 # HELP on currency swap

Can anyone explain to me how the answer to this question was calculated (or why it is calculated this way)? I cannot seem to see the logic behind it.

90 days ago the exchange rate for the Canadian dollar (C\$) was \$0.83 and the term structure was:

180 days

360 days

LIBOR

5.6%

6%

CDN

4.8%

5.4%.

A swap was initiated with payments of 5.3% fixed in C\$ and floating rate payments in USD on a notional principal of USD 1 million with semiannual payments.

90 days have passed, the exchange rate for C\$ is \$0.84 and the yield curve is:

90 days

270 days

LIBOR

5.2%

5.6%

CDN

4.8%

5.4%

What is the value of the swap to the floating-rate payer?

A) \$10,126. B) −\$2,708. C) \$3,472.

The present value of the USD floating-rate payment is: (1.028 / 1.013) = 1.014808 1.014808 × 1,000,000 = \$1,014,808

The present value of the fixed C\$ payments per 1 CDN is: (0.0265 / 1.012) + (1.0265 / 1.0405) = 1.012731 and for the whole swap amount, in USD is 1.012731 × 0.84 × (1,000,000 / 0.83) = \$1,024,932

−1,014,808 + 1,024,932 = \$10,126

I find it easier to solve currency swaps using actual number amounts.

So:

1. Price the Canadian swap by working out the discount factors and then using the 1- Z2 / (Z1 + Z2) formula (if you need a longer explanation on this, please let me know). I get 5.3%.

2. Work out the Canadian dollar equivalents. Principal: US\$1,000,000/0.83 = C\$1,204,819. Therefore interest payment = C\$31,928 (Canadian principal * swap rate of 5.3%, divided by 2 because the payments are semi annual.) Principal at maturity: C\$1,204,819 + C\$31,928 = C\$1,236,747

3. Discount these payments at the new interest rates after 90 days.

4. Convert your new total back into USD at the new exchange rate.

5. Value the floating-rate swap as normal.

Then compare the two values.

Does that make any sense?

This question is hard! OK, there are two sides you need to look at: The perspective of the floating rate payer, and the perspective of the fixed rate payer. Your objective is to find the PV of the payments made by each side.

The floating rate payer is borrowing \$1,000,000 USD, and paying USD floating rate interest (LIBOR). Note that since they’re borrowing and paying in USD, you will be discounting these cash flows by the LIBOR rates.

Since this is basically a floating rate note, we know that the value resets to par on each payment date. The next payment date is in 90 days, so the value of this note will reset to \$1,000,000 USD in 90 days, plus the interest payment that will be made.

The interest payment is based on the 6 month LIBOR at initiation, 5.6%. So (0.056/2)*1,000,000= 28,000. Thus, the value of this “note” is \$1,028,000 at time = 90 days from today. Discount this back to today at 90 day LIBOR (1,028,000/1.013)= \$1,014,808. This is the PV of what the floating rate payer will pay.

The fixed rate payer is borrowing \$1,204,819 CAD (you gotta do the conversion from \$1,000,000 USD), and paying \$31,928 CAD interest every 6-months (the question tells you the rate is 5.3% fixed, semi-annual payments. So just divide the fixed rate by 2, and multiply by \$1,204,819 CAD principal).

All you gotta do is discount the cash flows. The cash flows are: in 90 days, \$31,928 CAD interest is paid. In 270 days, \$31928 CAD interest is paid PLUS \$1,204,819 principal. Note that you’re using the CAD term structure to discount these, because they are CAD cash flows.

( \$31,928 CAD / 1.012 ) + (\$1,204,819 CAD + \$31,928 CAD)/1.0405 = CAD \$1,220,158 Of course, they want the answer in USD, so you convert it to USD using the current spot rate. 1,220,158*0.84= USD \$1,024,933. This is the PV of what the fixed rate payer will pay.

So the floating rate payer is receiving USD \$1,024,933, and paying USD \$1,014,808. This nets to +\$10,125, which is answer A.

Oh my goodness, thank you so much! It makes perfect sense now.

OR Much simpler

1. C\$1.201 notional @ initiation - do the rates thingie - convert it back to the current USD@90days later using .84/\$ rate. Make a note to yourself to add notional of 1.201 rather than 1 when you do the last discount rate ==> 0.99999

2. USD 1\$ notional - float cpn @ initiation 0.028 - use the 270 day LIBOR rate using 0.052*270/360+1 discounted ==> 0.98941

0.010124 X 1mill

Correct me if I am wrong. The step above is to calculate the fixed rate of a swap.

Fixed rate is already given. Floating rate is the inital 90-days LIBOR; also given.

Just some advice from a L3’er… you’ll see all this again in L3, so try not to forget this stuff, and you may as well learn it right now.

wow, thank you wayne for such a thorough explanation.