cfamaniac,
Short question, long answer (although by the way that you price the
question I think you realised that).
I'm going to start off by discussing the value at risk framework and
methodology. Having done that, I will then discuss the incoporation of
credit risk into the models. I will then try to answer your question
on the CBAS as best as possible.
Along the way I will use a mixture of personal knowledge (what are the
odds on there being a Google Answers researcher who has studied Value
at Risk), textbook information and links that either I personally
recommend or find from google searches.
I'm assuming given the specific question that you ask at the end that
you understand Value at Risk, however for completeness I will start
with a general description that would be suitable for someone with a
finance background but no specific exposure to Risk Management.
Value at Risk
-------------
Useful Resources :
Value at Risk by Philippe Jorion (ISBN 0-7863-0848-6)
Professional Risk Managers International Association (PRMIA)
http://www.prmia.org/
Gloriamundi (All about Value at Risk)
http://www.gloriamundi.org/
Value at Risk is a way of capturing the risk of a position, a
portfolio, an entire business into a single statistic. Specifically
Value at Risk has the following definition :
"VaR summarizes the expected maximum loss (or worst loss) over a
target horizon within a given confidence interval" (Jorion)
So to talk about Value at risk we need to specify the time period over
which the loss can happen (the horizon - a day, a month, a year, etc.)
and the likelihood of not seeing a loss during this period (the
confidence interval - typically 95% or 99%).
Consider a 1 month horizon with a 95% confidence interval.
Theoretically if you knew all of the possible outcomes that you could
have during the next month and ranked them based on how bad the loss
was - you would find the loss that split the outcomes into two sets.
In one set you would have 5% of the outcomes - all worse than the
chosen loss. In the other set you would have 95% of the outcomes - all
better than the chosen loss. This loss would be your value at risk.
If you just focus on Market Risk for a second, you can think of
calculating Value at Risk in the following way.
Suppose you hold a fixed portfolio of stocks. Take the historical
monthly returns of the stocks in your portfolio. Do a weighted sum of
the returns each month to get a series of monthly portfolio returns
for your portfolio. Sort these portfolio returns into ascending order.
Find the 5th percentile (ie. the lower tail of the distribution). Take
the corresponding return and multiply it by your portfolio value. This
number is a good estimate of the Value at Risk of your portfolio.
I say estimate, because unless you know the true distribution of your
portfolio, you will never get an exact number. Most of the methodology
in Value at Risk revolves around how best to estimate or approximate
the distribution of the returns.
Different types of risk
-----------------------
The above example focussed on Market risk - the most natural risk to
think about. This is how much money you stand to lose through price
movements on the market.
There are however several other types of risk. The most important are
:
Credit Risk - How likely it is that you will not recieve the true
value of your portfolio when you go to sell (or perhaps even before
that point). Value at Risk in this case quantifies how much you would
lose from this kind of loss.
Liquidity Risk - Losing money due to the lack of a buyer or seller
when you want to trade. Perhaps you want to sell a security you think
will do badly (an underperforming bond) but nobody wants to take you
up on the sale at the market price. In this case you either have to
lower your price to induce a trade or wait until the market becomes
more liquid (by which time the price may have decreased).
Operational Risk - Losses that occur due to business errors, criminal
activity or circumstances otherwise uncaptured in the above risks.
This is really a catch all risk group - very tough to measure (and
somewhat difficult to manage).
Market Risk, Credit Risk and Liquidity risk are most often captured by
models in a value at risk framework.
Modelling and Estimation Techniques in Value at Risk
----------------------------------------------------
The example I gave earlier was a very simplistic example - you just
create a set of historical returns and take a number from the
historical distribution of returns on your portfolio. This is not
always the most accurate or the fastest method - especially when
dealing with large portfolios.
The most typical methods used to measure value at risk are as follows
:
Delta-Normal method
You assume that the returns of the assets in your portfolio are
normally distributed. In other words, you can say everything you need
to know about the market risks of these stocks by estimating their
mean return, the variance of the stock returns and the covariance
between stock returns.
The first step is to estimate these statistics for all of the stocks.
Typically people use an equally weighted moving average to compute the
mean return and a factor model to estimate the variance-covariance
matrix for the stocks. (Or alternatively you buy data from RiskMetrics
- http://www.riskmetrics.com/)
To calculate the value at risk of the portfolio, you transform the
variance-covariance matrix of your stocks into a single portfolio
variance, and the mean returns of your stocks into a mean portfolio
return. You then calculate :
VaR = value * (mean -1.96 * sqrt(variance))
The advantages of this method are that it is simple and quick.
The disadvantages are that returns of stocks are not necessarily
normally distributed (certainly not true for derivatives and options)
and that there is no easy way to incorporate other risks such as
credit risk or liquidity risk into the model.
Historical Full Valuation Method
This is what we did in the example - you create portfolio returns
based on the historical returns of the assets in your portfolio. You
then calculate the 5th percentile of this historical distribution and
multiply it by your portfolio value.
The advantages of this method are that it is also relatively simple
and is more accurate than the normal distribution assumption.
The disadvantage is that to get an accurate statistic you need a long
time horizon to estimate over. If there is a structural change in the
market, this limits the horizon that you can use. It also requires
more computational power than the delta-normal model.
Credit risk and liquidity risk can be incorporated into this model in
that you can measure the historical credit losses and liquidity losses
and incorporate them into your model - however it is still not ideal.
Strutctured Monte-Carlo Simulation
Instead of just taking historical returns and averaging them,
construct simulated returns by using historical daily data from the
market and randomly selecting observations. So you would randomly pick
days from say the past 10 years of data and use the returns from that
day as your hypothetical returns in your sample path. Combine these
returns to create a hypothetical portfolio return.
After repeating this 10,000 times at least, you have a large sample of
"hypothetical" monthly returns for your portfolio. Calculate the 5th
percentile of this sample and multiply it by your portfolio value to
get the value at risk.
The advantages of this model is that you get a good accurate estimate
of value at risk from historical returns. It is also relatively simple
to modify the model to incorporate "hypothetical" credit and liquidity
risks.
The main disadvantages are the immense computation power required to
calculate the statistic and the difficulty in evaluating the accuracy
of your results.
Credit Risk Modelling
---------------------
RiskMetrics Credit Risk Technical Documentation :
http://www.riskmetrics.com/creditdocs.html
Default Risk papers :
http://www.defaultrisk.com/ps_models.htm
From Jorion :
"Credit Risk Depends on a number of factors : the current fair value
of the contracts, the potential future credit exposure, the extent to
which netting arrangements and colatteral can effectively reduce
exposure, tha the likelihood of default by the customer."
Credit risk modelling is most easily implemented in the Structured
Monte Carlo model. What you want to do is the following :
At each step on each sample path, consider the potential exposure you
have. For example, if you are on the pay fixed end of a swap contract
and the variable rate is high, you stand to lose the expected net
value of future positive payments if the counterparty defaults. You
would not lose the negative payments (ones you make).
Create a model of how likely it is that a counterparty would default.
This is by no means a simple issue. Ideally a model wants to
incorporate the current market conditions (which might affect the
counterparty's ability to pay), the current credit rating of the
counterparty (from moodys) as well as the size of the current exposure
to the counterparty. With a suitably parametrized model, you will have
a probability for each period and sample path of your simulation. I
will refer you to the papers above at defaultrisk.com and page 258 of
Jorion for an exact description of how to set up such a model (a
little bit technical to go into here - plus it's difficult to do
equations in this kind of an environment).
A suitable credit risk model will also take into account the joint
probability of more than one counterparty defaulting - holding
multiple bond swaps would decrease your overall risk as it is unlikely
that all counterparties would default at the same time. It will also
capture the likelihood of the creditworthiness of an investor changing
over time (A triple-A rated institution may over time fall to AA or
BBB in which case their likelihood of default increases).
You then want to reevaluate the price (and return) of your asset based
on the probability and size of loss in that sample path.
You will then have 10,000 (or more) sample paths that reflect not only
the losses due to the market risk, but also the expected losses from
default. Calculate value at risk as before, this time the statistic
will include the additional potential losses due to credit risk.
Callable Bond Asset Swaps
-------------------------
Google Search: callable bond "asset swap"
"asset swap"
http://my.dreamwiz.com/stoneq/products/callswap.htm
http://risk.ifci.ch/00010419.htm
I have no direct experience with callable asset swaps, so I will have
to rely on the descriptions from the web (what are the chances of
finding a resercher on Google Answers that also has CBAS experience :)
However the general methodologies remain the same.
An Asset Swap is another name for an interest rate swap. One where one
party pays a fixed stream of coupon payments in exchange for a
variable stream of coupon payments. It can be thought of as the net
difference of payments from buying a fixed coupon bond an selling a
floater. The notional amount is usuall not shifted (as the face values
of either half of the swap are normally the same.
Typically the floating coupon payments are set to some function of the
LIBOR rate for the period prior to payment. So if the floating bond
pays LIBOR + 2, then if at t = 2, the LIBOR rate is 7%, then the
coupon payment is 9% of the face value at t = 3. Swaps have a fixed
horizon, and may be cancelled by paying an amount equal to the current
value of the future expected payments.
A callable swap is a swap which can be terminated by one or both
parties prior to it's full term (perhaps for a penalty cost). So if
the expected cost of the remaining payments is higher than the cost of
exercising, a party may chose to exit the swap early to avoid payment
of the remaining amounts due.
To evaluate the risk involved in such an item, one would again
construct a montecarlo simulation of the underlying securities used to
drive the swap payments.
For example, one might want to simulate the LIBOR rate and the price
of the bonds involved in the swap. Along each path, one would start at
the final period and calculate the value of the position and the net
cashflows due. One would then determine based on this payment whether
it is optimal for either party to call the swap and also what the
expected risk would be if the counterparty defaulted (and the
probability of such an event happening based on the current market
conditions and counterparty characteristics). One would then work back
a period and decide based on the value of the payments in this period
and the expected value of future payments (as already calculated)
whether it would again be worthwhile calling the swap. Another
estimation of default risk would be done and the expected value
carried back another period. This is repeated until the expected value
of all cashflows is calculated for that sample path.
This is repeated 10,000 times (or more) to get a series of expected
values for the swap position. Value at Risk is then the 5th percentile
of the distribution of the net change in value of the position.
A callable bond asset swap is tricky because one has to evaluate the
probability of default at the same time as deciding whether the bond
should be called. These probabilities are obviously related, so it is
important that the model of default is consistent with the callability
of the bond.
There may be other specific issues that are important to the pricing
of a CBAS (such as the legal status of such contracts - 3rd party
guarantees or government oversight), but theoretically this is how one
would go about pricing one and evaluating it's risk.
The most important aspect of risk management is to apply a consistent
approach to estimating, pricing and measuring risk; testing your model
for deficiencies (use back testing to see whether the model works in
practice (the true losses should exceed the VaR 5% of the time if the
model works well) and documenting the assumptions made (in case it
becomes apparant that the market no longer supports this assumptions).
I hope this helps - the best I can hope to do is give you a solid
starting block from which to leapfrog off as well as a list of good
resources. Jorion's text is a great background reference, gloriamundi
is a great source of research papers, RiskMetrics is a great source of
software and data and PRMIA is where you go when you need to find a
true professional to talk to! Of course if any of this answer is
unclear, go for the answer clarification button and I will do my best
to be more clear.
Good Luck
calebu2-ga |
