Replicating portfolio (bonds)

[Number1]

Hello,
I've seen examples on how to setup a "replicating portfolio" to solve for the price of a bond with a known coupon rate given 2 other bonds with known prices and known coupons rates which pay on the same date annually. Can someone provide an example using 3 or more known bonds ? I have a feeling it's simple, but I just don't see it. Thanks

 

[David Harper CFA FRM]

see http://forum.bionicturtle.com/threads/law-of-one-price.5799/#post-16680
( I searched "law of one price")

 

[Number1]

Thanks for the link David, I didn't think to search for "law of one price". Is it possible to solve something like the below(expanding on the example you gave in the other post) ?

Coupon ----Price
2 - 7/8 ---- 98.40
4 - 1/2 ---- 99.80
6 - 1/4 ---- 101.30
7 - 1/4------- ???

Where the above all pay coupons on the same date .

Thanks for you very fast reply to my previous post !

 

[David Harper CFA FRM]

sure, same logic (but i *doubt* the prices are internally consistent: unless they've been priced consistently, you could get two different answers which I bet you would here). Still, law of one price says:

2.875% * X1 + 7.25% *(1 – X1) = 6.25%; i.e., X1 gives you allocation to 2 7/8 with same cash flow as 6 1/4. Then after you have X1:
X1*$98.40 + (1-X1)*P = $101.30; solving for (P) is implied price

but note you can also do:
4.5%* X2 + 7.25% *(1 – X2) = 6.25%; X2 gives you allocation to 4 1/2 with same cash flow as 6 1/4. Then after you have X2:
X2*$99.80 + (1-X2)*P = $101.30; solving for (P) is implied price ... which will be different, unless these numbers are priced consistently in the first place

 

[afterworkguinness]

Hi David,
If I understand correctly, you can allocate X% of any one of the bonds with (1-x) of the 7.35% bond ?

ie: You said 2.875% * X + 7.25% *(1 – X) = 6.25% and 4.5%* X + 7.25% *(1 – X) = 6.25%;

so then is this valid too: 6.25%*X + 7.25%*X =2.875% ?

Cheers

 

[David Harper CFA FRM]

Hi afterwork,

I haven't completely analyzed the 4-bond hypothetical, but I don't *think* yours would work. First, please note my two instances (X) are not the same variables; i just edited them to X1 and X2. The idea in the original question hinges on an interpolation of the price between two bonds that, together in a two-bond portfolio with a total weight of 100%, can replicate the coupon of the bond in the middle, so if you have the four bonds above, I can immediately see the following interpolations/replications (where each X% below is different):

• Mix of 2 - 7/8 (X%) and 6 - 1/4 (1-X%) that pay same coupon as 4 - 1/2
• Mix of 2 - 7/8 (X%) and 7 - 1/4 (1-X%) that pay same coupon as 4 - 1/2
• Mix of 2 - 7/8 (X%) and 7 - 1/4 (1-X%) that pay same coupon as 6 - 1/4
• Mix of 4 - 1/2 (X%) and 7 - 1/4 (1-X%) that pay same coupon as 6 - 1/4

One of which does not include the 7 1/4 bond, so in addition to the two I listed above, I *think* we could also add a third (where X3 is clearly different than X1 and X2):
2.875%*X3 + 7.25%*(1-X3) = 4.5%, the solving for P in X3*98.40 + (1-X3)*P = $99.80

But that's tentative on my part frankly: Number1 extended GARP's 3-bond question to 4-bonds and I'm not confident I've outlined all solutions (e.g., maybe overweight + short position = 100% should work, I don't know ... the 3-bond problem is somewhat easier I think). Thanks,

 

[Number1]

Hi David,
Are you saying the idea is you create a portfolios with X% of a lower coupon bond (B1) and (1-X)% of a higher coupon bond (B2) to replicate a bond with a coupon between B1 and B2 ?

 

[David Harper CFA FRM]

Hi Number1, Yes, that's the solution (to GARP's question, at least!). Going back to the original example of three bonds:

1. 2 - 7/8 @ $98.40
2. 4 - 1/2 @ [$99.80; i.e., the unknown to-be-solved-for]
3. 6 - 1/4 @ $101.30

mathematically, it's just an interpolation: 4.5 is (4.5 - 2.875)/(6.25 - 2.875) = 48.15% "of the distance between" 2.875 and 6.25 if you "start" from 2.875, or 51.85% if you "start" from 6.25. And 99.80 is the interpolated price; i.e., it is also 48.15% or 51.85% of the distance between the two "outer" prices. You need three bonds, you can't just take a ratio of two coupons.

The mathematical interpolation (merely) implements the law of one price:

• we can purchase one bond for $99.80 and get a $4.5 coupon; or, lacking an arbitrage we should be able to:
• allocate the same $99.80 to the two other bonds and receive two coupons that sum to the same $4.5. Thanks,

 

[Number1]

Awesome, thanks.