Backward induction - different approaches

CFA Level II
#1

I noticed that in CFAI curriculum by BI method 3rd node (last one before maturity FV value in last node) is discounted by previous node bond price + coupon and then coupon is added once again in this manner:

0,5 x ((FV + coupon)/discount factor + (FV+coupon)/discount factor))+ coupon = PV bond value in node 3

while, seems (or maybe I missed something) in Schweser same calculation does not include coupon payment added once again at the end:

0,5 x ((FV + coupon)/discount factor + (FV+coupon)/discount factor))= PV bond value in node 3

I am bit confused about this, so appreciate if someone can explain me what is the corect approach? Anyway, I will use CFAI approach.

#2

You probably realize the two methods are not interchangeable, i.e. they do not give the same result. Something must be wrong.

The approach with a coupon added at the end is correct though. So you get the present value of the principal and associated coupons plus the latest coupon that you received which is already at present value.

#3

Correct. I think that must be that I haven’t correct compared results since seems Schweser uses shortcuts. However, I wanted to share it to be 100 % sure.

#4

Don’t take my word for 100% assurance. I’d say 98% :slight_smile:

#5

Let’s wait what Magician says. Thanks anyway for your comment.

#6

The coupon is added to the discounted value of the expected value of the bond in the next period because you evaluate the bond at periods end.

I did all the examples in Schweser and all the QBank questions for reading 44 ad this is the method used.

#7

Did you make CFAI EOC or portal topic questions?

#8

Yes. I think I know what is confusing.

CFAI and Schweser use different notations.

Check example 5 exhibit 21 from CFAI. More precisely at time 2 upper node.

You will find C = 5 and V = 102.0721 . It was computed this way (105/1.08167) + 5 = 102.0721

Schweser uses another notation. Look at the example for LOA 44.d (page 168) In each box they include in this order

1)The present value of the bond

2)The coupon

3)The forward rate.

CFAI uses this notation

  1. Coupon rate

2)Sum of the present value and coupon rate.

#9

I will. Thanks a lot for this tip. I am quite sure that I must misunderstand some steps in the process.

#10

Without having Schweser’s notes I can only speculate here: they’re computing the present value of future cash flows to determine whether an option (say, a call option or a put option) would be exercised at that node. The decision depends only on the present value of the future cash flows; the current coupon payment will have to be paid either way.

#11

I checked and it is simply different.

Example Time 2 Discount 8 %, coupon 5 % FV 100 3 Y Bond

Schweser

0,5 ((100+5)1,08 + (100+5)1,08)= 97,222

CFAI

0,5 (105/1,08 + 105/1,08) +5 =102,222

#12

S2000

No option in this example.

#13

I looked at the CFA Institute study guide, and what I see is simply sloppy language.

If you look at Exhibit 9, the nodes at time 3 they distinguish between the value of the bond (V = 100) and the value of the coupon (C = 5). Similarly, the nodes at time 2 and time 1 have the coupon (C = 5) separated out.

However, when they show the calculations for time 2 values , they discount the time 3 values and add the coupon. The problem is that word “values”: in their calculation, they _ do not mean _ the value of the bond at time 2, they mean the value of the portfolio comprising the bond and the time 2 coupon payment.

It’s simply sloppy language on their part.

Rest assured, the value of the bond at time 2 is simply the average of the future cash flows discounted back to time 2; the coupon at time 2 is not part of the value of the bond.

Take a look at the articles I wrote on binomial interest rate trees: http://financialexamhelp123.com/binomial-trees-for-fixed-income/. I did it correctly.

#14

Thank you S2000.

It is clearly written plain language and concept is not hard to understand at all. Also I have found Fixed Income session the most interesting part and not difficult at all.

What confused me is that Schweser uses different calculation and on example on page 169 (Schweser notes - Fixed Income) in V0 period is also added coupon while in CFA I cucciculum is stated that in period 0 coupons are never been added. Also, there must be an error in last period before a bond maturity as i stated above.

Thus, I have to apply Schweser “approach” to solve EOC in Schweser and CFAI approach in solving EOCs in curriculum.

Since the exam is organized by CFAI, not by Schweser, I will continue use CFAI template for solving this.

I hope, there are no many mismatches between Schweser and CFAI in entire curriculum because I just do not have time to take a due dilligence.

#15

You’re quite welcome.

#16

I re-checked this issue and now seems that is not different approach between CFAI curriculum and Schweser.

However, what is exactly confusing me and such is since starting this topic is seems there are one approach for Binomial IR calculating for coupon bond (page 289, Example 3, CFAI curriculum) and another approach for coupon bond with embedded options (page 331, Exhibit 12 and 14 on page 332, CFAI).

First approach (no options) – there is coupon added (5) on Time 2 with syntax:

0,5 x (105/1,08+105/1,08)+ 5. Coupon is added at the end of each node except one in Time 0 where the syntax is 0,5 X (103,22/1,02 + 106,95/1,02) = Value of bond, this no coupon added at the end in Time 0, and is stated there shouldn’t be added at Time 0.

Second approach (with embedded options –put/call). I want to mention that I clearly understand the concept with options and correction the bond value in certain nodes due to option execution.

I just want to know why is this syntax applied here (different than syntax above):

In Time 2 no coupon added at the end just is simply discounted CF from last node (Time 3) and this is Bond FV value + coupon. Here is is syntax

0,5 x (104,250/1,055258+104,250/1,055258). Thus, no added coupon at the end.

Furthermore, here is added coupon in Time 0 thus

0,5 x (99,658+4,25/1,025+100+4,25/1,025)=101,54.

So, any help would be appreciated because it’s driving me nuts.

#17

When the bond has an embedded call option or put option, and the decision rule is to exercise the option whenever the strike price is better than the present value of the future cash flows (i.e., the call price is lower than PV(FCF) or put price is higher than PV(FCF)), then the first thing you do at a node is compute PV(FCF), without the coupon. You compare that to the strike price to see whether the option will be exercised or not, substituting the strike price if the option is exercised, then, finally, add the coupon.

The coupon gets added either way; the only difference between an option-free bond and a bond with an option is that the latter has to go through the decision rule first, then the coupon is added.

Once again, I urge you to read the articles I wrote; you’ll see that the treatments are consistent.

#18

Thank you. So, the key here is to differ Binomial tree for bond with no option with Binomial tree with option.

As always, S2000 responses quickly and with clear explanations. I took a look yesterday on your page and will do it later again but this time carefuly with additional attention.

If I pass this level by first attempt, the merit of this forum will be invaluable.