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Can someone solve this? CFA’s explanation doesn’t make sense to me.
Hi Tom,
I found CFA material pretty difficult to grasp , in a sense that they used too much to symbols and the formula looked cluttered, so I tried to study without memorizing the formula rather than trying to study via logic, thats the only way to go through Derivatives.
For this problem remember US company is taking loan in AUD and returning back the amount at expiration. And its a Fixed for fixed currency swap.
The swap rate for each currency is given and that will be the amount exchanged at each swap date(Quarterly, so 4 payments).
First lets start with US Dollars, since we are told to find swap at 60th day from initiation,
The value at initiation for each subsequent (90,180,270,360) swap of US dollar is (0.027695*100,000,000)=2769500
Now since this was the value which was supposed to be exchanged at each quarter at the beginning but now we are sitting at 60th we need to discount is to 60th day and that is where the PVA(present value of annuity) comes into play, it discount the payment to the present (60th) day.
1st payment = 0.999584*2769500=2768374.888
2nd payment= 0.998668*2769500=2765811.026
3rd payment= 0.998253*2769500=2764661.6835
4th payment= 0.998336*2769500=2764891.552
At expiry (4th payment, the principal is also exchanged, so that needs to be discounted as well)= 0.998336*100,000,000)=99833600
Now add all the currency quartely swap and the dollar payment at expiry , which gives you the value of US$ at 60th day=110897339.1495
Similar do this for AUD$, but dont forget to convert the currency in US using the spot exchange rate at 60th day: 1.13.
Value of swap at 60th day= Value of swap of US - Value of swap of AUD (this will be ur answer)
I believe the answer is 3rd option, I am writing this from office hence dont remember exactly please verify after doing the calculation.
Hope this helps.
Sorry I forgot to mention the Currency payment for each US$ at every quarter is multiplied by the PVA (30, 120,210,300), because remember you will make these payment only at end of each quarter but u have to find the value at 60th day, so u have to discount using the PVA of 30, 120 etc…
Hope this makes it clear.
Hey Muzaffark2,
My calculations seem to differ from yours. Below is your calculation and I’ve added where my calculations differ in bold. Let me know if you agree or if my calculations are off.
The value at initiation for each subsequent (90,180,270,360) swap of US dollar is (0.027695*100,000,000)=2769500
For the US side, I thought we should be multiplying the USD fixed rate by 90/360 because these are quarterly payments. This would be (.02497 * (90/360)) = .006243 . Next, I multiplied by the USD notional of $87,719,298. This would be (.006243 * $87,719,298) = $607,346.4895.
Now since this was the value which was supposed to be exchanged at each quarter at the beginning but now we are sitting at 60th we need to discount is to 60th day and that is where the PVA(present value of annuity) comes into play, it discount the payment to the present (60th) day.
1st payment = 0.999584*2769500=2768374.888 (0.999584 * $607,346.4895) = $547,360
2nd payment= 0.998668*2769500=2765811.026 (0.998668 * $607,346.4895) = $546,858
3rd payment= 0.998253*2769500=2764661.6835 (0.998253 * $607,346.4895) = $546,631
4th payment= 0.998336*2769500=2764891.552 (0.998336 * $607,346.4895) = $546,677
At expiry (4th payment, the principal is also exchanged, so that needs to be discounted as well)= 0.998336*100,000,000)=99833600
Based on using the USD notional amount this would be (0.998336 * $87,719,298) = $87,573,333
Now add all the currency quartely swap and the dollar payment at expiry , which gives you the value of US$ at 60th day=110897339.1495
Based on my calculations above this would be = $89,760,859
I similarly did this for the AUD side, except I used the quarterly AUD rate of (.027695 * 90/360) and AUD notional of $A 100,000,000.
Hi tom,
You did the same mistake I did first while solving …i agree the coupon should be quarterly but remember the present value if annuity is already taking quarterly payment into consideration…to prove it to you try calculating the PVA for any quarter and you will see what u mean. Hence if you divide the coupon payment into quarters you will be double counting and dividing coupon twice, hence you are getting a low value.
Hope this answers your query
I am sorry I used USD INSTEAD of AUD…I USED ALL THE RATES OF AUD BUT USED NOTIONAL AMOUNT OF US…THAT PART YOU ARE CORRECT…PLEASE RECTIFY IT…REST THE LOGIC REMAIN SAME…
I’m confused as to why we wouldn’t multiply the fixed rates of .027695 (A) and .02497 (USD) by 90/360. These are annual fixed swap rates, no? If not, how do we tell the difference on the exam?
Also, I’ve tried your calculations but I’m still ending up with an incorrect answer.
By that I mean for the USD side I’ve multiplied .02497 * $87,719,298 * each discount factor (added principal to the last pmt),
2,189,440 2,187,433 2,186,524
89,760,039
For the $A side I’ve multiplied .027695 * $A100,000,000 * each discount factor (added principal to the last pmt).
2,764,892 2,752,072 2,740,722 101,333,913
Then I add them together, convert the USD to $A @ 1.13 and get a difference of $746,114.76.
.
All the Libor rates are always given in annual time duration (asumme annual in exam unless otherwise given), and as I explained you are calculating PV and discounting using PVA which already takes into account the quarterly payment.
I have done the calculation and its matching, I Think you have committed a mistake with the coupon rates of USD, it is 0.2497%, which comes to be 0.002497 (0.2497/100).
1st : 219035.087*0.999584=218943.96
2nd : 219035.087*0.998668 =218743.33
3rd: 219035.087* 0.998253 =218652.43
4th: 219035.087 * 0.998336= 218670.612
PV ( Notional): 87573333.08
Total: 88448343.412 USD.
Into AUD: Total *1.13 = 99946628.055
Value of 60th day swap: 109591599- 99946628.055 =9644970.945
This is the answer, bro be careful with the calculation a small decimal error makes life hell in CFA, I also had to spend a lot of time scratching my head to find the decimal error in your calculation.
I hope now its clear, please not put too much pressure with just one problem solve and move on…lots of things to cover
Thanks for your help man! You were right it was that decimal error and not realizing the PVA already takes into account the quarterly pmt.
No prob man…Live long and prosper…
How did you get this 219035.087 number???
IMHO use a $1 or 1 AUD notional solve with that … .then move to the final Notional.
(makes the numbers smaller) - (but remember to put your calculator into full decimal mode).
Example for a Plain Vanilla Interest Rate swap:
Let’s say that two parties enter into a 1-year swap with quarterly fixed payments priced at 6.052%. At initiation, the 90-day LIBOR was 5.5%. 30 days later, we observe the following chart on a $30 million dollar notional
60-day LIBOR – 6.0% 150-day LIBOR – 6.5% 240-day LIBOR – 7.0% 330-day LIBOR – 7.5%
What is the value of the swap?
Step 1: Calculate the fixed payment at each payment period 6.052% X (90/360) = 1.513%
It’s easier to calculate in terms of $1.00. After all, we’re trying to find the net amount Day 90 - $0.01513 Day 180 - $0.01513 Day 270 - $0.01513 Day 360 - $1.01513 Remember the principal is returned in the last payment
Step 2: Calculate the present values of the fixed rate payments Now that we’re 30 days in, we need to calculate the discount factors for 60, 150, 240, and 330 days using the new LIBORs In 60 days - Z1 = 1 / 1 + (0.06 * (60/360) = 0.99010 In 150 days - Z2 = 0.97363 In 240 days - Z3 = 0.95541 In 330 days - Z4 = 0.93567 These are usually given so we don’t have to go through the process
CF1 = $0.0153 * 0.99010 = $0.01498 CF2 = $0.0153 * 0.97363 = $0.01473 CF3 = $0.0153 * 0.95541 = $0.01446 CF4 = $1.0153 * 0.93567 = $0.94999
PV of Fixed-Payer = $0.99399
Step 2: Calculate the floating rate payment at each period. There are four payments for each of the 4 quarters. Because 30 days have passed, the payments are at 60, 150, 240, and 330. However, we only need to know 1 payment, and we already know it. How can that be? Unlike the fixed-payment, the floating-rate payment is reset to the market value every payment date, therefore the par value is always equal to 100. As for the payment, well that was identified in the question 5.5%. Remember the floating-rate payment is always set for the next payment date, therefore on day 90 (60 days away) the value of the payment will be 5.5%. Also, we add the value of the principal as well as discount back at the 60-day (first payment) discount factor.
0.055*(90/360) = $0.01375 This is the floating rate payment undiscounted. Since it is 60-days out, we need to discount it back + the $1.00 bond value immediately after the payment is made.
CF1 = $1.01375 * 0.99010 = $1.00371 = PV of the floating-rate payer
Remember that it doesn’t matter what the floating rate coupon payments are at the last three settlement dates because the floating rate bond will be worth $1.00 plus the coupon of $0.01375 at day 90
Step 3: Determine the value to the fixed-rate payment The present value of the floating-payments are greater than the present value of the fixed-payments, therefore, the fixed payer stands to gain
Swap value to the fixed payer = 1.00371 – 0.99399 = 0.00972 Swap value to the fixed payer = $30,000,000 * 0.00972 = $291,630
Present value of the fixed payments + principal at the last payment period compared to the present value of the next floating payment (usually given) + principal at the next payment period. Just remember how to calculate the discount factors (which are usually given anyway) and to discount based on where you stand in the contract, 60 days away, 30 days away etc. And remember to stop valuing the floating rate after the next payment date.
Currency Swap using the same methodology … using an example again
Let’s consider a fixed-for-floating currency swap. The term is 1-year, the notional 5 million. The exchange rate at the beginning of the contract was 0.50 L/. The dollar is floating and the L is fixed at 6.8%. Payments are by quarter. Let’s say that 300 days have passed. At the last settlement date (30 days ago), the 90-day LIBOR for the and L;, was 5.6% and 6.4% respectively. Let’s say that after 300 days (meaning 60 days left), the LIBOR 60-day interest rate is 5.4% and the LIBOR 60-day L interest rate is 6.6%, and the exchange rate is 0.52 L/. Calculate the value of the swap to the L fixed payer, who receives floating.
Okay wow, let’s start with what we know. Both the fixed and the floating have one payment left 60-days from now that is equal to the coupon payment + principal. The fixed pays the 6.8% which was set at the beginning of the contract plus the par value principal back. Because there are 60 days left in the contract, we use the 60-day L LIBOR to calculate our discount rate (although chances are they will provide this).
Fixed Payment L Discount Factor: = 1 / 1 + [0.066 X (60/360)] = 0.98912 Payment = [0.068 X (90/360)] + 1 (principal) = 1.017 Discounted payment = 1.017 X 0.98912 = 1.00594
Now for the floating payment, we take the 90-day LIBOR rate (remember the floating payer is the guy) that was set at the last settlement day (30 days ago). The discount factor is the 60-day LIBOR $ rate.
Floating Payment $ Discount Factor = 1 / 1 + [0.054 X (60/360)] = 0.99108 Payment = [0.056 X (90/360)] + 1 = 1.014 Discounted Payment = 1.014 X 0.99108 = 1.00496
Calculate the values of the floating payment Floating Payment = 1.00496 X 50,000,000 = $5,024,776
Remember in the case of currency swaps, the two parties exchange the currency at the beginning of the swap because the idea is to hedge the holding effect (FSA name-drop there). Therefore, when we are calculating the value to the fixed or the floating payer, we must take the entire payment including the principal into account when netting the amounts
Calculate the values of the fixed payment. Now this is a little different because the fixed payment is in L, we need to first translate the notional amount over at the beginning exchange rate which was $0.50, multiply it by the discounted payment, and then translate it back at $0.52. This is extremely important to remember in currency swaps. If we were doing a fixed-for-fixed currency swap, this would be all that matters. However, in a fixed-for-floating currency swap, gains/losses are subject to both (a) fluctuations in the floating rate (b) appreciation/depreciation in the exchange rates.
Initial principal in $ X Initial Exchange Rate = 5million X 0.5 L/ = L 2.5 million Initial principal in ₤ X discounted payment = L2.5 million X 1.00594 = L2,514,850 L value of discounted fixed payment X new exchange rate = L2,514,850 X 1 / 0.52 = $4,836,227
As we can see the dollar has appreciated from 0.5 L/ to 0.52 L/, so the L fixed payer gains (on that front) because they are paying the depreciated currency and receiving the appreciated. Let’s think about the gains conceptually. To paint a picture, let’s assume that the floating payer is in the UK, and the fixed payer is in the US. At the initiation of the contract, there is $5 million dollars exchanged. The U.S. guy holds $5 million dollars worth of pounds, and the U.K. holds $5 million dollars. Also for simplicity, let’s assume for the moment that they both are paying fixed rates, contract value at initiation = 0. Then, over the course of the contract, as the pound depreciates and the dollar appreciates, the U.S. guy paying a fixed rate on the depreciated pound and receiving a fixed rate on the appreciated dollar is gaining. In the case of a fixed-for-floating, the gains/losses of the interest rate hedge and the currency hedge cumulate to the total gains/losses. So the first thing you do with a currency swap is value the swap like a plain vanilla swap. The next step would be taking into account the currency fluctuations, as I did above.
Value to the fixed = 5,024,776 – 4,836,250 = $118,549
A question might be in the case of a fixed-for-floating currency swap, what are the gains attributable to currency fluctuations and what are the gains attributable to interest rate fluctuations? Also the number above won’t add completely due to the rounding.
I’m getting a totally different result. I tried to put my calculations into an excel sheet. I would appreciate if someone could check it out and tell me where I’m wrong. Thank you
the file: https://files.fm/u/tnyy6mt7
Really? No one is able to solve this?