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 Post subject: Calculating yield to maturityPosted: Sun, 9 Oct 2011 13:21:35 UTC
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Joined: Thu, 18 Jun 2009 10:22:43 UTC
Posts: 44
Hi,

I'm required to calculate the yield to maturity for the following question:

An insurance company has invested in the following fixed-income security: \$5,800,000 of 10-year bonds paying 7% interest with a par value of \$6,000,000.

What I've done so far:

I've used the bond pricing formula -->

PV = A [(1 + (1+r)^-n)/r] + [FV/(1+r)^n]

where A = coupon payment, r =yield to maturity (YTM), FV = face value of bond, n = number of periods

I'm told that for the 10-year bonds: Par = \$6,000,000; PV = \$5,800,000, coupon = 7%

My equation so far:

5,800,000 = (6,000,000 x 0.07)*[ (1 - (1+r)^-10) / r] + [6,000,000 / (1+r)^10]
5,800,000 = 420,000[ (1 - (1+r)^-10) / r] + [6,000,000 / (1+r)^10]

Multiply both sides by [r(1+r)^10]
5,800,000r(1+r)^10 = 420,000((1+r)^10) [1 - (1+r)^-10)] + 6,000,000r
5,800,000r(1+r)^10 = 420,000((1+r)^10) + 420,000 + 6,000,000r
That last line is where I'm stuck.

I'm not sure where to go after that to solve for r.

Any help on solving r would be much appreciated

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 Post subject: Re: Calculating yield to maturityPosted: Mon, 10 Oct 2011 03:11:08 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sun, 24 Jul 2005 20:12:39 UTC
Posts: 3692
Location: Ottawa Ontario
no_idea wrote:
5,800,000 = 420,000[ (1 - (1+r)^-10) / r] + [6,000,000 / (1+r)^10]

This is correct...good work...and you can stop here!
Cannot be solved directly: iteration (fancy word for "hit and miss"!) required.
Go here: http://www.mathsrevision.net/gcse/pages.php?page=41

You can get more sites by googling "solving by iteration".

Anyway, once you pick your poison, you should get 7.4852937....

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 Post subject: Re: Calculating yield to maturityPosted: Mon, 10 Oct 2011 07:49:04 UTC
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Joined: Thu, 18 Jun 2009 10:22:43 UTC
Posts: 44
Thank you Denis!

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 Post subject: Re: Calculating yield to maturityPosted: Thu, 27 Oct 2011 14:05:04 UTC
 S.O.S. Newbie

Joined: Thu, 27 Oct 2011 10:02:14 UTC
Posts: 2
As stated in an earlier reply, such an equation can't be solved in closed form. Here is how you go about calculating YTM for your bond with Newton Raphson method, assuming the interest payments are twice a year

f(x) = 6000000 + -5800000 * (1+x)^20 + 210000 [(1+x)^20 - 1]/x

f'(x) = 20 * -5800000 * (1+x)^19 + 210000 * (20 x (1 + x)^19 - (1 + x)^20 + 1) / (x^2)

x = 0.1
f(x) = -20991749.8125
f'(x) = -572854768.254
x1 = 0.1 - -20991749.8125/-572854768.254 = 0.063355895812
Error Bound = 0.063355895812 - 0.1 = 0.036644 > 0.000001

x1 = 0.063355895812
f(x1) = -5805780.892
f'(x1) = -286126384.55
x2 = 0.063355895812 - -5805780.892/-286126384.55 = 0.0430649292828
Error Bound = 0.0430649292828 - 0.063355895812 = 0.020291 > 0.000001

x2 = 0.0430649292828
f(x2) = -1022856.6503
f'(x2) = -191072004.364
x3 = 0.0430649292828 - -1022856.6503/-191072004.364 = 0.0377116769648
Error Bound = 0.0377116769648 - 0.0430649292828 = 0.005353 > 0.000001

x3 = 0.0377116769648
f(x3) = -53810.0854
f'(x3) = -171291081.722
x4 = 0.0377116769648 - -53810.0854/-171291081.722 = 0.0373975328487
Error Bound = 0.0373975328487 - 0.0377116769648 = 0.000314 > 0.000001

x4 = 0.0373975328487
f(x4) = -173.2041
f'(x4) = -170189435.984
x5 = 0.0373975328487 - -173.2041/-170189435.984 = 0.0373965151349
Error Bound = 0.0373965151349 - 0.0373975328487 = 1.0E-6 > 0.000001

x5 = 0.0373965151349
f(x5) = -0.0018
f'(x5) = -170185877.368
x6 = 0.0373965151349 - -0.0018/-170185877.368 = 0.0373965151243
Error Bound = 0.0373965151243 - 0.0373965151349 = 0 < 0.000001

YTM = 3.74%
Annual YTM = 7.48%

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 Post subject: Re: Calculating yield to maturityPosted: Thu, 27 Oct 2011 15:37:50 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sun, 24 Jul 2005 20:12:39 UTC
Posts: 3692
Location: Ottawa Ontario
PJ, since rate ends up at ~7.4852937....
I think you need to go another "loop" to end up with 7.49 (if that's the rounding required);
agree?

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