Forward Rate Agreement Screencast

[mikey10011]

David,

In your screencast you calculated the value of a 3x9 FRA that locked in LIBOR@ F=3%. Three months later the prevailing 6-month spot rate was ST=4%. To calculate the amount to be settled in cash you discounted the net value H33 by 4% over 6 months

PV = 1/(1+ 4% x 0.5)

Jorian FRM Handbook (p. 187) presents a similar example except that he is calcuating the value of a 6x12 FRA that was locked in F=5%. Six months later the prevailing 6-month spot rate was ST=3%. To calcuate the amount to be settled in cash Jorian discounted the net value using a "4% value factor."

Question: Computationally what is the "4% value factor" in this case? And shouldn't Jorian be discounting the net value by ST=3% (and if so is this an errata by Jorian)?

---------------------------------

Also a side question, on calculating PV. Jorian FRM example 1.1 (pp. 5-6) asks to calculate the effective annual rate (EAR) of a bond that will be maturing in 1 month. He uses Equation (1.1)

PV = 1/(1 + EAR)^T

We are given PV=0.987 and he plugs in T=1/12 to solve for EAR. Now if we use EAR=4% for 6-month LIBOR per your screencast, why am I not using

PV = 1/(1+4%)^(6/12) ?

 

[mikey10011]

David,

I have been working through Hull and have more questions. (I am using an ancient edition so hopefully he has retained the same example and notation.) Specifically, he said that the forward rate agreement is an agreement to the following cash flows:

Time T1: -100
Time T2: +100 exp[Rk (T*-T)]

where Rk is the locked-in forward rate.

Like your screencast he derives the cash settlement at time T from the cash flows above as

-100 + 100 exp[Rk (T*-T) exp[-R (T*-T)] = -100 + 100 exp[(Rk-R) (T*-T)]

where R is the prevailing spot rate at time T.

Question #1a: Is he setting up the cash flows for someone writing or buying the FRA? I ask that because the term in the exponent Rk-R is opposite to what you had in the screencast (and in Jorian's FRM Handbook). Note that Hull also contradicts Jorian: "A long FRA position benefits from an increase in rates" (p. 188).

Question #2: Could you explain how Hull's setup matches Jorian's FRM Handbook jargon of a "short FRA position is similar to a long position in a bond" and Example 8.1 on borrowing to finance an investment? To make it concrete could you also say the same thing except using the numbers in your screencast?

Question #3: Could you number crunch in a spreadsheet your screencast numbers using Hull's equation above? For the FRM exam do you recommend using Harper/Jorian's annual compounding or Hull's continuous compounding?

Sorry for putting you through Hull's FRA but my next step is to reread Hull's interest rate swap valuation using FRAs.

 

[David Harper CFA FRM]

Hi Mikey,

Here is a link to a worksheet with three tabs (to see if we can resolve this). All three are same construct I used in the screencast, which was simply taken from Hull's approach.
1. FRA per Hull 4.3. This gives Hull's answer of 1,236 (I have the sixth and 7th editions of Hull, but not earlier sorry)
2. My screencast
3. Jorion's page 187, using this method, which I will continue to call Hull's method.

First observation:
You are correct that Jorion has an error (nicely done, no one's noticed this one so far, at least not on our site). There is no reason to discount at 4%. "shouldn’t Jorian be discounting the net value by ST=3% (and if so is this an errata by Jorian)?" Absolutely, yes!

Question #1 and #2. I think this is merely semantic. I checked my (highly reliable) CFA source on this and this is true: "the long position in FRA is long the rate;" i.e., will benefit if rates increase. But, please note, this is consistent with being short the bond: long FRM = long rate = short the bond (i.e., lower rate implies higher bond price, higher rate implies lower bond price).

Question #3. Sorry i don't have the prior Hull version, but notice, implied by the XLS, that *currently* Hull and Joroin are atypically somewhat "in agreement" here. Typically Hull is always continuous, but in the FRA he discounts (from payoff to settlement) at a DISCRETE FREQUENCY. I say somewhat b/c it appears to me Jorion is using annual while Hull is better by using a compound frequency "reflecting their maturity" As he says, if T2 - T1 is .5 years, then he uses semiannual; if 0.25, then quarterly.

(In my version, Hull does employ continuous compounding I notice. But this is to compute the value of the FRA itself; i.e., to PV the FRA contract to today. Please note this is different! Above we are PV'ing the payoff from settlement to today. In other words, for example, in an FRA 6 x 12, Hull uses discrete to calculate the payoff at 6--to PV from 12 to 6--but uses continuous to get the PV at time 0--to PV from 6 to 0.)

In regard to which (Jorian's apparent annual vs. Hull's specifically discrete), I like Hull's because (i) it is the assigned and (ii) it's a little more precise so you can't fault it while you could fault Jorion's for "rounding in the compounding"

Hope that is helpful

David