4.5 %	PV of Cash Flows	PVF/ PVAF @5 %	PV of Cash Flows
1-14	$                 37.50	10.2228	$               383.36	9.8986	$                   371.20
14	$           1,000.00	0.5400	$               539.97	0.5051	$                   505.07
PV of Cash Inflows			$               923.33		$                   876.27
PV of Cash Oiutflows			$               920.19		$                   920.19
NPV			$                    3.14		$                    -43.92

YTM per six months = Rate at which least +ve NPV + [ NPV at that rate / Change in NPV due to Inc of 0.5% in Int Rate ] * 0.5%
= 4.5 % + [3.14 / 47.06 ] * 0.5%
= 4.5 % + [0.07 * 0.5% ]
= 4.5 % + [0.0334 % ]
= 4.53 %

YTM Per anum = IRR per six months * 12 / 6
= 4.5334 % * 2
= 9.0668 %
i.e 9.07 %

PVAF = Sum [ PVF(r%, n) ]
PVF(r%, n) = 1 / ( 1 + r )^n
r - Int Rate per period
n - No. of Periods

How to calculate PVAF using Excel?
+PV(Rate,NPER,-1)
Rate = Disc rate
NPER - No. of Periods

Part B:

Duration:

Duration is a measure of the sensitivity of the price of a bond or other debt instrument to a change in interest rates.

Duaration = Sum [ Weight * Year ]

Particulars	Values
YTM per period	4.530%
Coupon Rate per period	3.750%
Time in periods	14
Period/ Year	Period

= [ ( 1 + Y ) / Y ] - [ [ ( 1 + Y ) + T ( C - Y) ] / [ C [ [ ( 1 + Y )^ t ] - 1 ] + Y ] ]
= [ ( 1 + 0.0453 ) / 0.0453 ] - [ [ ( 1 + 0.0453 ) + 14 ( 0.0375 - 0.0453 ) ] / [ 0.0375 [ [ ( 1 + 0.0453 ) ^ 14 ] - 1 ] +0.0453 ] ]
= [ ( 1.0453 ) / 0.0453 ] - [ [ ( 1.0453 ) + 14 ( -0.0078 ) ] / [ 0.0375 [ [ ( 1.0453 ) ^ 14 ] - 1 ] +0.0453 ] ]
= [ 23.0751 ] - [ [ ( 1.0453 ) + ( -0.1092 ) ] / [ 0.0375 [ [ ( 1.8594 ] - 1 ] +0.0453 ] ]
= [ 23.0751 ] - [ [ ( 0.9361 ) ] / [ 0.0375 [ [ 0.8594 ] +0.0453 ] ]
= [ 23.0751 ] - [ [ ( 0.9361 ) ] / [ 0.0322 ] +0.0453 ] ]
= [ 23.0751 ] - [ [ ( 0.9361 ) ] / [ 0.0775 ] ]
= [ 23.0751 ] - [ 12.0787 ]
= 10.9951 Periods
Duration in Years:
= Duration in Periods / 2
Duration in Years:
= Duration in Periods / 2
= 5.5 Years

Duration of Bond is 5.5 Years

Alternatively:

Period	Cash Flow	PVF@ 4.53 %	Disc CF	Weight	Wt * Period
1	$                37.50	0.9567	$      35.87	0.0390	0.0390
2	$                37.50	0.9152	$      34.32	0.0373	0.0746
3	$                37.50	0.8755	$      32.83	0.0357	0.1070
4	$                37.50	0.8376	$      31.41	0.0341	0.1365
5	$                37.50	0.8013	$      30.05	0.0326	0.1632
6	$                37.50	0.7666	$      28.75	0.0312	0.1874
7	$                37.50	0.7334	$      27.50	0.0299	0.2092
8	$                37.50	0.7016	$      26.31	0.0286	0.2287
9	$                37.50	0.6712	$      25.17	0.0273	0.2461
10	$                37.50	0.6421	$      24.08	0.0262	0.2616
11	$                37.50	0.6143	$      23.03	0.0250	0.2753
12	$                37.50	0.5876	$      22.04	0.0239	0.2873
13	$                37.50	0.5622	$      21.08	0.0229	0.2978
14	$                37.50	0.5378	$      20.17	0.0219	0.3068
14	$           1,000.00	0.5378	$    537.81	0.5843	8.1803
Duration in Periods					11.0006

Particulars	Values
Duration in Periods	11.0006
No. of periods in a Year	2
Duration in Years	5.5003