Valuing a Forward Rate Agreement with Interpolated Libor and Discount Factors

# Bond Interest Rate Swap Growth Rate


# Bond Interest Rate Swap Growth Rate












this should not be here because it shouldn't be here forever and eve

## Answer by Phil H (score 2)

https://quant.stackexchange.com/a/14983

## The FRA

A FRA is an agreement to exchange cash flows; the FRA in question is:

```
Start 15/9/14
End   15/5/15
```

which is 242 days. USD Money Market quoting is Actual/360, so the accrual factor here is 242/360 = 0.6722.

The FRA cashflows, therefore, are: on 15/9/14, Fix pays $\$1m * (0.6722 * 0.05) = \$33,611.11$, and Float pays $\$1m * (0.6722 * L)$, where L is the 9m Libor which fixed 2 days before, on 13/9/14. As it goes, this is the table of Libors from ICE (the new Libor administrators) for USD on 13/9/14:

```
Overnight 0.09030
1 Week    0.12050
1 Month   0.15500
2 Month   0.19550
3 Month   0.23360
6 Month   0.33090
1 Year    0.55530
```

Now, you may note there is no 8m. So you would have to look into the contract to determine how the fixing should be calculated in this circumstance. Suppose we do a common thing and interpolate between 6m and 1y. 6m is 15/9/14 to 16/3/15 (182d), and 1y is 17/9/15 (367d). So our value for 242 days linearly interpolated is 0.40368%.

This gives a Float cash flow of $\$1m * 0.6722 * 0.0040368 = \$2,713.63$.

The value of the FRA, on 13/9/14, paying fixed ('borrow ... 5%') and receiving float, is $\$-30,897.48$. What you haven't said is when you are valuing the FRA, but if we guess you mean to value it 1m before the fixing on 15/9/14, then the value would have been roughly -$31k depending on the market value of an 8m FRA at that time.

## Some misunderstandings

I suspect you have misunderstood the concept of a FRA; FRAs are not for borrowing, they are for hedging or creating exposure to a fixing risk. They are a Par instrument; that is, you don't enter a FRA and immediately pay or receive money - you enter the FRA believing the Fix rate to be at or near the fair expected rate on the fixing for that period. Back in textbook days you would use them to hedge against rates movement, and borrow cash when the day came.

To that end, the FRA also pays out the difference between the fixing and the FRA rate at the start of the FRA period, not at the end. So not only are you not borrowing money, the cashflows have all finished on the start date of the FRA.

Finally, FRAs fix on a fixing (Libor for USD), and that fixing no longer represents the true cost of funding. Libor is a mutually agreed number supposed to represent the rate at which a bank could borrow cash, unsecured, in the interbank market. But the market no longer permits significant unsecured borrowing, and the cost of funding varies.

Thus your 5% FRA, which has been out of the money since it was traded, will have been secured, probably in cash, accrued at the FedFund overnight rate, and will therefore have represented a loan TO the counterparty followed by a 'payment' which involved letting them keep the balance of some \$31k. If you had 'sold' the FRA to the counterparty for a similar amount, they would have paid you \$30k or so, and then immediately demanded that amount was put into your margin account with them as collateral.

## But the textbook...

The pre-crisis textbook theory would be that if Libor represents the cost of funding, and you use the ZCBs at 1m, 3m and 6m and bond at 1y to bootstrap a discount factor curve using the 6m ZCB to discount the initial payment from the 1y bond, you could do $f_{9m}/f_{1m}$ and convert that factor into a rate. The difference between that rate and the FRA rate would determine the value of the position.

## Really hedge it?

Remember the 8m rate was interpolated from 6m and 1y Libors. So the best way to hedge it (and to determine the value) would have been a combination of a 6m FRA and a 1y FRA, and a cash deposit. Good luck finding 12m Libor instruments to bootstrap a 12m curve from, but you will find 6m FRAs and IRS basis for the 6m curve.

Where are the bonds? Well, you can do something with a spread over the Treasuries to create a 3m Libor curve, and then apply some basis values to get to 6m, but you'll also need a funding curve.

But we're not in Kansas any more, Dorothy.

## Ye Olde Bootestrappings

If you want the textbook answer, here:

Assume 15/9/14 Spot. Thus

```
Date   Factor
Spot   1.0
```

ZCBs give

```
1m     0.996672
3m     0.989555
```

1y Quarterlies: Payment schedules:

```
      4%    16%
3m     10    40
6m     10    40
9m     10    40
1y   1010  1040
```

So we know that

$$10(a + b + c + d) + 1000d = 993.938, \;\;\text{(4%)} \\ 40(a + b + c + d) + 1000d = 1110.628 \;\;\text{(16%)}$$

Solve for $d$ first:

$$4.(993.938 - 1000d) = (1110.628 - 1000d) \\ 3000d = 4 \times 993.938 - 1110.628 \\ d = 0.9550413$$

1y 10% Semi at 1.051706, contains cashflows:

```
6m       50
1y     1050
```

So

$50b + 1050d = 1051.706$

Sub in $d=0.9550413$:

$b = 0.9782527$

Remember the 4% gave us:

$10(a + b + c + d) + 1000d = 993.938$

Sub in $a=0.989555$, $b = 0.9782527$, $d = 0.9550413$:

$c = 0.966821$

At last, a dfc:

```
Spot 1.0
1m   0.996672
3m   0.989555
6m   0.9782527
9m   0.966821
1y   0.9550413
```

Now, we want 1m to 9m, so the factor for the period is

$f_{1,9} = f_9/f_1 = 0.966821/0.996672 = 0.9700493$

Converting to an Actual/360 rate:

$$r = (f^{-1} - 1) * 365/242 \\ r = 4.657\%$$

Something like that, anyway.