Bond: yield to maturity is 10% on 10yr 8% semi-ann. coupon b

kmeline

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Aug 27, 2008
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so, I'm at a loss...

Q. The yield to maturity equals 10% on a 10yr 8% semiannual coupon bond. Question is what would I pay for this bond.

So, I think I know that YTM = 10%, k[sub:3h368mjb]d[/sub:3h368mjb]=8%, N=10, M=undefined par value, INT = 8%[*]M and V[sub:3h368mjb]b[/sub:3h368mjb]=INT/2(PVIFA[sub:3h368mjb]kd/2,2N[/sub:3h368mjb])+M(PVIF[sub:3h368mjb]kd/2,2N[/sub:3h368mjb]), but I'm clearly on the wrong path and not sure how to back into current yield and capital gains yield over the first year from here.

Any pointers??? THANKS!!
 
Re: Bond

YTM is (I believe) same as annual effective rate.

Since coupons are received semiannually, then we need the semiannual rate that equals 10% annual:
(1 + i)^2 = 1.10 ; i = .0488...... (a bit lesser than .05, as expected)

Make it a $100 bond, hence semiannual coupons of $4; cash flow looks like:
(0)?....(1)4....(2)4....(3)4 -------------------------------(19)4....20(4 + 100)

Present value of the coupons: c(1 - 1/(1+i)^n) / i = 4(1 - 1/(1+.0488...)^20 / .0488... = ~50.36

Present value of the $100: 100 / (1+i)^n = 100 / (1+.0488...)^20 = ~38.55
[note that this one may be calculated this way: 100 / (1.10)^10]

So purchase price = 50.36 + 38.55 = 88.91

Hope this helps...and hope I understood your problem properly...if not, TK will jump in!!
 
The Way I Learned It ...

Hi kmeline:

You wrote,
... V[sub:3hpxsrtc]b[/sub:3hpxsrtc]=INT/2(PVIFA[sub:3hpxsrtc]kd/2,2N[/sub:3hpxsrtc])+M(PVIF[sub:3hpxsrtc]kd/2,2N[/sub:3hpxsrtc]) ...

I'm not sure what the variable name V[sub:3hpxsrtc]b[/sub:3hpxsrtc] represents above, and its definition looks somewhat scary to me, but here's the formula that I used.

\(\displaystyle B = \sum_{I=1}^{N} \frac{C}{(1 + R)^I} + \frac{F}{(1 + R)^N}\)

\(\displaystyle B = \mbox{bond's price}\)

\(\displaystyle N = \mbox{number of coupon payment periods}\)

\(\displaystyle C = \mbox{coupon payment amount in dollars}\)

\(\displaystyle R = \mbox{required yield-to-maturity rate}\)

\(\displaystyle F = \mbox{future bond value at maturity}\)

This formula comes from the fact that the price of a bond is equal to the present value of its future cash flows.

Since the coupon payments are semi-annual, \(\displaystyle R\) is the yield-to-maturity for six months. In order to annualize \(\displaystyle R\), I used the following formula.

\(\displaystyle \mbox{Effective Annual Yield}\;=\;(\,1\;+\;\mbox{Periodic Rate})^{Payments\;per\;Year}\;-\;1\)

\(\displaystyle 0.10 = (1 + R)^{2}\;-\;1\)

\(\displaystyle R = 0.0489\)

I substituted this new value (4.89%) for \(\displaystyle R\) in the first formula; there are 20 payments over ten years; I picked $1,000 for the bond's value at maturity; 8% of $1,000 is $80, so each payment is $40.

\(\displaystyle B = \sum_{I=1}^{20} \frac{40}{(1 + 0.0489)^I} + \frac{1000}{(1 + 0.0489)^{20}}\)

\(\displaystyle B = 889.105\)

The bond's price is $889.11.

This seems to agree with Denis' calculation. So, it looks like the amount you would pay for the bond in your problem is about 88.9105% of the maturity value.

I also will be more confident after the actuary reviews this. :wink:

Cheers,

~ Mark
 
Re: Bond

Good call, Gentlemen. 2 for 2. The question should give us the value at redemption, just to avoid the ambiguity we see in the two solutions. $1000, $100, maybe it was $10,543.
 
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