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# Hull, Chapter 7 , Swaps

#### Unusualskill

##### Member
Hi,

I have question for Hull, Chapter 7, Swaps. I am quite confused about calculating the value of the floating rate bond.

Referring to this example:

Consider a $1 million notional swap that pays a floating rate based on 6-month LIBOR and receives a 6% fixed rate semiannually. The swap has a remaining life o f 15 months with pay dates at 3, 9, and 13 months. Spot LIBOR rates are as follows: 3 months at 5.4%; 9 months at 5.6%; and 15 months at 5.8%. The LIBOR at the last payment date was 5.0%. Calculate the value of the swap to the fixed-rate receiver using the bond methodology. (i)What is the difference between Spot LIBOR rates above and the LIBOR at the last payment date? (ii)Bfloating =[$1,000,000 + ($l,000,0 0 0 x 0.05/2)] x e-(0.054 x 0.25) =$ 1,011,255

Why only use 5%?

Any good explanations on these two topics?

(i)Calculate the value of a plain vanilla interest rate swap based on two simultaneous bond positions.
(ii)Calculate the value of a plain vanilla interest rate swap from a sequence of forward rate agreements (FRAs).

Thank you!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Unusualskill I input those assumptions into our swap pricer (XLS sheet is here https://www.dropbox.com/s/78dlwb9rsekop50/1028-swap-value.xlsx?dl=0). Hopefully this illustration below is helpful. Continuous discounting is assumed, although I don't think the question specifies (ideally it would say something like "Spot LIBOR rates with continuous compounding are as ...").
• Re (i) The 5.0% LIBOR is the (semi-annual) six-month LIBOR rate that prevailed three months ago; the rate is observed (determined) at the beginning of each six-month period but paid at the end. We don't need the 5.0% to discount, but we need it to know what the floater is paying in three months
• Re (ii) the floating rate note has a price equal to its par value on each coupon date (if the same rate that determines coupons also informs the discount rate, which is the case here: LIBOR). So for the floater-as-a-bond, we only need to present value the next coupon plus a par bond. There are many many discussion on this forum about this....
I hope that helps! Thanks,

#### Unusualskill

##### Member
Hi @David Harper CFA FRM ,

Thank you for the illustration. Now I understand about the floating rate note of the question.
But another question that I have doubt is:

For flxed leg of the swap, when we are using 5.4% , 5.6%and 5.8% to discount the fixed coupons, thus that means that:

From t=0 to 0.25, if we hold the bond over this period, we would earn 5.4%(annualized) return.
From t=0 to 0.75, if we hold the bond over this period, we would earn 5.6%(annualized) return.
From t=0 to 1.25, if we hold the bond over this period, we would earn 5.8%(annualized) return.

Then it is different from meaning of the LIBOR for floating leg.
From t=-0.25 to 0.25, we would earn 5%(annualized) return.

if the same meaning applies to 3 months at 5.4%, it would mean from t=0.25 to 0.5, over this period, we would earn 5.4%(annualized), which is not same idea as the way we discount here.

So my question is : what is meaning of the Spot LIBOR rates in the question: 3 months at 5.4%, 9 months at 5.6% and 15 months at 5.8%. Thank you.

#### Unusualskill

##### Member
Sorry, another clarification: Using FRA methodology for example 7.3:

Is R_forward=R_2+(R_2-R_1)(T_1/(T_2-T_1)=5.7% annualized forward rate?
Can I use this formula instead: [e^(0.056^0.75)]/[e^(0.054^0.25) ]-1= 2.84% to calculate forward rate and then multiplied with 1000000? The answer is slightly different.

Thank you.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Unusualskill

Sure. The 5.4%, 5.6% and 5.8% are current spot rates (aka, zero rates. Here is Tuckman's definition because spot rates are the fundamental building block of pricing: "A spot rate is the rate on a spot loan, an agreement in which a lender gives money to the borrower at the time of the agreement to be repaid at some single, specified time in the future."). In this context, the represent risk-free spot rates on the spot rate curve (Although Hull has argued a shift away from LIBOR and toward Overnight Indexed Swaps, OIS, for the risk-free rate). In this role, they are discounting future cash flows to present values (aka, time value).

In your example, LIBOR is also the index for the floating rate; the floating rate could pay based on any of several indexes and further could pay a margin on top of the index, but this counterparty is just paying LIBOR + zero. The floating rate payer is paying the six-month LIBOR at the beginning of each period, such that this six month LIBOR (0.5) is also part of the same term structure of zero rates. If we imagine going back in time six months, then the term structure at that time was:
• Three month (0.25 year) LIBOR = ?? (we don't know); six-month LIBOR = 5.0% per the given "The LIBOR at the last payment date was 5.0%"; nine-month LIBOR = ??? (we don't know)
• 3 months at = 5.4%; six-month LIBOR = ?? (we don't know); 9 months LIBOR = 5.6%; and 15 month LIBOR = 5.8%. These are just points on a curve, which may be different at any given point in time. Although notice how this curve appears to have shifted up since three months ago, because the current six month LIBOR is probably between 5.4% and 5.6%.
Re: [e^(0.056^0.75)]/[e^(0.054^0.25) ]-1= 2.84%, I think it might (?) mistakenly mix compound frequencies. The forward rate, F(0.25, 0.75), is the continuous forward rate implied by the two continuous spot rates. It is inferred from the assumption that these must be equal: exp(5.6%*0.75) = exp(5.40%*0.25)*exp(F*0.5); i.e., investing continuously at 5.6% over 0.75 years should have the same expected return as investing for 0.25 years at 5.4% then "rolling over" into the forward rate for 0.5 years. Then the retrieved continuous forward rate is translated into its semi-annual equivalent, as the basis for determining the floating rate payment. (this is the one imprecision IMO in the question: the 5.0% at the last coupon is presumably a semi-annual rate while the others are continuous, or perhaps they are all semiannual. Not sure.). I hope that's helpful!

#### ankit4685

##### Member
Hi @David Harper CFA FRM ,
In reference of Hull Ex 7.3 , Can you help me to understand the formula we are using to calculate Forward Rates (s.a ) , Is there any generic formula for this, not sure how we derived this
FR ( s.a ) = =2*(EXP(3.40%/2)-1)

#### Matthew Graves

##### Active Member
Subscriber
This formula is just converting continuously compounded rates to semi-annual, forward or otherwise. The place to start is by making the growth factors (1/Discount Factor) equivalent.

E.g. $\left(1+\frac{r_2}2\right)^{2t}=e^{r_1t}$

where r2 is the annualised semi-annual rate and r1 is the continously compounded rate.

If you re-arrange to solve for r2 you end up with the formula above.
$r_2=2\left(e^\frac{r_1}2-1\right)$

#### elena77

##### New Member
Subscriber
Hi David, (I moved my initial post in the archive to here)

for 7.10
I assume my calculation below misses the loss at the year 3, while final conclusion is the same at circa $0.413 (numbers in$ mio)
Floating rate payment: (10+0.4)*exp^(0.078441*0.5) = 9.61558, where 0.078441 is derived from conversion of 8% s.a. to c.c. i.e. 2ln(1+8%/2) =7.8441%
Fixed rate payment: using calculator - N=4, I/Y = 7.8441/2, PMT= 0.4, FV =10, resulting in PV= 10.02835
thereby loss to financial institution is calculated by : $10.02835-$9.61558 = $0.41297 , nearly equal to$0.413

Would you please clarify why the calculation above come to the same result as the correct answer?

Thank you.

Subscriber
Hi there,

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @elena77 I don't know why your approach is correct to the solution, it may be coincidence (and please note I can't prioritize permutations on Hull's EOC ... ).

If you want to apply the "as if" two bonds approach, which you are attempting to do, then the correct approach is simpler than you are doing:

PV Floating-rate leg = $10,450,000 PV Fixed-rate leg =$10,862,990 = $500,000 +$10,362,990; where 10,362,900 is given by N = 4; I/Y = 4; PMT = 500,000; FV = 10,000,000 and CPT PV = 10,362,900

and the PV(swap) = 10,362,900 - 10,450,000 = 412,990.

Notice this differs from the usual approach because the first cash flow is not X months in the future, it is modeled to be immediately (at time zero) hence the fixed rate bond is the "usual" 2-year bond (with first coupon in six months) plus the immediate \$500,000.

Thanks,

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