End of year derivative valuations improve for borrowers

The increase in interest rates over 2013 means that the 31 December 2013 valuations of borrower derivatives such as interest rate swaps will look much healthier compared to a year ago. The global economy certainly appears to have turned a corner through 2013 and this is being reflected in financial markets expectations for future interest rates i.e. yield curves are higher. As interest rates collapsed after the onset of the GFC many borrowers took advantage of what were, at the time, historically low levels. Base interest rates i.e. ignoring credit, were compelling and borrowers increased their fixed rate hedging percentages locking in swap rates for terms out to ten years. Unfortunately, as the global economy sank further into recession, interest rates fell further than most market participants expected. Consequently, derivatives such as interest rate swaps moved further out-of-the-money creating large negative mark-to-market positions.

The unprecedented steps taken by central banks in an effort to shore up business and consumer confidence, protect/create jobs and jump start lack lustre economies pushed interest rates lower for much longer. Through 2013 the aggressive monetary policy easing undertaken since 2008 (by the US in particular) has started to show signs that the worst of the Great Recession is behind us. The Quantitative Easing experiment from the US Federal Reserve’s Chairman Ben Bernanke appears to be a success (only time will confirm this). The labour market has strengthened, as well as GDP, in 2013 allowing a gradual reduction in Quantitative Easing to begin. Although the US Central Bank has been at pains to point out that the scaling back of QE does not equate to monetary policy tightening, merely marginally “less loose”,           the financial markets were very quick to reverse the ultra low yields that had prevailed since 2008.   The US 10-year treasury yield is the benchmark that drives long end yields across every other country so when bond markets in the US started to aggressively sell bond positions, prices dropped and yields increased globally. As the charts below show all the major economies of the world now have a higher/steeper yield curve than they did a year ago reflecting expectations for the outlook for interest rates. For existing borrower derivative positions the negative mark-to-markets that have prevailed for so long are either much less out-of-the-money, or are moving into positive mark-to-market territory.

Of the seven currencies that are included in the charts below, all display increases in the mid to long end of the curve i.e. three years and beyond, to varying degrees. Japan continues to struggle having been in an economic stalemate for 15-years so the upward movement in interest rates has been muted. The other interesting point is the Australian yield curve which shows that yields at the short end are actually lower at the end of the year than they were at the start of the year. Australia managed to avoid recession after the GFC, a beneficiary of the massive stimulus undertaken by China and the ensuing demand for Australia’s hard commodities. However, as China’s economy subsequently slowed and commodity prices fell, the recession finally caught up with Australia and the Official Cash Rate (OCR) has been slashed in 2013, hence short-term rates are lower than where they started the year.

As 31 December 2013 Financial Statements are completed there will be many CFOs relieved to see the turning of the tide in regards to the revaluation of borrower derivatives.

2012 to 2013 yield curve movements

Understanding Central Bank Liquidity Swaps

This is part 7 of a 10 part series on currency swaps and interest rate swaps and their role in the global economy. In part 7, we illustrated how companies use swaps in the global market place, but on a company-to-company basis. In part 8, we’ll explain the purpose of swaps on the central bank level and when they’re used.

As established earlier in this series, a currency swap is an agreement to exchange principal interest and fixed interest in one currency (i.e. the U.S. Dollar) for principal interest and fixed interest in another currency (i.e. the Euro). Like interest rate swaps, whose lives can range from 2-years to beyond 10-years, currency swaps are a long-term hedging technique against interest rate risk, but unlike interest rate swaps, currency swaps also manage risk borne from exchange rate fluctuations.

Banks and companies aren’t the only parties using currency swaps. A special type of currency swap, a central bank liquidity swap, is utilized by central banks (hence the name) to provide their domestic country’s currency (i.e. the Federal Reserve using the U.S. Dollar) to another country’s central bank (i.e. the Bank of Japan).

Central bank liquidity swaps are a new instrument, first deployed in December 2007 in agreements with the European Central Bank and the Swiss National Bank as U.S. Dollar funding markets ‘dried up’ overseas. The Federal Reserve created the currency swap lines to assist foreign central banks with the ability to provide U.S. Dollar funding to financial institutions during times of market stress. For example, if the Federal Reserve were to open up liquidity swaps with the Bank of Japan, the Bank of Japan could provide U.S. Dollar funding to Japanese banks (just as the Bank of England would provide liquidity to British banks, etc).

As the world’s most important central bank (next to the Bank of International Settlements, considered the central bank for central banks) in one of the world’s most globalized financial markets, the Federal Reserve has a responsibility of keeping safe financial institutions under its jurisdiction. Thus, when factors abroad (such as the European sovereign debt crisis) create funding stresses for U.S. financial institutions, the Federal Reserve, since 2007, has opened up temporary swap lines.

Generally speaking, currency liquidity swaps involve two transactions. First, like currency swaps between banks and companies (as illustrated in part 7), when a foreign central bank needs to access U.S. Dollar funding, the foreign central bank sells a specified amount of its currency to the Federal Reserve in exchange for U.S. Dollars at the current spot exchange rate.

In the second transaction, the Federal Reserve and the foreign central bank enter into agreement that says the foreign central bank will buy back its currency at a specified date at the same exchange rate for which it exchanged them for U.S. Dollars. Additionally, the foreign central bank pays the Federal Reserve interest on its holdings.

Unlike regular currency swaps, central bank liquidity swaps are rare and only occur during times of market stress. The first such occurrence, as noted earlier, was in December 2007, as funding markets started to dry up as the U.S. economy entered a recession as the housing market crashed.

More recently, on November 30, 2011, the Federal Reserve announced liquidity swaps with the Bank of Canada, the Bank of England, the Bank of Japan, the European Central Bank, and the Swiss National Bank, after the European sovereign debt crisis roiled markets throughout the fall. These swaps are set to expire in February 2013.

What necessitated the Federal Reserve’s most recent round of central bank liquidity swaps? The ongoing crisis in Greece, which in fact was onset by a series of ill-advised interest rate swaps with U.S. bank Goldman Sachs.

In part 8 of 10 of this series, we’ll discuss the role of interest rate swaps in more recent times: the Euro-zone crisis (as well as answer the question in part 5 about Goldman Sach’s role with Greece’s demise).

Floating-for-Floating and Fixed-for-Fixed Swaps: Domestic and Foreign Currency Transactions

This is part 4 of a 10 part series on currency swaps and interest rate swaps and their role in the global economy. In parts 1 and 2, we discussed the beginnings of swaps as well as the differences between interest rate swaps and currency swaps. In part 3, we discussed fixed-for-floating swaps. In part 4, we’ll discuss floating-for-floating and fixed-for-fixed swaps.

In the first 3 parts of this series on interest rate swaps and their role in the global economy, we’ve covered the broader strokes of interest rate swaps and currency swaps, with our most recent discussion focusing on fixed-for-floating swaps, or plain vanilla swaps. While similar, fixed-for-fixed swaps are slightly different from their plain vanilla counterpart.

Floating-for-floating rate swaps can be used to limit risk associated with two indexes fluctuating in value. For example, if company A has a floating rate loan at JPY 1M LIBOR and it has a floating rate investment that yields JPY 1M TIBOR + 60-basis points and currently the JPY 1M TIBOR is equal to JPY 1M LIBOR + 20-basis points. Given these metrics, company A has a current profit of +80-basis points. If company A thinks that JPY 1M TIBOR will decrease relative to the LIBOR rate or that JPY 1M LIBOR is going to increase relative to the TIBOR rate, it would initiate a floating-for-floating swap to hedge against downside risk.

Company A finds company B in a similar situation, each finding a comparable advantage to a floating-for-floating swap. Company A can swap JPY TIBOR + 60-basis points and receive JPY LIBOR + 70-basis points. By doing so, company A has effectively locked in profit of 70-basis points instead of holding +80-basis points unprotected to volatility in the base indexes.

A fixed-for-fixed swap is fairly straight forward. Let’s say an American firm, company C, is able to take out a fixed rate loan in the U.S. at 8%, but needs a loan in Australian Dollars to finance a construction project in Australia. However, the interest rate for company C is 12% in Australia. Simultaneously, an Australian company, company D, can take out a fixed rate loan of 9%, but needs a loan in U.S. dollars to finance a construction project in the U.S., where the interest rate is 13%.

This is where a fixed-for-fixed currency swap comes into play: company C (in the U.S.) can borrow funds at 8% and lend the funds to the Australian company for 8%, while company D (in Australia) can borrow funds at 9% and lend the funds to the U.S. company for 9%. The comparable advantage is equal for both company C and company D: both save 4% they would have otherwise had to have spent without fixed-for-fixed currency swaps.

In part 5 of 10 of this series, we’ve fielded some basic questions on interest rate swaps and will provide some clear, succinct answers to make this complex financial instrument a little more ‘plain vanilla.’

Interest Rate Swap Tutorial, Part 3 of 5, Floating Legs

Interest Rate Swap Example

For our example swap we will be using the following inputs:

  • Notional: $1,000,000 USD
  • Coupon Frequency: Semi-Annual
  • Fixed Coupon Amount: 1.24%
  • Floating Coupon Index: 6 month USD LIBOR
  • Business Day Convention: Modified Following
  • Fixed Coupon Daycount: 30/360
  • Floating Coupon Daycount: Actual/360
  • Effective Date: Nov 14, 2011
  • Termination Date: Nov 14, 2016
  • We will be valuing our swap as of November 10, 2011.
In the previous article we generated our schedule of coupon dates and calculated our fixed coupon amounts.

Calculating Forward Rates

To calculate the amount for each floating coupon we do the following calculation:

Floating Coupon = Forward Rate x Time x Swap Notional Amount

Where:

Forward Rate = The floating rate determined from our zero curve (swap curve)
Time = Year portion that is calculated by the floating coupons daycount method.
Swap Notional = The notional amount set in the swap confirmation.

In the next couple articles we will go through the process of building our zero curve that will be used for the swap pricing. In the meantime we will use the following curve to calculate our forward rates and discount our cashflows.

swap zero curve

The numbers at each date reflect the time value of money principle and reflect what $1 in the future is worth today for each given date.

Let’s look at our first coupon period from Nov 14, 2011 to May 14, 2012. To calculate the forward rate which is expressed as a simple interest rate we use the following formula:
simple interest formula
where:

forward rate discount factor

Solving for R
forward rate formula

In our example we divide the discount factor for May 14, 2012 by the discount factor for Nov 14, 2011 to calculate DF.

0.9966889 / 0.9999843 = 0.9967046

T is calculated using Actual/360. The number of days in our coupon period is 182. 182/360 = 0.505556

R = (1 – 0.9967046) / (0.9967046 x 0.505556) = 0.654%

Our first coupon amount therefore is:

Floating Coupon = Forward Rate x Time x Swap Notional Amount

$ 3,306.33 = 0.654% x 0.505556 x $1,000,000

Below is a table with our forward rate calculations & floating coupon amounts for the rest of our coupons.

swap forward rates

The final step to calculate a fair value for our complete swap is to present value each floating coupon amount and fixed coupon amount using the discount factor for the coupon date.

Present Value of Net Coupon is
(Floating Coupon Amount – Fixed Coupon Amount) x Discount Factor

interest rate swap

Our net fair value of this swap is $ 0.00 as of November 10, 2011.

So far in this tutorial we have gone through basic swap terminology, fixed leg coupon calculations, calculating forward rates for floating leg coupon calculations and discounted our cashflows to value a swap.

Thanks to our sister company Resolution for providing us with this series of posts.

Next Article: Present value of money & bootstrapping a swap curve

Interest Rate Swap Tutorial, Part 2 of 5, Fixed Legs

Interest Rate Swap Fixed Legs

Now that we know the basic terminology and structure of a vanilla interest rate swap we can now look at constructing our fixed leg of our swap by first building our date schedule, then calculating the fixed coupon amounts.

For our example swap we will be using the following inputs:

  • Notional: $1,000,000 USD
  • Coupon Frequency: Semi-Annual
  • Fixed Coupon Amount: 1.24%
  • Floating Coupon Index: 6 month USD LIBOR
  • Business Day Convention: Modified Following
  • Fixed Coupon Daycount: 30/360
  • Floating Coupon Daycount: Actual/360
  • Effective Date: Nov 14, 2011
  • Termination Date: Nov 14, 2016
  • We will be valuing our swap as of November 10, 2011.

Swap Coupon Schedule

First we need to create our schedule of swap coupon dates. We will start from our maturity date and step backwards in semi-annual increments. The first step is to generate our schedule of non-adjusted dates.

swap coupon dates unadjusted

Then we adjust our dates using the modified following business day convention.

swap coupon dates adjusted

Note that all the weekend coupon dates have been brought forward to the next Monday.

Swap Fixed Coupon Amounts

To calculate the amount for each fixed coupon we do the following calculation:

Fixed Coupon = Fixed Rate x Time x Swap Notional Amount

Where:

Fixed Rate = The fixed coupon amount set in the swap confirmation.

Time = Year portion that is calculated by the fixed coupons daycount method.

Swap Notional = The notional amount set in the swap confirmation.

Below is our date schedule with the Time portion calculated using the 30/360 daycount convention. More on daycounts can be found in this document titled Accrual and Daycount conventions.

Note the coupons which are not exactly a half-year due to the business day convention. If our business day convention was no-adjustment all the time periods would have been 0.5. This is a difference between swaps and bonds, as bonds will generally not adjust the coupon amounts for business day conventions, they will simply be 1/(# coupon periods per year) x coupon rate x principal.

swap schedule with daycount

The coupon amount for our first coupon will be 1.24% x 1,000,000 x 0.50 = $6,200.00. Below are the coupon amounts for all of the coupons.

swap coupon schedule

Now that we know our coupon amounts, to find the current fair value of the fixed leg we would present value each coupon and sum them to find the total present value of our fixed leg. To do this we calculate the discount factor for each coupon payment using a discount factor curve which represents our swap curve. We will build our discount factor curve later in this tutorial series.

Thanks to our sister company Resolution for providing us with this series of posts.

Next Article: Swap floating legs including calculating forward rates

Interest Rate Swap Tutorial, Part 1 of 5

This is the first in a series of articles that will go from the basics about interest rate swaps, to how to value them and how to build a zero curve.

Introduction to Interest Rate Swaps

An interest rate swap is where one entity exchanges payment(s) in change for a different type of payment(s) from another entity. Typically, one party exchanges a series of fixed coupons for a series of floating coupons based on an index, in what is known as a vanilla interest rate swap.

The components of a typical interest rate swap would be defined in the swap confirmation which is a document that is used to contractually outline the agreement between the two parties. The components defined in this agreement would be:

Notional –  The fixed and floating coupons are paid out based on what is known as the notional principal or just notional. If you were hedging a loan with $1 million principal with a swap, then the swap would have a notional of $1 million as well. Generally the notional is never exchanged and is only used for calculating cashflow amounts.

Fixed Rate – This is the rate that will be used to calculate payments made by the fixed payer. This stream of payments is known as the fixed leg of the swap

Coupon Frequency – This is how often coupons would be exchanged between the two parties, common frequencies are annual, semi-annual, quarterly and monthly though others are used such as based on future expiry dates or every 28 days. In a vanilla swap the floating and fixed coupons would have the same frequency but it is possible for the streams to have different frequencies.

Business Day Convention – This defines how coupon dates are adjusted for weekends and holidays. Typical conventions are Following Business Day and Modified Following. These conventions are described in detail here.

Floating Index – This defines which index is used for setting the floating coupons. The most common index would be LIBOR. The term of the index will often match the frequency of the coupons. For example, 3 month LIBOR would be paid Quarterly while 6 month LIBOR would be paid Semi-Annually.

Daycount conventions – These are used for calculating the portions of the year when calculating coupon amounts. We’ll explore these in more detail in our discussions on fixed and floating legs. Details of different daycounts can be found here.

Effective Date – This is the start date of a swap and when interest will start accruing on the first coupon.

Maturity Date – The date of the last coupon and when the obligations between the two parties end.

Thanks to our sister company Resolution for providing us with this series of posts.

Next Article: Constructing fixed legs including calculating coupon amounts.

Interest Rate Swap Tutorial, Part 5 of 5, building your swap curve

Swap Curve

In the final article in this series, we will continue to build out our discount factor curve using longer datedpar swap ratesPar Swap rates are quoted rates that reflect the fixed coupon for a swap that would have a zero value at inception.

Let look at our zero curve that we have built so far using LIBOR rates.

zero curve

We are now going to build out this curve out to 30 years using par swap rates. These rates are as of Nov 10, 2011, and reflect USD par swap rates for semi-annual LIBOR swaps. The daycount convention is 30/360 ISDA.

par swap rates

Also keep in mind that these rates reflect the settlement conventions, so the one year rate is for an effective date of Nov 14, 2011 and termination of Nov 14, 2012. If we were to price a one year swap from the curve we have built so far, we can derive the 6mo discount factor, but we are currently missing the 1year factor. Since we know the swap should be worth par if we receive the principal at maturity, then the formula for a one year swap is:

1 year par swap rate resized 600

Notice that the T’s would be adjusted for holidays & weekends and are calculated using the appropriate discount factor. We can rearrange our formula to solve for df(1year).

swap bootstrapping

Using our example data:

discount factor par swap

We calculate the missing discount factor as: 0.99422634. But, this for a swap which settles on November 14th, and we are building our curve as of November 10th. So we need to multiple this by the discount factor for November 14th to present value the swap to November 10th. So the discount factor we use in our curve for Nov 14, 2012 is 0.9942107.

We continue by calculating discount factors for all the cashflow dates for our par swap rates. The next step is to calculate the discount factor for May 14, 2013. Our first step is to calculate a par swap rate for this date as it is not an input into our curve. We linear interpolate a rate between our 1 year and 2 year rates.

1.5 year par swap rate = 1 year + (2 year – 1 year)/365 x days

= .58% + (.60%-.58%)/365 x 181 = 0.589918%

We now can solve for the missing discount factor, continuing our bootstrapping through the curve.

zero curve construction

Thanks to our sister company Resolution for providing us with this series of posts.

Interest Rate Swap Tutorial, Part 4 of 5, Swap Curve Construction

Zero Curve

In the previous articles we described basic swap terminology, created coupon schedules and calculated fixed and floating coupon amounts. We also present valued our cashflows and calculated forward rates from our Zero Curve. A zero curve is a series of discount factors which represent the value today of one dollar received in the future.

In this article we are going to build up the short end of our discount factor curve using LIBOR rates. 

Here are the rates we are going to use. They represent USD Libor as of November 10, 2011.

ON

0.1410%

T/N

0.1410%

1W

0.1910%

2W

0.2090%

1M

0.2490%

2M

0.3450%

3M

0.4570%

4M

0.5230%

5M

0.5860%

6M

0.6540%

7M

0.7080%

8M

0.7540%

9M

0.8080%

10M

0.8570%

11M

0.9130%

Our first step will be to calculate the start & end dates for each of our LIBOR. Our TN settles in one day, and the other rates all settle in two days. We also will need to calculate the exact number of days in each period. Keep in mind that November 12th was a Saturday so our TN rate ends on the Monday, November 14th.

libor curve

Our formula for converting rates (simple interest) to discount factors is

simple interest discount factor

Where R is our LIBOR rates and T is our time calculated by the appropriate daycount convention, which in this case is Actual/360.

So our first discount factor reflecting the overnight rate is:

overnight rate

which equals: 0.999996083348673.

Bootstrapping

For our subsequent rates, they settle in the future. So when we calculate their discount factors, we will need to discount again from their settle date. See the image below to see the time frame each rate represents.

zero curve bootstrapping

Because we need the previous discount factors to calculate the next discount factor in our curve, the process is known asbootstrapping.

To calculate the discount factor for TN:

TN rate LIBOR

Which equals; 0.999988250138061 x 0.999996083348673 = 0.999984333532754

We continue the process for each time period, to build up the short end of our curve.

libor discount factors

We have shown how to convert LIBOR rates into a discount factor curve, while taking into consideration the settle dates of the LIBOR rates.

Thanks to our sister company Resolution for providing us with this series of posts.

Next Article: Building the long end of the curve using Par Swap Rates.