# Pricing a bond using a Binomial Tree

Hello,

I have a question on the aforementioned topic. It seems that the CFA curriculum provides the following formula:

Bond value at a node = 0.50 * [(VH + C) / (1 + i)] + [(VL + C) / (1 + i)]

But on their examples and in the practice problems it seems they use:

Bond value at a node = 0.50 * [(VH) / (1 + i)] + [(VL) / (1 + i)] + C

Schweser also seems to use the first one, but they don’t produce the same answers. Would someone please clarify?

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The first one’s correct.

The value of the bond at a given node does not include the value of the coupon at that node.

When using backward induction, you need to include the coupon while discounting (you discount VH + C indeed). So basically

Value at Node X = 0.5 ((VH+C)/(1+r)^x + (VL+C)/(1+r)^x) = X

But when you discount X back to get the value of the bond before Node X (lets say Node W), you will need to include the coupon at Node X when discounting to Node W.

This is yet another topic on this. I have done the readings through CFAI and Schweser plus EOC questions and never came across a scenario where the second model was used. Am I missing something (ie, thankfully not missing something)? Lol.

Yeah this is not clear; I dont get it either. I also rememberedpretty well last they did all the exercises using the first formula. I checked the book from last year and exercises and that’s true, they use to do it by including the coupon first and then discounting thr whole. Now they use the second fromula in the exercises althought the lectures gives you the first one. I dont get it. Magician can you help please?

EDIT: Actually the formula that you gave is wrong afilippi. Bond value at a node = 0.50 * [(VH) / (1 + i)] + [(VL) / (1 + i)] + C It should be 0.50 * [(VH +C ) / (1 + i)] + [(VL +C ) / (1 + i)] + C CHeck the last exercise in reading 43 of the CFAI. Exercise 10.

Ok, bumping this. Now that I’m going through this a little more in depth it seems the CFAI material is extremely confusing on this topic. In reading 43, example 3 they use:

0.50 * [(VH +C ) / (1 + i)] + [(VL +C ) / (1 + i)] + C

To solve the values at each node. However that does not jive with their equation (1) just above the example, which is:

Bond value at a node = 0.50 * [(VH + C) / (1 + i)] + [(VL + C) / (1 + i)]

Using this gives a different answer than the first equation.

No wonder there’s been so many questions around this. Something is off here and I’m not sure which is right.

I’m lost too. This does not make sense. Which formula should we use? Is the CFAI wrong?

It’s p***ing me off cause I had written this entire binomial section off as something I knew and did not have to really focus on any longer. Now I’ve spent hours trying to figure out this disconnect. D****t!

I think you all have it you are just getting slightly confused over one thing. When you use backward induction and a binomial tree you obviously take the par value and the coupon in the last period and discount to the second last period using the relevant interest rates and use the formula:

0.50 * [(VH + C) / (1 + i)] + [(VL + C) / (1 + i)]

So if the par value is 100 and the coupon is 5, with the two relevant interest rates being 6% and 4%, you get:

0.5 * [(100 + 5)/1.06) + [(100 + 5)/1.06)] = 99.0566 and 0.5 * [(100 + 5)/1.04) + [(100 + 5)/1.04) = 100.9615 – let’s call this node “time period 2”

THESE VALUES are the values of the bond at time period 2!!! 99.0566 and 100.9615

Only when you want to find the value of the preceding node, i.e. time period 1, THEN you add the coupon of 5, and discount that back to the preceding node to find those values.

So to conclude – coupons are only added to the value of a bond at a particular node if you intend on discounting that back to the previous period.

Looking at the test questions on the site of the CFA institute, we see again as explanation the second formula… Frankly, it`s quite confusing. But I use the first formula as it is logical for me (I even draw a line with expected cash flows, look for the correct forward rates - because sometimes they give you an additional year to make you crazy..) and that`s it. Tnx, Magician for all your priceless help you are giving to us!

Good to see the history on this thread, coming from 2015 and everyone is now CFA or L3 passed!

I asked S2000 about this and I did not get a clear answer there is a DIFFERENCE in Schweser and in the Curriculum how the bond is valued!

this would be incorrect if you look at the examples in the curriculum and EOCs but in schweser this is the approach

this would be incorrect if you look at the examples in the curriculum and EOCs but in schweser this is the approach

For the value of the bond at a given node, you don’t add the coupon payment.

For the value that you will discount back to the previous time, you do add the coupon payment.

Once the coupon is paid it is still part of your investment value, but it is no longer part of the value of the bond.