Why do these 2 ways of calculating a bond value give different results?

CFA Level I Fixed Income
#1

Hello everyone,

Let’s say I have a 3-year, 5% bond with par value = 100, discount rate is 4%. Now I want to calculate the bond value right at the middle of the first year (full price) and I have done this by these 2 ways using the financial calculator (Professional BA II Plus):

* 1st method:

N = 2, PMT = 5, I/Y = 4, FV = 100 => PV at the end of year 1 = -101.8861

Bond value right at the middle of the first year = (101.8861 + 5)/(1+4%)1/2 = 104.8104

* 2nd method:

N = 2.5, PMT = 5, I/Y= 4, FV = 100 => Bond value right at the middle of the first year ( - PV ) = 102.3350 (this is approximately equal to the flat price)

I believe the answer of the 1st method is correct but why does the 2nd method give the different answer? What does the financial calculator (BA II plus) understand by those inputs?

Please help me clarify this :(. Thank you.

#2

It’s because your cash flows are messed up in the first example. The bond pays the principal in 3 years not 2. Since the cash flow is coming earlier in the first example it has more value (time value of money) which is causing the bond to be priced higher.

The second example looks right to me.

#3

I didn’t really get what you meant … But let’s do the normal way of calculating the full price, it’s like this:

N = 3, PMT = 5, I/Y = 4, FV = 100 => PV of the bond at the beginning of the first year = -102.7751

Full price = 102.7751 x (1+4%)1/2 = 104.8104 which is same as the answer given by the first method.

So the first example should be correct, shouldn’t it?

#4

Why are you multiplying by 1.040.5?

In the first example you have the cash flow of the principal coming in on year two when it should be coming in on year 3

Here is the break down

Year 1 = $5/1.04

Year 2 = $5/1.042

Year 3 = $105/1.043

PV = Sum of the above cash flows = 102.78

The cash flow of $100 is being discounted at 1.043 which is correct.

in your first example you are receiving the cash flow in year 2

Year 1 = $5/1.04

Year 2 = $105/1.042

Then you’re adding $5 and discounting at 1.040.5. I’m not sure why you’re doing this?? it doesn’t make sense to me

You can’t sell your bond and then expect to receive a coupon afterwards

#5

Let me break down the first example as follows:

Year 1/2

Year 1: 5

Year 2: 5

Year 3: 105

First, I discounted cash flows of year 2 and 3 to bring them back to year 1, which is why N=2, PMT = 5, I/Y= 4, FV = 100, then PV of these cash flows are 101.8861. Because there’s another $5 on year 1, then I added 5 to 101.8861, which is 106.8861. Then I brought this sum back to year 1/2 by dividing it by 1.040.5.

I think this way is the same as discounting all cash flows to bring them back to year 0 and then multiplying them by 1.040.5.

#6

I get what you’re doing.

You’re breakdown doesn’t line up with what you’re doing though. In your breakdown you say

Year 2 = 5

but what you’re doing is

Year 1 = 5/1.04

Year 2 = 105/1.042

PV = 101.89

Year 3 = 5/1.040.5… then you’re adding this cash flow and i’m saying that’s where the problem is

Calculate the cash flows out and look at the timing of the flows.

To answer your question of why the two answers are different it’s because the timing of your cash flows are different.

#7

Which example do you think is correct?

Here’s attached the illustration of example 1.

https://drive.google.com/file/d/19qbubfF3Hjp45YxzYjGDOSveFAH6UQOD/view?usp=sharing

#8

Referring to your 2nd method. What you set as N affects your PMT directly.

When you set N = 2.5 and PMT = 5, the calculator will assume there is 2.5 × 5 = 12.50 in total coupon instead od 15.

You can check this by setting I/Y = 0 and CPT PV to see the total cash flow is 112.50.

That’s why you cant get the answer by using method 2.

#9

There’s a BOND worksheet on your BA II that would help your between coupon situation. :bulb: I think TVM expects an integer for N, which might be causing your difficulties.

Also, the CFA is very heavy on what’s called prospective valuation: you discount all future cash flows to your valuation date to come up with a PV. There is also what is called a retrospective valuation: you accumulate your PV and deduct the accumulated value of cash inflows to get to the subsequent PV. Both the prospective and retrospective methods will give you the same value.

#10

He/she wants to value the bond 6 months before expiration so putting in 2.5 is ok. The problem is that the timing of flows is off because he is counting an inflow of 100 at year 2 when the cash inflow should be at year 3 and discounted from year 3 back that is what is causing a major difference in price

#11

Oh I’ve got it, thank you, but could you break down the timing of cash flows in this case as understood by the calculator (when input N=2.5)?

#12

Thank you for telling me that function of TVM.

#13

You can refer to my illustration of example 1. The inflow of 100 is on year 3.