The outsourcing company usually claims that keeping their own IT support team would be more expensive than outsourcing. I believe this is a fallacy, and here is why.

Firstly, consider an organisation (let’s say it’s a bank) that has its own IT support team. This team is shown on Figure 1.

An internal IT support team.

Say, the team consists of 20 people, which is just enough to keep the IT infrastructure up and running. Obviously, his team needs a manager. The manager is responsible for task allocation, making sure that not all of the team take holidays at the same time, conducting end of year performance reviews — in short, the manager does not do anything useful for the bank’s IT systems. The manager represents the overhead which is needed for the team to work well. In other words, the bank pays salaries to 21 people, but only 20 of them do some useful work.

Next, let us consider the case where the bank has outsourced the IT support to another company, the solution provider. The solution provider is a big company that serves many different clients. Suppose our bank still needs 20 people to maintain its IT infrastructure. Like before, these 20 people need a manager. However, the solution provider is a big company that consists of many such teams. Their managers, in turn, have higher-level manager, that oversees multiple teams. These higher-level managers report to other, even higher, managers, and so on until the pyramid converges to one person, the president of the company. This is shown on Figure 2.

Outsorced IT support.

Here, the bank that uses the services of the 20 people, pays not only their and their manager’s salaries, but also a fraction of the salaries of all the higher management up to the top (otherwise, where would their salaries come from?)

Paying just a bit more for the same amount of work would not be such a big deal, but it does not stop here. There is one more figure that stands between the bank and the solution provider: the account manager. The job of the account manager is to make the bank pay as much as possible for the services it receives. Guess where the account manager’s salary comes from? Also from the fees that the bank pays to the solution provider. The bank, in fact, feeds its worst enemy.

But wait, before the bank decided to outsource its IT support, they made calculations which showed that the fees of the solution provider are lower than the costs of having IT support inside the company. How did they do these calculations? The trick is to design outsourcing such that the solution provider will have to do less work than an internal support team would have done. Let us look at an example.

One morning at the bank that I am consulting, my Microsoft Access application crashed and damaged the database file. ”No big deal,” thought I, “the files are on a network server, which is backed up nightly. I can ask for the last night’s backup.” It should take no longer than an hour to restore a file from a backup, right?

I phoned the bank’s IT helpdesk and explained the problem. “Ok,” said the person on the other end of the line, “I will make a ticket for the backup/restore team. You can expect the problem to be solved within 24 hours.” “Can it be done quicker?” asked I anxiously. “I need this file to complete an urgent task.” “All I can do is make a ticket,” answered the helpdesk person, “the rest is up to the backup and restore team, and they have 24 hours, according to the SLA.”

The bank has an agreement (Service Level Agreement, or SLA) with the solution provider, which specifies the time allotted to solving IT problems. As long as the solution provider conforms to the SLA, they have little incentive to hurry up with doing their job.

Bottomline: outsourcing IT support looks attractive on paper, but in reality the bank gets much less work done for the money they pay.

I am not trying to say that outsourcing is generally a bad idea. In many cases it is exactly the thing to do. Every two years, the banks needs to repaint the facade of its office building. It hires services of a painting company. The painting company does the job in a week and paint some other facades before it is time to repaint this bank again. There is no reason to keep the painters on the bank’s payroll.

Now let us look at three more arguments that are often used to justify outsourcing, and see why they do not apply to IT support.

1) Doing IT support is not the bank’s core business.

This argument would only work if one believes that a
company doing only its core business is good for the company. This is simply not true. Say, the core business of a railroad is to bring passengers from one place to another. If the railroad decides not to sell hotogs on the platforms because ”selling hotdogs is not our core business”, someone else will do it and take all the money.

2) IT solution providers have more expertise in IT support than the bank.

This is not true either. They just hire anybody who is willing to work for the salary they pay, train them quickly and send them to the customers. When the bank owns its IT support, it have much more control of the quality of its workforce.

3) Outsourced work can be done in other countries where wages are lower.

This was true several years ago, but by now the people who can do the work became expensive regardless of their geografical location. The people that remained cheap are probably not a good hire even for a low salary. Besides, IT support requires the engineer’s physical presence at the customer’s site so often that one has to keep a fairly big team  co-located with the customer anyway.

So, why do so many companies outsource their IT support? The reasons are plentiful, for instance:

1) It looks good on paper. The manager who has outsourced IT support can claim that he has saved a lot of money for the company.

2) It shifts the responsibility to the solution provider. If an internal IT support group messes things up (people do make mistakes every now and then), their manager gets the blame. If an external solution provider messes things up, they are to blame, not the manager who outsourced the work to them.

3) Due to the herd instinct. Outsourcing is popular, so people tend to do this because everyone else does. 20 years ago a vast majority of males smoked for exactly the same reason.

Sadly, many companies continue to outsource their IT support, and are likely to continue to do so in the future.

Risky zero rate calculation example.

In his other blog entry the author argues that, in fact, any software algorithm can be reduced to mathematical formulae, which are not patentable.

If we can find matrix such that

• is Hermitian,
• can be decomposed as ( denotes conjugate transpose of ), where is invertible,

then there exists a decomposition where is real. Note: in this post, I discuss a general case where elements of are complex numbers. All of that is also applicable, as a special case, to real-valued matrices. For them, “Hermitian” means “symmetrical”, and “unitary” means “orthogonal”. The elements of an intensity matrix of a Markov chain are, of course, real.

To find the eigendecomposition of a non-Hermitian matrix , we start with the eigendecomposition of the matrix . We use abbreviation . Matrix is Hermitian, which is easy to see by checking that using the fact that .

Suppose we have found the eigendecomposition

(using our favourite matrix manipulation package). Then, by definition of , we have

.

Multiplying both sides of this equation by on the left and on the right, we get

.

Let , then . Thus,

,

which is the eigendecomposition we were looking for.

Problem

Consider a five-year semi-annual vanilla payer swap in Euro: every six months (coupon period) we pay interest over six months at a fixed rate K and receive interest over the same period at Euribor 6-month rate fixed at the start of the period (fixing date). The number of days in the coupon period is calculated using Actual/360 convention, the notional amount is 1. The swap starts today (9 April 2010) and the fixed rate K equals today’s par swap rate, so mark-to-market value of the swap is zero now. There is no netting or collateral agreement for this trade. For simplicity, we ignore all other details (for instance, the fact that Euribor actually fixes 2 days before the start of the coupon period, or that the coupon payment is rolled over if it falls on a weekend). We want to calculate the credit value adjustment (CVA) for this swap. By definition, CVA is the difference between the risk-free value of the trade and its value that takes into account the possibility of the counterparty’s default [1].

Inputs

We have:

1. Discount curve P(t) for Euro. P(t) is today’s price of a zero-coupon riskless bond that pays 1 euro at time t.
2. Implied volatility cubes C(t, τ, k) and S(t, Ï„, k) for Euro. S(t, τ, k) is the swap rate volatility implied by the price of a swaption with tenor Ï„, strike k and expiry date t. C(t, Ï„, k) is the interest rate volatility implied by a caplet with tenor τ, strike k and expiry date t.
3. Recovery rate R. If the counterparty defaults, we expect to recover E*R, where E is our exposure to the counterparty at the moment of default. In other words, our losses will be (1-R)*E.
4. Probability PD(t) of the counterparty’s default. PD(t) is the probability that the counterparty defaults between now and time t, as viewed by the market (for instance, derived from CDS prices as shown in [2]).

Calculation

Assuming independence between exposure and counterparty’s credit quality, CVA is given (see [1]) by the formula:

where is the end date of the trade and is the discounted expected exposure computed under an equivalent martingale measure (e.g. risk-neutral measure).

Recovery rate R and default probability PD are given, so we only need to calculate the expected exposure .

We recall that the exposure at time t is the maximum of the trade’s mark-to-market value (MtM) and zero:

Consider a contract that gives its holder the right, but no obligation, to enter the swap at time t. Essentially it is an option to acquire the swap at no cost at time t. This option is equivalent to receiving max(MtM(t),0) at time t. Therefore, its value today is the discounted expectation of max(MtM(t),0) calculated under an equivalent martingale measure, which is the definition of . Thus,

the discounted expected exposure on a trade at time t equals the value of the option to acquire the trade at no cost at time t.

Now let us fix time in the future and calculate today’s value of the option to acquire the swap at . The MtM of the swap is defined by the value of the outstanding cashflows. If the swap’s coupon dates are and for some i, then at one already knows the cashflow that is due at . Indeed, the floating rate was fixed at , so it is known at . Remaining cashflows are not known yet, but they can be priced as a swap starting at  and ending at . We see that the payoff of the option in question depends on:

• Euribor fixing at
• Forward swap rate at

To avoid having to price a path-dependent derivative, we replace the option to enter the swap at with the sum of two options: a cap starting at  and ending at  and a swaption starting at  and ending at . It is easy to see that the sum of the payoffs of the cap and the swaption is greater or equal to the payoff of the original option, so the sum of the values of the cap and the swaption give us a conservative estimate of the expected exposure.

We start with pricing a cap starting at  and ending at . Its underlying rate is Euribor 6-month and strike is K. As shown in [3], we can use Black’s formula to price the cap. We take implied volatility

,

calculate the forward rate

and use Black’s formula to get the cap value

Next, we price the swaption part in a similar way, again taken from [3]. The implied volatility is

the forward swap rate is

and the swaption price is

The total upper boundary for discounted expected exposure at is .

Defining the quantity

we get the upper boundary for CVA as a sum

We take CVA approximation

Conclusion

Although we were able to compute an analytical approximation for CVA, we had to make numerous assumptions to simplify the task. For derivatives more complex than swaps (or even for a portfolio of swaps) analytical computation of CVA would not be feasible. Instead, we will have to compute CVA by Monte Carlo simulation, as outlined in [1]. An example of CVA calculation with Monte Carlo is shown here.

References

1. Zhu, Steven H. and Pykhtin, Michael, A Guide to Modeling Counterparty Credit Risk. GARP Risk Review, July/August 2007. Available at SSRN: http://ssrn.com/abstract=1032522

2. Beck, Ronald. The CDS market: a primer. Available online: http://www.dbresearch.com/PROD/DBR_INTERNET_EN-PROD/PROD0000000000185396.pdf

3. Brigo, Damiano and Mercurio, Fabio, Interest Rate Models – Theory and Practice: With Smile, Inflation and Credit. Springer; 2nd edition (August 2, 2006).

where is the current MtM, is the add-on, is the notional amount and is the trade-specific coefficient.

This method works reasonably well for a single trade.

If we have several trades covered by a netting agreement, we have a problem. We cannot just calculate the exposure on a netted portfolio as a sum of exposures per trade: exposures are not additive. One way to tackle this problem is to use the 40-60 rule. This rule, however, is so seriously wrong that it becomes alarming how many people use it without thinking of its shortcomings.

Consider a netted portfolio of n trades with MtMs and per-trade add-ons . The 40-60 rule says that the exposure on the netted portfolio is

where

(if all MtMs are negative, NGR=1).

In other words, NGR (net to gross ratio) is defined as the net replacement cost of the portfolio divided by the sum of the replacement costs for each transaction.

The 40-60 rule for exposure calculation is described in the Treatment of potential exposure for off balance sheet items (1995), issued by the Basel Committee on Banking Supervision. The Committee, to my knowledge, does not provide any justification for this rule.

Worse yet, it is known that the rule is wrong. Back in 2001, ISDA published their Response to the Basel Committee on Banking Supervision’s Consultation on the New Capital Accord. There they conclude that “the aggregation rule fails to measure the true risk in a portfolio” (page 52). To substantiate this claim, they give a simple example (Annex A) showing that the 40-60 rule can grossly over- or underestimate the exposure. In other words, the value given by the 40-60 rule has nothing to do with the real exposure on the netted portfolio.

If this method is so bad, why is it so widely used? I don’t have a good answer to this question.

How can we forecast the future volatilities? We can either

• use the volatility implied by options (caps, floors and swaptions for interest rate, FX options for exchange rate)

or

• use historical volatility of the risk factors.

Which one will better predict the volatility in the future? The analysis by De Jong et al. suggests that the volatility implied by the options is a poor predictor, because it consistently overestimates realised volatility. That means, we have to use historical volatility in exposure calculations. Their results, however, are based on fairly old data; it would be interesting to repeat their experiments with recent time series.

On the other hand, when we calculate CVA, we must use implied volatilities, by definition of CVA.

Indeed, if I had to rely on numerical results in finance, I would be afraid.

Take counterparty risk as an example. There are three popular numerical measures of counterparty risk: exposure at default, loss given default and probability of default. Firstly, it would be wrong to hope that just three numbers fully describe our risk. Secondly, numerical results inevitable depend on the mathematical model that we use. A model is always imprecise, but we seldom know by how much we are off. Thirdly, once we have chosen a model, we have to calibrate it. We can calibrate to market data, to historical data, or some mix of those. In either case, we essentially try to predict the future based on market consensus (i.e. public opinion) or history. Neither is a reliable predictor, as everyone knows.

Does it mean that numerical results are useless? Not at all. For example, correlation is not a perfect measure of statistical dependency; still, it is useful in many cases. Likewise, it is useful to know our potential exposure on each counterparty. After all, we need to know how much cash we need to reserve to cover our losses in case of default, and potential exposure gives us a number. We are not sure that this number is exact (in fact, we are sure it isn’t), but at least we have a starting point.

Here is an example. Suppose we have a netting agreement with a certain counterparty, and at the moment we have a single trade with them: a semi-annual receiver swap of 6 month EURIBOR versus 4.08% fixed from 19 June 2009 to 19 June 2015. Below is the potential exposure profile of the swap at 95% confidence.

We see that the peak of the potential exposure profile is around 38. We want to add another trade to this portfolio: a semi-annual payer swap of 6 month EURIBOR versus 4.5% fixed from 19 June 2011 to 19 June 2017. Firstly, let us see the exposure profile of the new swap alone:

The peak exposure is around 34.

If we simply add the two peak exposures, we get 38+34 = 72. However, if we calculate exposure profile of the two netted swaps with a Monte Carlo simulation, we will see that the total exposure of the netted portfolio is much lower:

The peak is around 14, which is lower than each of the two swaps separately. Indeed, under the netting agreement the swaps partially compensate each other, so adding the second swap actually lowers our exposure to the counterparty.

Why would we have two trades that cancel each other out? It often happens in practice. Say, the receiver swap was done a while ago, and it still has a few years to go. Right now a trader needs a payer swap, so this swap is added to the portfolio.

To summarise: we have seen that the sum of exposures on individual trades can be very far from the total exposure on these trades covered by a netting agreement. The exposure profiles above are screenshots of MarketSimulator.