# Lies, damned lies and valuations

With the passing of 30 June we have entered another busy period for year-end valuations. One of the most common questions we are asked at these important balance dates is “why is there a difference between the bank valuation and the Hedgebook valuation (or any other system’s valuation for that matter)?” The question is most commonly posed by auditors. It is probably not surprising that auditors want a perfect reconciliation between the client’s information and their own independent check but alas it will never come to pass. In this article we consider a selection of reasons that can lead to differences in valuations.

Although the modelling of interest rate swap valuations is relatively unchanged over many, many years there are subtle differences that will result in no two valuations being the same. From an interest rate swap perspective the most likely source of valuation differences is the construction of the zero curve. The zero curve is used to estimate the future cashflows of the floating leg of the swap, as well as the discount factors used to net present value the future values of the cashflows (both fixed and floating legs).

The underlying interest rate inputs into the zero curve construction (deposit rates, bank bills, LIBOR, futures, swap rates, etc.) may be slightly different between one system and another. Unlike official rate-sets such as BKBM, BBSW, LIBOR, EURIBOR, CDOR, etc. there is no one source for zero curves.

The mathematical technique to combine the various inputs into a zero curve can also differ (linear interpolation, cubic spline). These types of differences can lead to discrepancies between one valuation and another. Although on a percentage of notional basis the discrepancies are small, the monetary differences can become material if the notional of the swap is big enough i.e. a \$500 difference on a \$1 million interest rate swap becomes a \$50,000 difference on a \$100 million swap – a number that will draw attention but in reality is still immaterial.

The timing of the market snapshot for closing rates can be different, too. For example, Hedgebook uses New York 5pm as the end of day for valuation purposes, therefore, if the Hedgebook valuation is compared to a system that uses, say, Australia 5pm as its rate feed, then any movement in the intervening period will cause differences in valuations.

What we have talked about above is premised on the fact that two identical deals are being valued against each other. By far the most common reason for different valuations lies in human error around the inputting of a deal. From an interest rate swap perspective, the rate-set frequency (monthly, quarterly), accrual basis, business day conventions, margins on the floating leg are all possible areas which can result in valuation differences. The most common input error we come across relates to amortising interest rate swaps (changing face value and/or interest rate over the life of the swap). Very often there is no way for an auditor to realise that a swap is an amortising structure just by looking at the bank valuation. Often it is just the face value of the swap at the current valuation date that is shown on the bank valuation. It is the schedule at the end of the original bank confirmation that is required to accurately input and value such a structure.

Of course the true valuation of a derivative is determined by the price at which it can be sold/closed out which will be different to a valuation for accounting purposes. Valuations for accounting purposes are based on mid rates and, therefore, take no account of bid/offer spreads. Some of the changes we are seeing in the International Financial Reporting Standards are trying to provide greater consistency and more explicit definitions of fair value (IFRS 13). At least the “risk-free” component of an interest rate swap is a well-established methodology. The same cannot be said for the credit component (CVA), for which there is a myriad of approaches. It will be interesting to see how differences are reconciled and treated by auditors, as there is even less likelihood of two valuations being the same.

# IFRS 13: Fair value measurement – Credit Value Adjustment

The purpose of this blog is to examine IFRS 13 as it relates to the Credit Value Adjustment (CVA) of a financial instrument. In the post GFC environment, greater focus has been given to the impact of counterparty credit risk. IFRS 13 requires the valuation of counterparty credit risk to be quantified and separated from the risk-free valuation of the financial instrument. There are two broad methodologies that can be considered for calculating CVA: simple and complex. For a number of pragmatic reasons, when considering the appropriate methodology for corporates, the preference is for a simple methodology to be used, the rationale for which is set out below.

IFRS 13 objectives

Before considering CVA it is worthwhile re-capping the objectives of IFRS 13. The objectives are to provide:

–          greater clarity on the definition of fair value

–          the framework for measuring fair value

–          the disclosures required about fair value measurements.

Importantly, from a CVA perspective, IFRS 13 requires the fair value of a liability/asset to take into account the effect of credit risk, including an entity’s own credit risk. The notion of counterparty credit risk is defined by the risk that a party to a financial contract will fail to fulfil their side of the contractual agreement.

Factors that influence credit risk

When considering credit risk there are a number of factors that can influence the valuation including:

–          time: the longer to the maturity date the greater the risk of default

–          the instrument: a forward exchange contract or a vanilla interest rate swap will carry less credit risk than a cross currency swap due to the exchange of principal at maturity

–          collateral: if collateral is posted over the life of a financial instrument then counterparty credit risk is reduced

–          netting: if counterparty credit risk can be netted through a netting arrangement with the counterparty i.e. out-of-the money valuations are netted with in-the-money valuations overall exposure is reduced

CVA calculation: simple versus complex

There are two generally accepted methodologies when considering the calculation of CVA with each having advantages and disadvantages.

The simple methodology is a current exposure model whereby the Net Present Value (NPV) of the future cashflows of the financial instrument on a risk-free basis is compared to the NPV following the inclusion of a credit spread. The difference between the two NPVs is CVA.  The zero curve for discounting purposes is simply shifted by an appropriate credit spread such as that implied by observable credit default swaps. To give a sense of materiality, a NZD10 million swap at a pay fixed rate of 4.00% with five years to maturity has a positive mark-to-market of +NZD250,215 based on the risk-free zero curve (swaps). Using a 200 basis point spread to represent the credit quality of the bank/counterparty the mark-to-market reduces to +NZD232,377. The difference of -NZD17,838 is the CVA adjustment. The difference expressed in annual basis point terms is approximately 3.5 bp i.e. relatively immaterial. In the example we have used an arbitrary +200 bp as the credit spread used to shift the zero curve. In reality the observable credit default swap market for the counterparty at valuation date would be used.

The advantages of the simple methodology is it is easy to calculate and easy to explain/demonstrate. The disadvantage of the simple methodology is takes no account of volatility or that a position can move between being an asset and a liability as determined by the outlook for interest rates/foreign exchange.

The complex methodology is a potential future exposure model and takes account of factors such as volatility (i.e. what the instrument may be worth in the future through Monte Carlo simulation), likelihood of counterparty defaulting (default probability) and how much may be recovered in the event of default (recovery rate). The models used under a complex methodology are by their nature harder to explain, harder to understand and less transparent (black box). Arguably the complex methodology is unnecessary for “less sophisticated” market participants such as corporate borrowers using vanilla products, but more appropriate for market participants such as banks.

Fit for purpose

An important consideration of the appropriate methodology is the nature of the reporting entity. For example, a small to medium sized corporate with a portfolio of vanilla interest rate swaps or Forward Exchange Contracts (FECs) should not require the same level of sophistication in calculating CVA as a large organisation that is funding in overseas markets and entering complex derivatives such as cross currency swaps. Cross currency swaps are a credit intensive instrument and as such the CVA component can be material.

Valuation techniques

Fair value measurement requires an entity to explain the appropriate valuation techniques used to measure fair value. The valuation techniques used should maximise the use of relevant observable inputs and minimise unobservable inputs. Those inputs should be consistent with the inputs a market participant would use when pricing the asset or liability. In other words, the reporting entity needs to be able to explain the models and inputs/assumptions used to calculate the fair value of a financial instrument including the CVA component. Explaining the valuations of derivatives including the CVA component is not a straightforward process, however, it is relatively easier under the simple methodology.

Summary

IFRS 13 requires financial instruments to be fair valued and provides much greater guidance on definitions, frameworks and disclosures. There is a requirement to calculate the credit component of a financial instrument and two generally accepted methodologies are available. For market participants such as banks, or sophisticated borrowers funding offshore and using cross currency swaps, there is a strong argument for applying the complex methodology. However, for the less sophisticated user of financial instruments such as borrowers using vanilla interest rate swaps or FECs then an easily explainable methodology that simply discounts future cashflows using a zero curve that is shifted by an appropriate margin that represents the counterparty’s credit should suffice.

# 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).

# 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. 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: where: Solving for R 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. 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 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. Then we adjust our dates using the modified following business day convention. 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. 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. 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

## 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. 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. 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: 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). Using our example data: 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. 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. Our formula for converting rates (simple interest) to discount factors is 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: 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. 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: 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. 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.