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xingwei86524
Topic Author
Posts: 19
Joined: April 11th, 2018, 8:06 pm

### libor in arrears

Hello,

For an exercise, I am asked to compare the fixed leg payment "K" in two FRA products.

1).  The first FRA is a regular product, where the floating leg (i.e. 3 month libor) is set in one years time, and paid at time 1 year + 3 months.  Fixed Leg "K" is set today.

2).  The second FRA is of libor in arrears form, where the floating leg is both set and paid at one years time.  Fixed Leg "K" is set today.

I am asked how does the fixed leg "K" compare in these two FRA products.

I think that the fixed payment "K" in the libor in arrears FRA should be higher than the regular FRA product.  The reason is because the libor in arrears payment has higher value today than the regular libor payment.  Thus, we need a higher K in order for the two legs to net out to zero.

Am I correct?

Many Thanks

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: libor in arrears

you've left out a whole bunch of necessary details about exactly what is being paid and when,
but the interest rate on the floating leg would be the same in both cases hence if the contracts are what they seem to be the fixed rate should be the same in both cases

>> 1).  The first FRA is a regular product, where the floating leg (i.e. 3 month libor)
>> is set in one years time, and paid at time 1 year + 3 months.  Fixed Leg "K" is set today.

so the floating leg is the 3 months from 1 year to 1 year + 3 months?
and the fixed leg is also the 3 months from 1 year to 1 year + 3 months? If not, what exactly is it?

>> 2).  The second FRA is of libor in arrears form, where the floating leg is both set and paid at one years time.
>> Fixed Leg "K" is set today.

so the floating leg is the 3 months from 1 year less 3 months to 1 year ?
and the fixed leg is also the 3 months from 1 year less 3 months to 1 year ? If not, what exactly is it?

xingwei86524
Topic Author
Posts: 19
Joined: April 11th, 2018, 8:06 pm

### Re: libor in arrears

Sorry.  In both cases, the floating rate is the three month libor rate from 1 year to 1 year + 3 months, and it is observed in one year.

However, in the first case, payment happens at time 1 year + 3 months.  In the second case, payment happens at time 1 year (i.e. it both sets and pays at time =one year)

The fixed leg coupon "K" for both product is set today.  The question is, for which of these two products should we expect K to be higher?  Or should K be the same?

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: libor in arrears

I think it should be the same

xingwei86524
Topic Author
Posts: 19
Joined: April 11th, 2018, 8:06 pm

### Re: libor in arrears

Why?

Let's consider today's value (t=0) of the floating leg payment in both FRAs.  Because a libor in arrears payment happens at time one year, which is earlier than a regular libor payment at time one year plus three months.  Its value today must be higher than that of the regular libor payment.  Knowing this, in order for today's value of both contracts to be zero, I would think the fixed coupon payment "K" for the in arrears FRA has to be higher.  Agree?

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: libor in arrears

do both fixed payments take place at the same time or is one at 1 year and the other at 1 year + 3 months?
that was one of the details

xingwei86524
Topic Author
Posts: 19
Joined: April 11th, 2018, 8:06 pm

### Re: libor in arrears

sorry.

for the first contract (i.e. regular FRA), fixed and floating payment takes place at one year plus three months.

for the second contract (i.e. in arrears), fixed and floating payment takes place at one year.

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: libor in arrears

then your argument that a floating payment which takes place earlier must be worth would also hold for the fixed payment

xingwei86524
Topic Author
Posts: 19
Joined: April 11th, 2018, 8:06 pm

### Re: libor in arrears

so are you saying that yes K2>K1?

K2: fixed coupon of second product
K1: fixed coupon of first product

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: libor in arrears

no, I'm saying K2=K1

xingwei86524
Topic Author
Posts: 19
Joined: April 11th, 2018, 8:06 pm

### Re: libor in arrears

but I don't understand why.

Let the t=0 value of the libor in arrears payment at time one year be lia(0).
And let the t=0 value of the libor payment at time one year plus three month be li(0).

We know that lia(0) is alway greater than li(0)  ( i.e. lia(0) > li(0 )

Then in order for both contracts to have zero value to at time =0, how can K2=K1?

ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Re: libor in arrears

the same argument could be made about the fixed leg being more
Let's suppose you but both of these at the same time
1) floating set in one year, paid in 1 year + 3 months
Fixed leg K1 set now

2) in arrears, floating set and paid in 1 year
Fixed leg K2 set now

If the Libor rate at the end of the year is L,
1. will pay (L-K1)/4 after 1 year + 3 months
2. will pay (L-k2)/4 after 1 year

K1 is chosen so that you expect (L-K1)/4 to be zero
K2 is chosen so that you expect (L-K2)/4 to be zero

So K1=K2

xingwei86524
Topic Author
Posts: 19
Joined: April 11th, 2018, 8:06 pm

### Re: libor in arrears

but you would not be taking the expectation under same measure?

1) under the 1 year + 3 month measure
2) under the 1 year measure.

bearish
Posts: 4518
Joined: February 3rd, 2011, 2:19 pm

### Re: libor in arrears

This was a standard quant interview question in the 90’s... The point of the question is that Libor is a convex function of the price of a traded instrument (a somewhat hypothetical zero coupon bond) so there will be a Jensen’s inequality effect making the present value of the immediate pay (set in arrears) cash flow greater. The standard, set in advance and paid in arrears, Libor cash flow is linearized by discounting from the pay date to the set date. Of course, non-Libor discounting will introduce some third order effects, but I would be rather surprised if they made a measurable impact in this particular case.

xingwei86524
Topic Author
Posts: 19
Joined: April 11th, 2018, 8:06 pm

### Re: libor in arrears

Hi Bearish,

Thanks for your response.

So what is your conclusion about K2 > K1?
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