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# Have I valued these Forward Rate Agreements correctly?

#### blackbird926

##### New Member
Q1. Suppose a firm has purchased a 3s6s Forward Rate Agreement at a rate of 4% on a notional principal of €10m. Suppose at t=3, the 3 month spot rate is 4.7%. Calculate the payment made under the FRA.

I worked out as follows:

10m * ((.047/4 - .04/4)/1+.047/4) = 17296.76303; The payment made under the FRA at t=3 is €17296.76.

Q2. From Q1, suppose at t=1 the spot 2 month rate is 4.2% while the 2f5 forward rate is 4.35%. Calculate the value of the FRA at t=1.

I worked out as follows: 10m(.04-.0435/1.0435) = -33540.97

Discount back to period t=1.

-33540.97/(1+.042/6) = -33540.97/1.007

= -33307.81529

The value of the FRA at t=1 is €-33307.82.

For the second question in particular, I am unsure as to whether I have worked this out correctly. I would appreciate any assistance. Also, it's a pleasure to join the site as a new member and I look forward to contributing to discussions.

Best regards and many thanks,

Michael

#### Aleksander Hansen

##### Well-Known Member
Q1. Suppose a firm has purchased a 3s6s Forward Rate Agreement at a rate of 4% on a notional principal of €10m. Suppose at t=3, the 3 month spot rate is 4.7%. Calculate the payment made under the FRA.

I worked out as follows:

10m * ((.047/4 - .04/4)/1+.047/4) = 17296.76303; The payment made under the FRA at t=3 is €17296.76.

Michael

€10MM x (0.04 - 0.047) x 0.25 = (€17,500)
(€17,500)/[1 + (0.047 x 0.25)] = (€17,297)
?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
I agree with ahansen w.r.t. payment at T3 = +/- 17,500

For the valuation at 1 month, I'd calculate (in the same way) the t3 payment = (4.35% - 4%)*0.25*10 = +/- EUR 8,750. Then, careful not to discount 3 months but two months to PV(T1) = 8,750/(1+4.2%*2/12) = 8,689; i.e., 2/12 of 4.2% spot
... it would also be okay to discount continuously (unlike the FRA rate which is necessarily quarterly): PV(T1) = [(4.35% - 4%)*0.25*10]*exp(-4.2%*2/12) ~ = same 8,689.

Welcome blackbird!

#### brian.field

##### Well-Known Member
Subscriber
I am curious as to whether this is a legitimate question or more of a philosophical question. It looks like an FRA is discounted from T2 to T1 at LIBOR. I am wondering why this is the case. If one argued that LIBOR is the appropriate discount since it a market-rate, couldn't one also argue that the fixed rate is an appropriate rate at which to discount the FRA? (Using similar market-rate based arguments?)

#### brian.field

##### Well-Known Member
Subscriber
Also, since the FRA has a value of 0 at inception, it seems like one could argue in favor of either the fixed rate or LIBOR as the appropriate rate at which to discount!

#### brian.field

##### Well-Known Member
Subscriber
Actually, it looks like the discount rate is a risk-free rate not necessarily equal to a LIBOR of the correct period length.

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi Brian,

In many countries such as Thailand, the Treasury yield curve is illiquid across maturities. On the other hand, the LIBOR/Swap Yield curve is liquid across maturities worldwide. The LIBOR swap zero curve is obtained by bootstrapping and is almost risk-free. That is why it is used to discount the Cash flows to obtain the value of the FRA. The Fixed rate cannot be used for discounting because FRA's are OTC - the fixed rate is unique to that particular FRA.

Thanks!
Jayanthi

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Thanks Nicole - appreciate it

Jayanthi

#### brian.field

##### Well-Known Member
Subscriber
Thanks for your message Jayanthi. While I agree with what you are saying, generally, it isn't really addressing my philosophical question.

Consider today at time t=0 and an FRA that requires a fixed payment of 5% and a floating payment of 1YLIBOR from t=3 to t=4.

At t=0, we do not know that actual 1YLIBOR rate that will apply in 3 years but we do know the 1Y forward 1YLIBOR rate payable in 3 years. We establish the FRA at t=0, and at this time, the fixed rate is set so that the FRA's value is 0 at t=0. By the time we get to t=3, the FRA may not equal 0.

My point is that at t=3, we will know that value of the FRA at t=4 and it is customary to discount the t=4 value back to t=3 with a risk-free rate. I thought that I read that this discount rate was the appropriate tenor LIBOR.

My question is as follows:

If one can argue that LIBOR is the appropriate discount rate, why couldn’t one argue that the fixed rate is ALSO an appropriate discount rate since the FRA is designed so that its value is the same under both rate schemes initially. This is the philosophical part. I think the answer lies in the fact that when the FRA is executed, the Fixed Rate is set to a level based on the FORWARD LIBOR rate AND the Forward LIBOR rate may not turn out to be the ACTUAL LIBOR rate.

To illustrate in another way, let’s assume that at t=0, we create an FRA that is to be in place from t=0 to t=1. This is essentially an immediate FRA. Then, we know that the ACTUAL 1Y LIBOR (and it will equal the 1 year forward 1YLIBOR commencing at t=0 by definition).

Then, in order for the FRA to have a value of 0 at inception, the Fixed Rate would have to equal the LIBOR rate, and in kind, we could say that the discount rate is the FIXED Rate OR the LIBOR rate!

Make sense?

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi Brian,

Your argument is analogous to the coupon rate = YTM on a fixed rate bond at the time of issuance. However, with time, while the coupon rate is fixed, the YTM being the market rate will change. Hence, we use the YTM for discounting. I am going to let David answer this!

Thanks!
Jayanthi

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