Synthetic CDO Valuation Using Quasi-Analytic Methods

Overview

A collateralized debt obligation (CDO) is a security that is backed by or linked to a diversified pool of credits. The credits can be assets, such as bonds or loans, or simply defaultable names, such as companies or countries. In general, there are two types of CDOs: cash CDOs and synthetic CDOs. A cash CDO is backed by “true” assets, such as bonds or loans. Its payoffs, either coupons or principals, come from the actual cash flows of the assets in the pool. Synthetic CDOs are not backed by cash flows of assets. Instead, they are linked to their reference entities by credit derivatives, such as credit default swaps. The payoffs of most synthetic CDOs are only affected by credit events, e.g., defaults, of the reference entities and are not related to the actual cash flows of the pool.

A common structure of CDOs involves slicing the credit risk of the reference pool into a few different risk levels. A “slice” of credit risk, the credit risk between two risk levels, is called a tranche. The tranche that absorbs the first loss piece is often called an equity tranche. The remaining tranches are called mezzanine or senior tranches. Tranches with higher credit risks support the tranches with lower credit risks.

A typical feature of a cash CDO is the so-called cash flow waterfall structure where a tranche is paid for its coupon and/or principal only if all the tranches with lower credit risk have been paid off. Such a feature requires a match between the incoming coupons and principals of the pool and the payouts of the tranches. Most synthetic CDOs don’t have a cash flow waterfall structure. Synthetic CDOs are only concerned with the default loss. This simplified structure makes structuring, managing and valuing synthetic CDOs easier than cash CDOs. The important implication of such a simplified CDO structure is that tranche customization is possible. Since the payoffs of one tranche don’t depend on the payoffs of other tranches, it is easy to structure CDOs with any types of tranches, based on an investor’s risk appetite. This makes synthetic CDOs attractive to both CDO investors and managers and has helped the rapid development of the CDO market.

Synthetic CDOs without a cash flow waterfall structure are sometimes called single tranche CDOs. In the rest of this document all CDOs discussed are single tranche CDOs.

CDO tranches, CDSs on tranches and CDO notes

The risk levels of a synthetic CDO are determined by the total accumulated loss of the reference pool. A tranche is defined as a certain loss range. The lower bound of the range is called an attachment point and the upper bound a detachment point. For example, a 5-10% tranche has an attachment point of 5% and a detachment point of 10%. When the accumulated loss of the reference pool is no more than 5% of the total initial notional of the pool, the tranche will not be affected. However, when the loss has exceeded 5%, any further loss will be deducted from the tranche’s notional until the detachment point, 10%, is reached. At this point, the tranche is wiped out.

The most common credit derivatives of synthetic CDOs are credit default swaps (CDS) and credit-linked notes on CDO tranches. A CDS on a CDO tranche is, to a certain extent, similar to a single-entity credit default swap (for single entity CDSs, see the Credit Default Swaps FINCAD Math Reference document). It has a payoff leg and a premium leg. The buyer of a CDS on a tranche will be compensated by the seller for any loss to the tranche and in return the buyer pays a periodic premium to the seller. While the basic structure is similar, there are two fundamental differences between a tranche CDS and single entity CDS:

1. In a single-entity CDS the seller will take the full loss of the reference credit while in a CDS on a CDO tranche the seller is protected by the tranches with higher credit risk. In more detail, as long as the total loss of the reference pool does not exceed the attachment point of a tranche, the tranche is not affected. Moreover, the loss to a tranche is limited to its notional, not the total loss of any of the credits in the pool.

2. In a single-entity CDS the premium notional is fixed during the life of the CDS while in a CDS on a CDO tranche the premium notional changes. When a tranche suffers a loss, the buyer will be paid up to the notional of the tranche and at the same time, for most CDOs, the lost amount will be subtracted from the tranche’s notional. Thus the premium payment reduces, until the notional reaches 0 when the CDS ends.

A similar comparison can be made between a tranche CDS and a basket CDS, such as a first-to-default CDS (for details on basket CDSs see the Basket Default Swaps FINCAD Math Reference document). Like a single entity CDS, a basket CDS allows only a single credit event and the investor will suffer the full loss if a credit event occurs. The only difference between a basket CDS and a single-entity CDS is that in a basket CDS the credit risk to the investor is from multiple entities. We can say that basket CDSs are more like single-entity CDSs than tranche CDSs. Particularly, one should not confuse a first-loss CDS with a first-to-default CDS. A first-loss CDS is a CDS on the equity tranche of a CDO and, as is pointed out above, it is significantly different from a first-to-default CDS.

It should be to be pointed out that a first loss CDS is a CDS on the loss of a portfolio of entities up to a certain level. It doesn’t have to be associated with a formal CDO structure, although it can always be viewed as a CDS on the equity tranche of a CDO, i.e., a CDO that has only one tranche, the equity tranche.

Synthetic CDOs consisting only of credit default swaps are unfunded CDOs. Some synthetic CDOs issue notes linked to CDO tranches. These CDOs are funded or partially funded. The buyer of a CDO note pays a certain amount, the note’s principal, upfront and receives periodic coupon payments from the CDO that are based on the principal and a fixed coupon rate. If the tranche doesn’t suffer any loss during the life of the note, the investor will receive a full repayment of the amount invested at the beginning. However, if there are defaults in the reference pool and the pool’s accumulated loss exceeds the attachment point of the linked tranche, the principal will be reduced and thus the future coupon and the principal repayment will be reduced.

A similar comparison to the above can be made between a CDO note (also known as a credit-linked note or a tranche-linked note) and a single-entity credit-linked note. For details on single entity credit-linked notes see the Credit-Linked Notes and Rating Sensitive Notes FINCAD Math Reference document.

Some CDO products, such as CDOs on the CDS indices CDX and ITraxx, are exchange traded and are standardized. For a standardized CDO, each entity in the reference pool has the same notional and the same recovery rate. While standardized CDOs are a special class of CDOs, the homogeneity of their reference pools greatly simplifies their valuation, and more importantly, speeds up their calculation.

A typical CDO has a reference pool of more than 100 reference entities. Valuation of a CDO with such a large reference pool is a challenging task. Monte Carlo simulation has been a useful tool in CDO valuation, but its performance has always been an obstacle for wide use in the rapidly developing CDO market. Non-Monte Carlo methods have proven to be more effective than Monte Carlo simulation, thanks in part to the standardization of the CDO market. Such non-Monte Carlo methods have been referred to in the literature as quasi-analytic, semi-analytic, or closed-form methods.

FINCAD provides tools to value both CDSs on tranches and CDO notes. CDSs on tranches are valued with both Monte Carlo and quasi-analytic methods. CDO notes are valued with Monte-Carlo simulation. This document mainly discusses quasi-analytic methods. For discussions on the application of Monte Carlo simulation to CDOs see the Synthetic CDO Valuation Using Monte Carlo Simulation FINCAD Math Reference document.

Formulas & Technical Details

Consider a CDO tranche with attachment and detachment points, respectively. Let denote the accumulated loss of the pool at time (the current time is 0). Let denote the (full) notional of the tranche at time. Then:

Credit default swaps on tranches

Consider a CDS on a tranche with its notional being the tranche’s size. Let be the time value of the premium leg (also called a fixed leg or a fee leg) of a credit default swap on the tranche. Let be the annual premium coupon rate and the premium coupon payment time points. The premium payment amount at a premium payment date is the product of the premium rate and the current notional of the tranche. For most synthetic CDO deals, when an entity defaults prior to payment date, the accrued interest must be paid at the default time. For other less common synthetic CDOs there is no such accrued premium payment. Accordingly, the present value of the premium payments can be calculated as

where: is a risk-free discount factor curve and the constant if the accrued premium is paid when an entity defaults prior to a premium payment date; and , otherwise.

Let be the time t value of the payoff leg (also called a default leg or a floating leg or an asset leg) of the CDS. Then:

If we assume that compensation is made at the next premium coupon payment date when default occurs, then the above formula can be detailed as:

The par CDS spread is then:

CDO notes (Tranche-linked notes)

For a tranche-linked note, at any coupon payment date, if the pool loss hasn’t eaten into the tranche yet, the investor will receive a predetermined coupon based on the initial principal. However, when the tranche suffers a loss, the note’s notional will be reduced and therefore both the future coupon payments and the redemption will be reduced.

Consider a tranche-linked note with a principal equal to the tranche’s size. Let be the time value of the note. Suppose the annual coupon rate of the note is . Then the present value of the note is:

The one-factor Gaussian (normal) copula model

Default times of different reference entities are correlated. Ideally, a full correlation matrix should be used in modeling the correlation between reference entities of a reference pool. Unfortunately, this is difficult due to computational intensity, calibration complexity and rarity of default data. The current popular model on default correlation is the so-called one-factor Gaussian (normal) copula model. Under this model the credit quality of entity is represented by a Gaussian random variable:

where

are independent Gaussian random variables with mean 0 and standard deviation 1 (N(0, 1)) and

is a constant between 0 and 1. The constant is called the factor loading of entity . A higher value of represents a higher credit quality of the entity. Given a default probability of the entity one can then find a threshold value of such that:

Since is an N(0,1) variable, , where is the cumulative distribution function of an N(0,1) variable. With the Gaussian copula model we can determine the credit correlation easily by simply using the joint probability:

where

is the correlation of and

is the cumulative standard bivariate normal distribution function.

A special case of the Gaussian copula model is the one for which all factor loadings are equal. For this case each pair of the reference entities has the same correlation, and the factor loading is the square root of the correlation:

The Fast Fourier Transform (FFT)

In the valuation of a CDO, the key is to calculate the loss distribution of the reference pool. Let be the default time of entity and the loss amount when the entity defaults. Then the total loss of the reference pool up to a given time horizon is:

where

is the number of entities in the pool and

is the indicator function of

When the reference entities are homogeneous, i.e., all entities have the same notional and the same recovery rate, the estimation of is relatively simple: the loss distribution can be determined by the number of defaults. However, for a more general case, to determine the loss amount at a default time we have to trace down the defaulted entity and hence the simple counting method no longer works. For this case, a more general method is required.

Note that to calculate the distribution of we only need to determine its characteristic function:

.

For this a natural choice is the FFT. When the characteristic function has been determined, the reverse FFT can be used to back out the loss distribution. For details on calculating loss distributions using FFT see the paper by A. Debuysscher and M. Szego [5].

The recursion method

The FFT discussed above is general and convenient for extension, e.g., to the case of random recovery rates. The downside of an FFT for CDO valuation is its resource intensity in computation, particularly, when the losses given default of the CDO’s reference entities vary significantly from entity to entity.

More recently, another type of methods, the so-called recursion or convolutional methods, have been proposed (for details see Andersen [1] (2003) and Luo [7] (2006)). Research has shown that the recursion method is significantly faster than the FFT. For a comparison of computational speed of the FFT and the recursion see Luo [7] (2006).

The main idea of a recursion method is that when the loss from default of each entity in a pool is an integer and if the defaults of all entities are independent, then the loss distribution of an -entity pool can be derived from the loss distribution of a sub-pool with entities and the default probability of the entity convolutionally. Under a factor copula model the defaults of the reference entities in a pool are conditionally independent given the common factor. Thus the recursion method can be used to calculate the conditional loss distribution of the reference pool. When all the conditional loss probabilities are known, integrating over the common factor one can obtain the absolute loss distribution of the reference pool.

In general, losses from defaults are not integers. The first step in using a recursion method is to approximate non-integer losses points by integer loss points. Such an algorithm is proposed in Andersen [1] (2003).It is slightly modified in FINCAD implementation to improve accuracy.

Remark 1: The computational speed of a synthetic CDO depends heavily on the homogeneity of its reference pool for either a FFT method or a recursion method. If a reference pool is homogeneous, i.e., the loss given default of all entities in the pool is the same, both methods are fast, although the recursion method is even faster than the FFT. For a generic pool, the computational speed varies significantly with the level of homogeneity of the pool. The lesser the loss given default varies from entity to entity, the faster the calculation. In FINCAD implementation the so-called grouped recursion method is used. In this method a reference pool is approximated by homogeneous subgroups (see Luo (2006) for details).

Remark 2: In the calculation of risk statistic DVOX (see the section Calculation of Risk Statistics for its definition) on individual entities a general method is to shift the CDS spread curve X basis points of the entity first and then revalue the CDO with the shifted default curve. A tranche’s DVOX of the entity is calculated as the difference of the bumped fair value and the original fair value. When the number of entities is large, revaluation needs to be done for each entity and therefore for a large CDO it can be very time consuming to calculate all the DVOXs. For the FFT method FINCAD uses this generic method for DVOX calculation. For the recursion method, a smarter algorithm proposed by Brasch (2004) is adapted. This method avoids revaluing the CDO by using the results that are already been calculated during the calculation of the CDO’s fair value to derive the DVOXs of individual entities directly. This makes the DVOX calculation of individual entities almost costless.

Tranche correlation and base correlation

CDO tranches are actively traded in the market and, especially for standardized tranches, market quotes are available for par CDO tranche spreads. A natural question to ask is: based on a one-factor Gaussian copula model, can we back out correlations from quoted tranche spreads? In more detail, given the spread of a CDO tranche is there a correlation that gives the tranche a par spread equal to the given spread? Such a correlation is known as a tranche correlation. Research shows that such a correlation does exist for reasonable par CDS spreads, but, in general, is not unique except the equity tranche. For the equity tranche (the tranche with an attachment of 0), the implied correlation is unique (for a proof see the paper by Burtschell et al [4]). The implied correlation of an equity tranche is referred to as a base correlation. For a non-equity tranche its base correlation can also be defined. It is the correlation that applies to the tranche that has an attachment point of 0 and the same detachment point of the given tranche. It should be pointed out that a tranche correlation and a base correlation are tranche dependent. For different tranches their correlations can be different.

For a non-equity tranche, the tranche that has an attachment point of 0 and a detachment point equal to that of the given tranche is called in this document a base tranche of the given tranche. CDO quotes are available for regular tranches, but not for base tranches other than the equity tranche. Hence only for the equity tranche its base correlation can be calculated directly from the market quotes. For other tranches their base correlations cannot be calculated directly and a so-called bootstrapping algorithm must be used.

To explain the idea of bootstrapping base correlations, consider the second tranche of a CDO, i.e. the tranche that is beside the equity tranche. Suppose the base correlation of the equity tranche has been determined (with a root finding method, e.g., the bisection method), and we want to determine the base correlation of the second tranche. Note that the loss to the second tranche is the difference of the loss to its base tranche and the loss to the first (equity) tranche. Thus the payoff and premium of the second tranche can also be expressed as the differences of the base tranche and the equity tranche. Since the payoff and premium of the equity tranche can be determined from its base correlation, by matching the calculated par spread and the quoted par spread of the second tranche, we can solve for the base correlation of the second base tranche using a root finding procedure. For more details on bootstrapping, the reader is referred to the paper by McGinty et al [9].

Remark: Although theoretically the base correlations of a CDO can be uniquely determined from tranche par spreads, in implementation, this may not be possible when some tranche values are not sensitive to a change in correlation. A simple example is a CDO that has a single tranche with its detachment point to be close to 100%. In this case the tranche loss is essentially the total loss of the reference pool and thus the correlation of the reference entities has little effect on the value of the tranche. For this case any number between 0 and 1 can be a candidate of the base correlation.

Base correlation mapping

Base correlations are quoted for standardized tranches. Rigorously speaking, such correlations can only be used to value CDOs that have the same reference baskets and the same tranche structure. However, since the value of a CDO tranche is determined by the loss distribution of the reference portfolio, through proper loss profile matching one can reasonably make use the quoted base correlations by mapping correlations of standardized tranches to bespoke tranches. More generally, one can map base correlations of different CDOs through loss profile matching.

There can be many different ways of matching loss profiles of CDOs. Here are the two most commonly used ones in the industry.

1. Match expected loss ratio. For a given base correlation define the expected loss ratio of a portfolio as:

where

is the portfolio loss for a certain time horizon,

is the indicator of and

stands for the expectation with respect to the risk-neutral probability measure.

Suppose and are two credit portfolios and the base correlation of portfolio corresponding to detachment is . Then there is a unique point such that:

where

and are the expected loss ratios of portfolios and , respectively. For this loss profile matching an obvious choice is to select as the base correlation of portfolio at the detachment . For more details see the paper of Pugachevsky [10].

This way given a base correlation curve of a benchmark synthetic CDO, a bespoke base correlation curve can be built. If the bespoke base correlations at certain detachment points need to be decided, interpolation can be used. The most popular interpolation method is the quadratic method. Note that in FINCAD no extrapolation is allowed. In the interpolation when a detachment point of a bespoke CDO is beyond the base correlation curve range mapped through the above process, the base correlation at the closest end point of the curve will be picked.

2. Shift the base correlation curve. Suppose the base correlation curve (as a function of its detachment point) of portfolio is known. To determine the base correlation curve one possibility is simply by shifting:

where

is the expected loss of portfolio and

is a scaling factor.

FINCAD Functions

CDO valuation given tranche or base correlations

aaCDO_ST_DS_std_p(d_v, contra_d, npa_ini, num_ref_ini, num_dflt, tranche_type, tranche_corr, corr_type, freq_pr, acc, d_rul, dp_type, dp_bskt, intrp_tb, rate_recover, hl, dfstd, intrp, accu_level, calc_para, stat, tranche_list):

Calculates the fair values and other statistics of a credit default swap on standardized CDO tranches using a one-factor Gaussian copula model. The correlation can be either a tranche or a base correlation. Multiple tranches can be valued simultaneously. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCDO_ST_DS2_std_prem_cf(d_v, contra_d, npa_ini, num_ref_ini, num_dflt, tranche_type, tranche_corr, corr_type, freq_pr, acc, d_rul, dp_type, dp_bskt, intrp_tb, rate_recover, hl, dfstd, intrp, accu_level, calc_para, output_type, tranche_list):

Calculates expected cash flows and their present values of a credit default swap on standardized CDO tranches using a one-factor Gaussian copula model. The correlation can be either a tranche or a base correlation. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCDO_ST_DS_p(d_v, contra_d, ref_bskt, tranche_type, tranche_corr, corr_type, freq_pr, acc, d_rul, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para, stat, tranche_list):

Calculates the fair values and other statistics of a credit default swap on synthetic CDO tranches using a one-factor Gaussian copula model. The correlation can be either a tranche or a base correlation. Multiple tranches can be valued simultaneously. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCDO_ST_DS2_prem_cf(d_v, contra_d, ref_bskt, tranche_type, tranche_cpn, corr_type, freq_pr, acc, d_rul, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para, output_type, tranche_list):

Calculates expected cash flows and their present values of a credit default swap on synthetic CDO tranches using a one-factor Gaussian copula model. The correlation can be either a tranche or a base correlation. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

CDO valuation given factor loadings

aaCDO_ST_DS_fl_p(d_v, contra_d, ref_bskt, tranche_type, tranche_cpn, freq_pr, acc, d_rul, modl_type, modl_para, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para, stat, tranche_list):

Calculates the fair values and other statistics of a credit default swap on synthetic CDO tranches using a one-factor copula model. Different reference entities can have different factor loadings. Multiple tranches can be valued simultaneously. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCDO_ST_DS2_fl_prem_cf(d_v, contra_d, ref_bskt, tranche_type, tranche_cpn, freq_pr, acc, d_rul, modl_type, modl_para, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para, output_type, tranche_list):

Calculates expected cash flows and their present values of a credit default swap on synthetic CDO tranches using a one-factor Gaussian copula model. Different reference entities can have different factor loadings. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCDO_ST_DS_dgen_fl_p(d_v, contra_d, ref_bskt, tranche_type, tranche_dgen, cpn_tbl_tran, pr_fix, freq_pr, acc, d_rul, modl_type, modl_para, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para, stat, tranche_list):

Calculates the fair values and other statistics of a credit default swap on synthetic CDO tranches using a one-factor Gaussian copula model. Different reference entities can have different factor loadings and the premium leg can have variable notionals and coupons. Multiple tranches can be valued simultaneously. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCDO_ST_DS2_dgen_fl_prem_cf(d_v, contra_d, ref_bskt, tranche_type, tranche_dgen, cpn_tbl_tran, pr_fix, freq_pr, acc, d_rul, modl_type, modl_para, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para,output_type, tranche_list):

Calculates expected premium cash flows and their present values of synthetic CDO tranches using a one-factor Gaussian copula model. Different reference entities can have different factor loadings and a premium leg can have variable notionals and coupons. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

Calculation of sensitivities with respect to individual entities

aaCDO_ST_DS_risk(d_v, contra_d, ref_bskt, tranche_type, tranche_corr, corr_type, freq_pr, acc, d_rul, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para, tranche_list, out_type, entity_list):

Calculates for each entity the credit spread and other sensitivities of credit default swaps on synthetic CDO tranches. A one-factor Gaussian copula model with either a tranche or a base correlation is used in CDO modeling. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCDO_ST_DS_fl_risk(d_v, contra_d, ref_bskt, tranche_type, tranche_cpn, freq_pr, acc, d_rul, modl_type, modl_para, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para, tranche_list, out_type, entity_list):

Calculates for each name the credit spread and other sensitivities of credit default swaps on synthetic CDO tranches. A one-factor Gaussian copula model with variable factor loadings is used in CDO modeling. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCDO_ST_DS_dgen_fl_risk(d_v, contra_d, ref_bskt, tranche_type, tranche_dgen, cpn_tbl_tran, pr_fix, freq_pr, acc, d_rul, modl_type, modl_para, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para, tranche_list, out_type, entity_list):

Calculates for each name the credit spread and other sensitivities of credit default swaps on synthetic CDO tranches that have variable notionals and coupons. A one-factor Gaussian copula model with variable factor loadings is used in CDO modeling. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

Calibration of base correlations

aaCDO_ST_DS2_base_ic(d_v, contra_d, ref_bskt, tranche_sprd, freq_pr, acc, d_rul, dp_type, dp_bskt, intrp_tb, hl, dfstd, intrp, accu_level, calc_para, tranche_list):

Given par spreads of a credit default swap on synthetic CDO tranches, calculates the base correlations of the tranches. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCDO_ST_DS2_std_base_ic(d_v, contra_d, num_ref_ini, num_dflt, tranche_sprd, freq_pr, acc, d_rul, dp_type, dp_bskt, intrp_tb, rate_recover, hl, dfstd, intrp, accu_level, calc_para, tranche_list):

Given par spreads of a credit default swap on standardized CDO tranches calculates the base correlations of the tranches. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

Calculation of loss distributions

aaCredit_loss_prob2_cum(ref_bskt, prob_bskt, intrp_prob, modl_type, modl_para, low_bound, up_bound, unit_type, time_points, accu_level, calc_method).

Calculates the cumulative loss distribution of a basket of defaultable entities using a one-factor Gaussian copula model. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCredit_loss_prob2_disc(ref_bskt, prob_bskt, intrp_prob, modl_type, modl_para, time_points, accu_level, calc_method, output_type):

Calculates the loss distribution at discrete loss points of a basket of defaultable entities using a one-factor Gaussian copula model. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

aaCredit_tranche_loss_prob_cum(ref_bskt, prob_bskt, intrp_prob, tranche_bound, tranche_type, modl_type, modl_para, time_pts, accu_level, calc_method, output_type):

Given a basket of defaultable entities and tranches of credit losses, calculate cumulative loss distributions and expected loses of the tranches using a one-factor Gaussian copula model. Two calculation methods, a fast Fourier transform and a recursion method, are provided.

Base correlation mapping

aaCDO_ST_baseCorr_map(base_corr, intrp_gen, ref_bmark, ref_custom, intrp_prob, time_m, detach_custom, map_type, rescale, accu_level, calc_method):

Map a benchmark base correlation curve to a bespoke base correlation curve.

Description of Inputs

Functions valuing synthetic CDOs given tranche or base correlations

aaCDO_ST_DS_std_p

aaCDO_ST_DS2_std_prem_cf

aaCDO_ST_DS_p

aaCDO_ST_DS2_prem_cf

Input Argument	Description
d_v	Valuation date
contra_d	The array contra_d is a table of default swap date(s). It can have 2 to 7 entries. The first four entries hold in sequence the terminating date, effective date, odd first coupon date and next-to-terminating coupon date for premium payments. The last three entries hold switch values in sequence: the effective date adjustment method 1. adjust effective date, 2. do not adjust effective date terminating date adjustment method 1. adjust terminating date, 2. do not adjust terminating date) and date generation method 1. backward, 2. forward, 3. IMM dates for premium payments. An IMM date is the 20^th of a month. For example, if the coupon frequency is quarterly, the IMM dates are: March 20, June 20, Sept. 20 and Dec. 20. In the table only the terminating date and the effective date are required. The default values of the third and fourth entries are 0 and that of the last three entries are 1.
ref_npa	Notional of a reference entity. It is used in the standardized tranche functions only.
num_ref_ini	Initial number of reference entities. It is used in the standardized tranche functions only.
num_dflt	Number of defaulted entities up to the valuation date, inclusive. Used in the standardized tranche functions only.
tranche_type	Type of tranche (or loss layer) definition - a switch: 1. proportional (to the initial pool principal), 2. actual dollar amount. These two types are mutually convertible. For example, when a proportional tranche boundary value is converted to an actual dollar amount, the initial tranche notional is multiplied by the boundary value.
tranche_corr	A 5- to 9-column tranche data table. The required columns are (attachment point, detachment point, correlation, notional, coupon). The correlation can either be a tranche correlation or a base correlation, which is determined by the parameter corr_type. The notional can be any number between 0 and the size of the tranche. If it is 0, it will be calculated as the size of the tranche. When a portion of a tranche, instead of the whole tranche, is traded, the actual traded initial notional should be entered here. For the optional columns 6 - 9, their default values are 1 (buy protection), 1 (premium notional is adjusted for loss), 0 and 0, respectively. . The upfront fee in column 8 is the ratio of the upfront payment amount to the outstanding notional of a tranche. An upfront payment is a premium that is paid from the protection buyer to the protection seller at the valuation date. if contra_d(2) < = d_v or at the effective date if contra_d(2) > d_v, i.e., if the CDO is forward-start. If a CDO is not forward-start, an upfront fee is only used in the par or implied spread calculation and is not included in the premium valuation. For a forward-start CDO the upfront payment is included in the premium value of the CDO. Column 9 holds tranche prices, which are used only for the calculation of implied tranche spreads.
corr_type	Correlation type - a switch: 1. tranche correlation; 2. base correlation. The tranche correlation of a tranche is a correlation that can be applied to this tranche. A base correlation of a tranche is a correlation that can be applied to its base tranche, the tranche having an attachment point of 0 and a detachment point being the same as that of the given tranche.
freq_pr	Premium payment frequency, a switch.
acc	Accrual method - a switch.
d_rul	Business day convention - a switch.
dp_type	Default curve type – a switch: 1. a CDS spread curve, 2. a default probability curve
dp_bskt	A default curve, either a default probability or a CDS spread curve, can be time-based (the first column >0 and <1000) or date-based (the first column >=1000). If it is a default probability curve or a time-based CDS spread curve, the first column has time (in years) or dates. If it is a date-based CDS spread curve, the first two columns have the effective and terminating dates of the CDS spreads. The next m columns have default probabilities/CDS spreads, where m is the number of reference entities. These columns must be ordered in the same way as in the reference data table. Times and dates (the terminating dates for a CDS spread curve) must be increasing. For a time-based curve, the time-accrual method can be put in the third entry of the table intrp_tb. Probabilities must be nonnegative, <=1 and nondecreasing. CDS spreads must be nonnegative. For a date-based default probability curve, the probabilities in the first row must be 0 and the first date must be the valuation date if there is one. Excel users may pass term strings to represent time intervals. Term strings must be in the format kB, kD, kW, kM, or kY where k is an integer and B, D, W, M and Y stands for market day, calendar day, week, month and year respectively, e.g., 3M = 3 months.
intrp_tb	A default curve parameter table. The array intrp_tb can have one to six entries. The first entry has the default probability curve interpolation method. The second entry has the default probability curve bootstrapping method (see switch sw_1040 in aaCredit_dfltprob_DSSpred2). It is used only if the default curve type is "par CDS spread curve" ( dp_type =1). If this entry is missing and dp_type=1, the bootstrapping method will be set to method 1 (assuming constant density). The third entry stores the time accrual (day counting) method if the default curve is time-based. If this entry is missing and the default curve is time-based, the time accrual method will be set to 30/360. If the default curve is not time based, this entry will be ignored. The fourth to sixth entries store the effective and terminating date adjustment types and the date generation method of a CDS curve, respectively. These entries are used only when dp_type = 1. Their default values are 1.
rate_recover	Recovery rate of all the credits in the credit pool. Used in the standardized tranche functions only.
hl	Holiday list
dfstd	Discount factor curve. It may be input as a 2-column, multi-row table (column 1 = date, column 2 = discount factor), or as a single cell containing a rate. If input as a single rate, there are three format choices: 1. a rate (1 row, 1 column); or 2. a rate and a rate quotation basis (1 row, 2 columns); or 3. a rate, a rate quotation basis, and an accrual method (1 row, 3 columns). If the basis or accrual is not provided, they are assumed to be annual, actual/365. A 2-column flat-rate discount factor curve is constructed internally using the rate, basis, and accrual method. The last date in the discount factor curve >= last "date" in the default curve and the terminating date of the swap
intrp	Interpolation method of the discount factor curve, a switch.
accu_level	Level of accuracy, a switch. There are four levels of accuracy. For valuation functions the accuracy levels are determined according to the levels of accuracy in integration. The four accuracy levels correspond to four different numbers of points, 10, 30, 50 and 100, respectively, that are used in the Gaussian quadrature of integration. For base correlation calibration, they are determined according to accuracy in integration and levels of tolerance in convergence. The four levels correspond to the above numbers of Gaussian quadrature points and the four tolerance levels, 0.01, 0.001, 0.0001 and 0.00001, respectively.
calc_para	In a fair value calculation function it is a calculation parameters and premium accrual type table. It can have one or three entries. The first entry has the calculation method (1 = FFT, 2 = a recursion method). The second entry is the bump size (in basis points) for DVOX and/or delta calculation. The third entry has the premium accrual type (1 = pay accrued premium upon default, 2 = pay no accrual premium upon default). The default value of the second entry is 1 (basis point) and that of the third entry is 2. In a cash flow function it is a one or two entry table. It is similar to above but without the second entry, the entry of a DVOX bump size.
stat	A list of indicators for fair value and other statistics. It can be any subset of {1,…23}.
tranche_list	List of tranche indicators. It can be any (ascending) subset of {1, 2, the number of rows of the tranche table}.
ref_bskt	Reference data table. It can have 2 or 3 columns. The first two columns have the notional and recovery rate of a reference entity. The optional third column has the default status of an entity (0 = alive, 1 = defaulted). When the column is missing, its default value is 0.
output_type	Output type of a premium cash flow table, a switch, 1. a 6-column table of no-default cash flow table (tranche ID, date, notional, coupon, no-default interest, accrued interest); 2. a 6-column cash flow table (tranche ID, date, no-default interest, expected interest, present value, accrued interest); 3. a 16-column extended cash flow table (tranche ID, effective date, terminating date, notional, coupon, no-default int., no-default fixed premium, total no-default cash flow, accrued interest, expected interest., expected fixed premium, total expected cash flow, present value of interest, present value of fixed premium, total present value, discount factor). The fixed premium column has a non-zero value only if the effective date > the valuation date and the upfront column of the tranche table has a positive value. 4. a default probability curve

For other functions, the definitions for most of their input parameters are the same as those given in the above table. In the following, descriptions will only be given for those parameters that don’t appear in the above table or have different meanings.

Functions valuing synthetic CDOs given factor loadings

aaCDO_ST_DS_fl_p

aaCDO_ST_DS2_fl_prem_cf

aaCDO_ST_DS_dgen_fl_p

aaCDO_ST_DS2_dgen_fl_prem_cf

Input Argument	Description
tranche_cpn	A 4- to 8-column tranche data table. It is similar to the table tranche_corr (see the input parameter table above) except that it doesn’t have a correlation column
modl_type	Model type. There is only one selection in this release: normal (Gaussian) copula model.
modl_para	An array of factor loading(s). It can have up to N (the number of reference entities, i.e., rows of ref_bskt) elements. The ith element is the factor loading of the ith entity. If the table has m< N elements, the factor loading of any jth (j>=m+1) element is set to the value of the mth element. Note that if all the pairs of the entities in the basket have the same correlation, the factor loading of all the entities is the same, which is the square root of the absolute value of the correlation.
tranche_dgen	A tranche table. The required columns are (attachment point, detachment point). The columns 3 - 5 are optional. Their default values are 1 (buy protection), 1 (premium notional is adjusted for loss) and 0% (no upfront fee), respectively. See tranche_cpn above for the description of an upfront fee in column 5.
cpn_tbl_tran	A notional and coupon table. It has 2N+2 columns (effective date, terminating date, notional of tranche 1, coupon of tranche 1, notional of tranche N, coupon of tranche N), where N is the number of tranches. The terminating date must be ascending and the terminating date of the ith row must be the same as the effective date in the (i+1)th row. The notional of a tranche cannot be larger than the tranche size.
pr_fix	A fixed premium payment table. It can be a single entry table or an N+1-column table, where N is the number of tranches. If it is single entry table, all tranches have no fixed premium payments. If it is an N+1-column table, the first column has payment dates and the (i+1)th column the fixed premium payment of the ith tranche, i=1...N. Dates must be ascending and fixed premium payments must be nonnegative. Note also that an upfront payment of a tranche can be treated as a fixed payment, but will be used in valuation only if its payment date is later than d_v.
stat	A list of indicators for fair value and other statistics. It can be any subset of {1, 2,…21}.
output_type	Output type of a premium cash flow table, a switch, 1. an 8-column table of no-default cash flow table (tranche ID, date, notional, coupon, no-default interest, no-default fixed payment, total cash flow, accrued interest); 2. a 5-column expected cash flow table (tranche ID, date, expected interest, expected fixed payment, total expected cash flow). 3. a 5-column present value cash flow table (tranche ID, date, present value of interest, present value of fixed payment, total present value); 4. a 6-column interest cash flow table (tranche ID, date, no-default interest, expected interest, present value, accrued interest); 5. a 16-column extended cash flow table (tranche ID, effective date, terminating date, notional, coupon, no-default interest, no-default fixed payment, total no-default cash flow, accrued interest, expected interest, expected fixed payment, total expected cash flow, present value of interest, present value of fixed payment, total present value, discount factor). 6. a default probability curve

Functions for calculating sensitivities for each entity in the reference pool

aaCDO_ST_DS_std_risk

aaCDO_ST_DS_risk

aaCDO_ST_DS_fl_risk

aaCDO_ST_DS_dgen_fl_risk

Input Argument	Description
out_type	An output type. In aaCDO_ST_DS_std_risk, it is a switch with three values: 1. delta of CDS spread; 2. DVOX of CDS spread. 3. Jump-to-default value (value of default), in aaCDO_ST_DS_risk, it has four values: 1. delta of CDS spread; 2. DVOX of CDS spread; 3. rho of correlation; 4. Jump-to-default value (value of default) and in aaCDO_ST_DS_fl_risk and aaCDO_ST_DS_dgen_fl_risk, it has five values: 1. delta of CDS spread; 2. DVOX of CDS spread; 3. rho of correlation; 4. rho of factor loading; 5. Jump-to-default value (value of default)
entity_list	A list of entities of the reference pool. In aaCDO_ST_DS_std_risk, the entity list is any subset of {1…, number of entities minus number of defaulted entities}, noting that in this case no reference table is given and only the information on the survived references is available. In other functions the list can be any subset of {1, 2,...number of entities}.

Functions for calibrating base correlations

aaCDO_ST_DS2_std_base_ic

aaCDO_ST_DS2_base_ic

Input Argument	Description
tranche_sprd	A tranche table with par spreads or implied spreads given prices. It can have 3 to 5 columns. The required columns are (attachment point, detachment point, par/implied CDS spread). For the optional forth and fifth columns (trading position, upfront payment) their default values are 1 (buy protection), 1 (premium notional is adjusted for loss) and 0%, respectively. The upfront payment in column 5 is a fee that is paid from the protection buyer to the protection seller at the valuation date. An upfront fee, a number between 0 and 1, is the ratio of the upfront fee amount to the outstanding notional of the tranche.
accu_level	Level of accuracy, a switch. There are four levels of accuracy. They are determined according to the levels of accuracy in integration and accuracy of convergence in root finding.
calc_para	It is the same as that of a cash flow function. See the input parameter description of the function aaCDO_ST_DS2_prem_cf for details

Functions for calculating loss distributions

aaCredit_loss_prob2_cum

aaCredit_loss_prob2_disc

aaCredit_tranche_loss_prob_cum

Input Argument	Description
ref_bskt	A two-column reference entity data table (notional, recovery rate)
intrp_prob	The interpolation method of a probability curve; a switch: 1. linear, 2. exponential For both interpolation methods, if the default probability curve is not long enough, the exponential extrapolation method is always used to extend the curve.
prob_bskt	A basket default probability curve. It is a time-based default probability curve (time in years, default probability of entity i, i=1, 2 to N), where N is the number of entities (rows) in the reference entity table ref_bskt.
low_bound	The lower bound of loss range
up_bound	The upper bound of loss range
unit_type	Unit type of loss bounds, a switch: 1. the lower and upper bounds are proportional to the total notional of the reference entity pool; 2. the bounds are in dollar amount.
time_points	Time points (in years) at which loss probabilities are to be calculated
calc_method	Calculation method. a switch 1. a fast Fourier transform 2. a recursion method
output_type	Output type, a switch (used in aaCredit_loss_prob2_disc only) 1. remove negligible loss points 2. keep all loss points Note: If the losses given default of the reference entities are not the same, many extra loss points may be used in the calculation of the loss probabilities. These points are not actual loss points, but may still have very small probability values due to approximation or round off errors. When output type 1 is selected, any loss points with a loss probability less than 0.00000000001 will not be included in the output table.
tranche_bound	A tranche definition and loss range table. It has four columns. The first two columns have the tranche attachment point and detachment point. The third and fourth columns have the lower bound and upper bound of the tranche loss. The two bounds are the range of the loss for which the probability is calculated.

The function for base correlation mapping

aaCDO_ST_baseCorr_map

Input Argument	Description
base_corr	A base correlation table (detachment point, base correlation). Note that a detachment point must be >0 and rigorously increasing, and a base correlation must be >=0 and <1.
intrp_gen	Interpolation method of a base correlation curve – a switch.
ref_bMark	The reference data table of the benchmark CDO. It can be a two-column reference entity data table (notional, recovery rate) or a three column and one row table where the third column has the number of entities in the basket. Note that for a three column table, all the reference entities must have the same notional and the same recovery rate.
ref_custom	Similar to ref_bMark, but it is the reference table for the bespoke CDO.
prob_bmark	Default probability curve of the benchmark reference basket. It can be an N+1 column table with N the number of entities, where the first column has the time horizon and column i+1 has the default probability of entity i. or a two-column table where the second column has the default probabilities of each entity in the basket. Note that for t he second type of table all the entities must have the same default probabilities.
intrp_prob	See above
time_m	Time to maturity of the CDOs
detach_custom	Detachment points of the custom CDO at which base correlations are to be calculated.
map_type	Correlation mapping type, a switch. See the section Base correlation mapping for details.
accu_level	See above
calc_method	See above

Description of Outputs

Functions that calculate fair values and other statistics

aaCDO_ST_DS_std_p

aaCDO_ST_DS_p

aaCDO_ST_DS_fl_p

Output Statistics	Description
1	Fair value
2	Value of payoff
3	Value of premium
4	Accrued interest of premium
5	fair value minus accrued interest
6	Implied spread given price. Case 1: The input tranche data table does not have a column of quoted prices. For this case when the effective date <= the valuation date, the implied spread of a tranche is the (par) CDS spread that makes the fair value minus the accrued interest equal the amount of the tranche’s upfront payment; otherwise, it is the CDS spread that makes the fair value minus the accrued interest equal 0. Case 2. The input tranche data table has a column of tranche prices. For this case when the effective date <= the valuation date, the implied spread of a tranche is the CDS spread that makes the fair value minus the accrued interest equal the amount of the tranche’s upfront payment plus the given tranche price; otherwise, it is the CDS spread that makes the fair value minus the accrued interest equal the tranche price.
7	Ratio of par or implied spread and reference credit spread aggregate. This is the ration of the par spread of a tranche and the sum of the par CDS spreads of reference entities at maturity.
8	Attachment point
9	Detachment point
10	Tranche size
11	Tranche notional
12	Tranche loss up to the valuation date, inclusive
13	Outstanding tranche notional. This is the difference of the initial tranche notional and the accumulated loss to the tranche.
14	DVOX of credit default spreads
15	Rho of recovery rate
16	Rho of correlations ( rho of factor loadings in aaCDO_ST_DS_fl_p)
17	BPV of risk-free rate
18	Theta
19	Number of days that premium accrues
20	Next coupon date
21	Previous coupon date
22	Number of remaining cash flows
23	Implied upfront fee given premium coupon rate. If the CDO is not forward-start, it is the upfront fee per one unit notional of a tranche that makes the upfront payment amount, the fee times the outstanding notional of a tranche, equal the tranche’s fair value minus its accrued interest; otherwise it is the upfront fee that makes the fair value of the tranche minus its accrued interest have a value of 0. If the trading position of a tranche is selling protection, the upfront payment must be multiplied by -1 in the calculation.

The function aaCDO_ST_DS_dgen_fl_p has all the above outputs except the statistics 6 and 7.

The outputs are arranged in a two-dimensional array. Different rows correspond to different statistics. The columns correspond to the tranches in the tranche list tranche_list. The order of the tranches remains the same as in the list.

Functions that calculate CDO premium cash flows

aaCDO_ST_DS2_std_prem_cf

aaCDO_ST_DS2_prem_cf

aaCDO_ST_DS2_fl_prem_cf

table_type	Switch	Description
1	No-default cash flow table	A six column table of cash flows (tranche ID, date, notional, coupon, no-default interest, accrued interest) given that no default occurs during the life of the CDO.
2	Expected cash flow table	A six-column expected and present value cash flow table (tranche ID, date, no-default interest, expected interest, present value, accrued interest)
3	Extended cash flow table	A 16-column extended cash flow table (tranche ID, effective date, terminating date, notional, coupon, no-default interest, no-default fixed payment, total no-default cash flow, accrued interest, expected interest, expected fixed payment, total expected cash flow, present value of interest, present value of fixed payment, total present value, discount factor). The columns corresponding to fixed cash flows store the upfront fee payment only. These columns have zero values unless the CDO is forward-start, i.e., contra_d(2)>d_v.
4	Default probability curve	If the input default curve is a CDS spread curve, it is the default probability curve generated from the spread curve. Otherwise, it is simply the default probability curve standardized from the input curve.

For aaCDO_ST_DS2_dgen_fl_prem_cf, its output table has the following six types

table_type	Switch	Description
1	No-default cash flow table	An 8-column table of no-default cash flow table (tranche ID, date, notional, coupon, no-default interest, no-default fixed payment, total cash flow, accrued interest)
2	Expected cash flow table	A 5-column expected cash flow table (tranche ID, date, expected interest, expected fixed payment, total expected cash flow).
3	Cash flow present value table	A 5-column cash flow present value table (tranche ID, date, present value of interest, present value of fixed payment, total present value);
4	Interest cash flow table	A 6-column interest cash flow table (tranche ID, date, no-default interest, expected interest, present value, accrued interest
5	Extended cash flow table	A 16-column extended cash flow table (tranche ID, effective date, terminating date, notional, coupon, no-default interest, no-default fixed payment, total no-default cash flow, accrued interest, expected interest, expected fixed payment, total expected cash flow, present value of interest, present value of fixed payment, total present value, discount factor). The columns corresponding to fixed cash flows store the upfront fee payment only. These columns have zero values unless the CDO is forward-start, i.e., contra_d(2)>d_v.
6	Default probability curve	See the above table for descriptions.

Functions that calculate sensitivities on individual reference entities

aaCDO_ST_DS_std_risk

aaCDO_ST_DS_risk

aaCDO_ST_DS_fl_risk

aaCDO_ST_DS_dgen_fl_risk

These functions calculate sensitivity statistics with respect to individual entities. Their outputs are two-dimensional arrays. The rows correspond to the entities given in entity_list and the columns to entity IDs (the first column) and tranche IDs given in the table tranche_list. The tranche order is the same as that in the input array tranche_list.

Functions that calculate implied base correlations

aaCDO_ST_DS2_std_ic

aaCDO_ST_DS2_ic

The outputs of these base correlation calibration functions are two dimensional arrays. The rows correspond to the entities and the columns to tranche IDs given in the table tranche_list. The tranche order is the same as that in the input array tranche_list.

Functions for calculating loss distributions

aaCredit_loss_prob2_cum

aaCredit_loss_prob2_disc

aaCredit_tranche_loss_prob_cum

For aaCredit_loss_prob2_cum, its output table, a 1 by N table with N being the number of points in the input array time_points, gives for any time point in this array the probability that the total accumulated loss from time 0 to this time point will fall between the range low_bound and up_bound.

For aaCredit_loss_prob2_dis, the loss probabilities are calculated for a set of loss units. It is an M by N table, where M is the number of loss units and N is the number of points in the input array time_points . Note that M depends on the reference data table ref_bskt. If this table is far from being homogeneous, M can be very large. If the output type is set to “remove negligible loss points”, the extra loss points, which are not the “true” loss points of the reference pool, will not be included in the output table. See more details on the input parameters table of these functions.

For aaCredit_tranche_loss_prob_cum, its outputs are similar to those of the function aaCredit_loss_prob2_cum except that the loss probabilities are for losses of tranches instead of losses of the whole reference portfolio.

Function that maps base correlations of benchmark tranches and bespoke tranches:

aaCDO_ST_baseCorr_map

The output of this function is a base correlation curve for the bespoke CDO tranches.

Calculation of risk statistics

DVOX

1. DVOX on all credit spread curves of the entities in the reference pool.

This statistic is an output of a fair value calculation function, e.g., aaCDO_ST_DS_std_p. It is defined as the change in the fair value per X basis point shift in all the par CDS spread curves of the entities in the reference pool. In more detail, let be the fair value of a CDO derivative. For every entity in the reference pool, add X basis points to its default curve (if the default curve is a default probability curve derive a par spread curve first) and build a new default curve. Then combine these default curves together to form a basket default curve and use this basket default curve as an input to revalue the derivative. Let be the new fair value. The DVOX is then calculated as follows:

2. DVOX on a single reference entity.

This statistic is calculated in functions, that calculate risk statistics exclusively, e.g., aaCDO_ST_DS_std_risk. It is defined similarly as above, but only the default curve of the specified entity is shifted X basis point.

See Remark 2 in the section Recursion Method on fast calculation of DVOXs on individual reference entities.

Delta

Deltas are calculated in functions that output risk statistics exclusively. The delta on the par CDS spread of a reference entity is defined as:

where DVOX at par of a CDS is the DVOX of a CDS with its premium rate being the par premium rate of the CDS.

Rho of recovery rate

1. Rho of recovery rate of the whole reference pool.

This statistic is calculated in fair value calculation functions only. It is the change in the fair value of a CDO derivative per 1% change in the recovery rate. In more detail, let be the fair value of the derivative at a recovery rate . Then:

If the recovery rates of the reference entities differ, the recovery in the above formula should be replaced with a vector of recovery rates and to bump it simply add 0.01 to every component of the vector.

2. Rho of recovery rate of a single reference entity.

This statistic is calculated in functions that only calculate risk statistics. It is defined similarly as above, but only the recovery rate of the specified entity is bumped 1%. If a function, e.g., aaCDO_ST_DS_std_risk, has only one recovery rate as an input for all the reference entities, such a risk statistic is not calculated.

Rho of correlation

This statistic is an output of functions that take in a correlation and calculate fair values. It is the change in the fair value of a CDO derivative per 1% change in the correlation. In more detail, let be the fair value of the CDO derivative at the correlation Then:

Rho of factor loading

1. Rho of factor loadings on all the reference entities.

This statistic is an output of functions that take in factor loadings and calculate fair values. It is the change in the fair value of the CDO derivative per 10% increase in the factor loadings of all the reference entities. In more detail, let be the fair value of the CDO derivative. For all the entities in the reference pool add 0.1 to their factor loadings. Then revalue the derivative to get a new fair value . The rho of factor loading is then calculated as follows:

2. Rho of factor loading on a single reference entity.

This statistic is an output of functions that have a factor loading table as an input and calculate risk statistics only. It is defined similarly as above, but only the factor loading of a specified entity is bumped 10%.

Theta

The theta of a CDO derivative is the change in the fair value of the derivative per one day increase of the valuation date. Let be the fair value of the derivative. Then:

BPV

The BPV (basis point value) of a risk free discount factor curve is the change in the fair value of a CDO derivative when the risk-free discount factor curve is shifted up one basis point. To shift up a discount factor curve simply add one basis point to every point of the corresponding spot rate curve of the discount factor curve.

Jump-to-default value

The jump-to-default value, also known as the value on default (VOD), of a tranche position of a synthetic CDO is the sensitivity of the CDO resulting from an instant default of an entity in the reference basket, keeping all other credit spreads in the reference basket unchanged. It is calculated as the sum of the loss amount of the entity which is assumed to default instantly and the change of the CDO’s fair value after and before the default of the entity. For example, for a protect seller position the loss amount is the negative of the loss given default of the reference entity. For the second part, calculate the fair value of the tranche and then recalculate it with the default status to default of the entity for which its VOD is to be calculated. The second part is simply the difference of the tranche’s value of the second and the first calculation.

Examples

Context examples are given below for various types of CDOs on valuation, calibration of implied base correlations and delta hedging.

Example 1: CDOs with no upfront fee payments and no historical defaults

Consider a standardized synthetic CDO with three tranches, 0%-3%, 3%-10% and 10%-15%. The reference pool has 125 names and each has a notional of 1000000. An investor bought the whole equity tranche and to hedge it he sold the whole mezzanine tranche 3-10%. For the equity tranche, whenever a reference name defaults he will be compensated for the loss and at the same time the notional of his premium payment will be reduced by the lost amount. For the mezzanine tranche, when the accumulated loss reaches 3% of the total notional of the pool, he has to pay to the buyer of the tranche any loss to the tranche and at the same time he will receive less premium payment: the premium notional of the mezzanine tranche will be reduced by the loss to this tranche. Suppose all the reference names have the same flat CDS spread, 0.5% and the same recovery rate, 40%. Other details of the CDO and the required data are shown in the following six tables.

aaCDO_ST_DS_std_p

Argument	Description	Example Data	Switch
d_v	Valuation date	1-Dec-2006
contra_d	CDS date table	see below
ref_npa	Notional of a reference entity	1000000
num_ref_ini	Initial number of reference entities	125
num_dflt	Number of defaulted entities up to the valuation date	0
tranche_type	Type of a tranche (or loss layer) definition	1	proportional
tranche_corr	Tranche table	see below
corr_type	Correlation type	2	base correlation
freq_pr	Premium payment frequency	3	quarterly
acc	Accrual method	2	act./360
d_rul	Business day convention	2	next business day
dp_type	Default curve type	1	par CDS spread curve
dp_bskt	Par CDS spread curve	see below
intrp_tb	Default curve parameter table	see below
rate_recover	Recovery rate	0.4
hl	A holiday list	0
dfstd	The discount factor curve	see below
Intrp	Interpolation method of the discount factor curve, a switch.	1	linear
accu_level	Level of accuracy	2	accuracy level 2
calc_para	Calculation parameter table	see below
stat	List of statistics	{1, 2,…,23}
tranche_list	Tranche list	see below

CDS Date Table

Terminating date	Effective date	First coupon date	Next to last coupon date	Effective date adjustment	Terminating date adjustment	Date generation method
20-Dec-2010	1-Dec-2005	0	0	2	2	3
				Switch: no adjustment	Switch: no adjustment	Switch: IMM date generation

Tranche Table

Attachment	Detachment	Base Correlation	Notional	Coupon	Position	Switch
0	0.03	0.35	0	0.1	1	Buy protection
0.03	0.1	0.38	0	0.02	2	Sell protection
0.1	0.15	0	0	0	1	Buy protection

Reference Entity CDS Spread Table

Effective Date	Terminating Date	CDS Spread of Entity 1	CDS Spread of Entity 2	CDS Spread of Entity 3	CDS Spread of Entity …
1-Oct-2006	1-Oct-2010	0.005	0.005	0.005	…

Default Curve Parameter Table

interpolation of default probability curve	bootstrapping method	accrual method of time-based default curve	effective date adjustment	maturity date adjustment	date generation method
1	1	4	2	2	3

Risk-free Discount Factor Curve

Date	Discount factor
1-Dec-2006	1
1-Jun-2007	0.971285862
1-Dec-2007	0.943396226
1-Dec-2008	0.88999644
1-Dec-2009	0.839619283
1-Dec-2011	0.747258173
1-Dec-2016	0.558394777
1-Dec-2020	0.417265061

Calculation Parameters and Premium Accrual Type Table

Calculation method	Bump size	Premium accrual type
2	1	1

Tranche List

1

2

To value the tranche portfolio call the function aaCDO_ST_DS_std_p,

Results

Output Statistics	Description	Tranche 1	Tranche 2
1	Fair value	125825.44	-30885.52
2	Value of payoff	1278041.58	-679223.77
3	Value of premium	-1152216.14	648338.25
4	Accrued interest of premium	-75000.00	35000.00
5	Fair value minus accrued interest	200825.44	-65885.52
6	Par spread	11.86%	2.21%
7	Ratio of par spread and reference credit spread aggregate	18.98%	3.54%
8	Attachment point	0.00	3750000
9	Detachment point	3750000.00	12500000
10	Tranche size	3750000.00	8750000
11	Tranche notional	3750000.00	8750000
12	Tranche loss up to the valuation date, inclusive	0.00	0
13	Outstanding tranche notional	3750000.00	8750000
14	DVOX of credit default spreads	21067.43	-17928.86
15	Rho of recovery rate	4453.10	60.88
16	Rho of correlations or factor loading	-22976.17	-4168.75
17	BPV of risk-free rate	-18.02	34.01
18	Theta	-1030.13	660.26
19	Number of days that premium accrues	72.00	72.00
20	Next coupon date	20-Dec-2006	20-Dec-2006
21	Previous coupon date	20-Sep-2006	20-Sep-2006
22	Number of remaining cash flows	17.00	17.00
23	Implied upfront fee given coupon	5.355345%	0.752977%

To round check par spreads simply replace the coupons of tranches 1 and 2 in the input tranche data table with the output par spreads and set the statistic list to the single number 5 (the statistic fair value minus accrued interest) and then rerun the function. The outputs for both tranche 1 and 2 should be 0.

To round check the implied upfront fee, simply multiply the size and the implied upfront fee of a tranche. For example for tranche 1 this is:

which equals the fair value minus accrued interest of tranche 1. To round check tranche 2 a negative sign should be applied to the result since the tranche has a position of selling protection.

To view the cash flows of the premium legs call the function aaCDO_ST_DS2_std_prem_cf with table_type = 3. Here is the output premium cash flow table:

Results

ID	Date	No-default Interest	Expected Interest	Present Value	Accrued Interest
1	20-Dec-2006	-94791.67	-94708.61	-94424.71	-75000.00
1	20-Mar-2007	-93750.00	-91271.52	-89701.93	0.00
1	20-Jun-2007	-95833.33	-89988.17	-87143.66	0.00
1	20-Sep-2007	-95833.33	-86953.60	-82985.84	0.00
1	20-Dec-2007	-94791.67	-83251.38	-78308.26	0.00
1	20-Mar-2008	-94791.67	-80692.29	-74829.76	0.00
1	20-Jun-2008	-95833.33	-79133.92	-72322.41	0.00
1	22-Sep-2008	-97916.67	-78457.84	-70628.49	0.00
1	22-Dec-2008	-94791.67	-73766.69	-65438.28	0.00
1	20-Mar-2009	-91666.67	-69387.95	-60711.15	0.00
1	22-Jun-2009	-97916.67	-72103.14	-62151.35	0.00
1	21-Sep-2009	-94791.67	-67904.06	-57678.98	0.00
1	21-Dec-2009	-94791.67	-66117.40	-55346.14	0.00
1	22-Mar-2010	-94791.67	-64404.07	-53170.42	0.00
1	21-Jun-2010	-94791.67	-62758.82	-51089.56	0.00
1	20-Sep-2010	-94791.67	-61176.74	-49097.29	0.00
1	20-Dec-2010	-94791.67	-59653.40	-47187.91	0.00
2	20-Dec-2006	44236.11	44235.64	44103.03	35000.00
2	20-Mar-2007	43750.00	43705.37	42953.77	0.00
2	20-Jun-2007	44722.22	44564.33	43155.66	0.00
2	20-Sep-2007	44722.22	44403.63	42377.45	0.00
2	20-Dec-2007	44236.11	43731.00	41134.44	0.00
2	20-Mar-2008	44236.11	43517.73	40356.04	0.00
2	20-Jun-2008	44722.22	43761.20	39994.42	0.00
2	22-Sep-2008	45694.44	44453.69	40017.63	0.00
2	22-Dec-2008	44236.11	42772.14	37943.08	0.00
2	20-Mar-2009	42777.78	41104.96	35964.88	0.00
2	22-Jun-2009	45694.44	43619.11	37598.74	0.00
2	21-Sep-2009	44236.11	41937.13	35622.19	0.00
2	21-Dec-2009	44236.11	41647.29	34862.48	0.00
2	22-Mar-2010	44236.11	41353.03	34140.04	0.00
2	21-Jun-2010	44236.11	41053.95	33420.45	0.00
2	20-Sep-2010	44236.11	40749.82	32703.70	0.00
2	20-Dec-2010	44236.11	40441.00	31990.24	0.00

Note that in the call to the above cash flow function, the parameter table calc_para must be replaced with the following table

Calculation Parameters and Premium Accrual Type Table

Calculation method	Premium accrual type
2	1

Example 2: CDOs with an upfront fee payment

Suppose in Example 1 the tranches are to be traded and the equity tranche has a 5% upfront fee, a fee that is paid to the protection seller at the valuation date. Then the tranche table should be modified as follows:

Tranche Table

Attachment	Detachment	Correlation	Notional	Coupon	Position	Notional Adjustment Method	Upfront Fee
0	0.03	0.35	0	0.1	1	1	5%
0.03	0.1	0.38	0	0.02	2	1	0
0.1	0.15	0	0	0	1	1	0

Since for a non-forward-start CDO an upfront payment of a tranche is not counted as part of the fair value in a FINCAD CDO function, the outputs are the same as those of example 1 except for the par CDS spreads and the ratios of par spread aggregate which are given below:

Results

Output Statistics	Description	Tranche 1	Tranche 2
6	Par spread	10.12%	2.21%
7	Ratio of par spread and reference credit spread aggregate	16.20%	3.54%

Note that the outputs of tranche 2 are the same as those in example 1. This is obvious since tranche 2 doesn’t have an upfront fee.

Example 3: Implied spreads given tranche prices

Suppose in Example 2 quoted prices of tranche 1 and tranche 2 are 140000 and -250000, respectively To calculate the implied spreads that values the two tranches to these quotes. We can call the same function with the input tranche table being replaced with the following tranche table:

Tranche Table

Attachment	Detachment	Base Correlation	Notional	Coupon	Position	Notional adjustment method	Upfront fee	Quoted price
0	0.03	0.35	0	0.1	1	1	0.05	140000
0.03	0.1	0.38	0	0.02	2	1	0	-250000
0.1	0.15	0	0	0	1	1	0	0

and the statistics list being replaced with the single number 6. The results are shown below

Results

Output Statistic	Description	Tranche 1	Tranche 2
6	implied spread given price quote	8.82%	1.40%

Example 4: CDOs with historical defaults

Suppose in Example 1 five of the reference entities have defaulted. To value the CDO, in the input parameter table of Example 1 set the number of defaults to 5 and replace the default curve with a curve of 120 entities. Suppose the CDS contract dates are given as follows

CDS Dates Table

Terminating date	Effective date	First coupon date	Next to last coupon date	Effective date adjustment	Terminating date adjustment	Date generation method
1-Sep-2010	1-Sep-2004	0	0	1	1	3

Then call the same function to get the following results:

Results

Output Statistics	Description	Tranche 1	Tranche 2
1	Fair value	229881.17	-685813.37
2	Value of payoff	406159.91	-1275745.18
3	Value of premium	-176278.74	589931.81
4	Accrued interest of premium	-15000.00	35000.00
5	Fair value minus accrued interest	244881.17	-720813.37
6	Par spread	0.2518	0.0460
7	Ratio of par spread and reference credit spread aggregate	0.4197	0.0766
8	Attachment point	0.00	3750000.00
9	Detachment point	3750000	12500000
10	Tranche size	3750000	8750000
11	Tranche notional	3750000	8750000
12	Tranche loss up to the valuation date, inclusive	3000000	0
13	Outstanding tranche notional	750000	8750000
14	DVOX of credit default spreads	4560.18789	-25625.675
15	Rho of recovery rate	9039.54836	16855.5286
16	Rho of correlations or factor loading	-8428.8721	9191.3896
17	BPV of risk-free rate	-26.877809	141.064217
18	Theta	-253.6743	952.160623
19	Number of days that premium accrues	72	72
20	Next coupon date	20-Dec-2006	20-Dec-2006
21	Previous coupon date	20-Sep-2006	20-Sep-2006
22	Number of remaining cash flows	16	16
23	Par upfront fee ratio	0.32650823	0.08237867

Example 5: Calibration of base correlations

Suppose in example 1 the par spreads of tranches 1 and 2 are 12% and 2%, respectively, but the base correlations are unknown. To calculate the base correlations that are implied from the par spreads, we can call the function aaCDO_ST_DS2_std_base_ic. The input tables are given in the following three tables.

aaCDO_ST_DS_std_base_ic

Argument	Description	Example Data	Switch
d_v	Valuation date	1-Dec-2006
contra_d	CDS date table	see example 1
num_ref_ini	Initial number of reference entities.	125
num_dflt	Number of defaulted entities up to the valuation date	0
tranche_sprd	Tranche table	see below
freq_pr	Premium payment frequency	3	Quarterly
acc	Accrual method	2	act./360
d_rul	Business day convention	2	next business day
dp_type	Default curve type	1	par CDS spread curve
dp_bskt	Par CDS spread curve	see example 1
intrp_tb	Default curve parameter table	see example 1
rate_recover	Recovery rate	0.4
hl	Holiday list	0
dfstd	The discount factor curve	see example 1
intrp	Interpolation method of the discount factor curve, a switch.	1	Linear
accu_level	Level of accuracy	2	accuracy level 2
calc_para	Calculation method and premium accrual type table	See below
tranche_list	Tranche list	see example 1

Tranche Table

Attachment	Detachment	Spread
0	0.03	0.12
0.03	0.1	0.02

Calculation Method and Premium Accrual Type Table

Calculation method	Premium accrual type
2	1

Other input tables are the same as those in example 1. The results are shown below:

Output Table Results

Tranche ID	Base Correlation	Number of Internal Function Calls	Convergence (0 = converged)
1	0.3442	13	0
2	0.4145	26	0

Note that the calibrated base correlations are slightly different from those in the input tranche table in example 1. This is because the spreads in the tranche table of this example are slightly different from the par spreads of the tranches (see the outputs of example 1 corresponding to the statistic 6).

Example 6: A CDO of variable reference notionals and recovery rates

Suppose in example 1 the CDO is not standardized and its reference entity pool has 10 names with their notionals and recovery rates given as in the following table:

Notional and Recovery Rate Table

Notional	Recovery Rate
1000000	0.4
1000000	0.4
1000000	0.4
2000000	0.4
2000000	0.4
2000000	0.3
2000000	0.3
3000000	0.3
3000000	0.3
3000000	0.3

Suppose also that CDS spread curves of the entities are:

Par CDS Spread Curve

		Names
Dates		1	2	3	4	5	6	7	8	9	10
Effective Date	Maturity Date
1-Dec-2006	1-Dec-2010	0.02	0.02	0.01	0.03	0.04	0.02	0.05	0.03	0.02	0.02

Moreover, the coupon rates of tranche 1 and tranche 2 are 5% and 2%, their tranche correlations are 0.35 and 0.38, and their upfront fees are 30% and 10%, respectively. To value the CDO, call the function aaCDO_ST_DS_p with the following input tables:

aaCDO_ST_DS_p

Argument	Description	Example Data	Switch
d_v	Valuation date	1-Dec-2006
contra_d	CDS date table	see example 1
ref_bskt	Reference notional and recovery rate table	see above
tranche_type	Type of a tranche (or loss layer) definition	1	proportional
tranche_corr	Tranche table	see below
corr_type	Correlation type	1	tranche correlation
freq_pr	Premium payment frequency	3	Quarterly
acc	Accrual method	2	act./360
d_rul	Business day convention, a switch.	2	next business day
dp_type	Default curve type	1	par CDS spread curve
dp_bskt	Par CDS spread curve	see above
intrp_tb	Default curve parameter table	see example 1
hl	holiday list	0
dfstd	The discount factor curve
intrp	Interpolation method of the discount factor curve, a switch.	1	linear
accu_level	Level of accuracy	2	accuracy level 2
calc_para	Calculation parameter table	See example 1
stat	Stat list	{1, 2,…,23}
tranche_list	Tranche list	1, 2

Tranche Table

Attachment	Detachment	Tranche Correlation	Notional	Coupon	Position	Notional adjustment method	Upfront fee
0	0.1	0.35	0	0.05	1	1	0.3
0.1	0.2	0.38	0	0.02	2	1	0.1
0.2	0.3	0	0	0	1	1	0

Results

Output Statistics	Description	Tranche 1	Tranche 2
1	Fair value	684586.66	-336820.42
2	Value of payoff	960530.33	-472931.00
3	Value of premium	-275943.66	136110.58
4	Accrued interest of premium	-20000.00	8000.00
5	Fair value minus accrued interest	704586.66	-344820.42
6	Par spread	0.07	0.04
7	Ratio of par spread and reference credit spread aggregate	0.27	0.16
8	Attachment point	0.00	2000000.00
9	Detachment point	2000000.00	4000000.00
10	Tranche size	2000000.00	2000000.00
11	Tranche notional	2000000.00	2000000.00
12	Tranche loss up to the valuation date, inclusive	0.00	0.00
13	Outstanding tranche notional	2000000.00	2000000.00
14	DVOX of credit default spreads	2341.28	-1876.69
15	Rho of recovery rate	5504.05	871.72
16	Rho of correlations or factor loading	-8042.14	727.70
17	BPV of risk-free rate	-113.00	73.29
18	Theta	-515.42	335.54
19	Number of days that premium accrues	72.00	72.00
20	Next coupon date	39071.00	39071.00
21	Previous coupon date	38980.00	38980.00
22	Number of remaining cash flows	17.00	17.00
23	Implied upfront fee given premium coupon	0.35229333	0.172410211

Example 7: A CDO with variable reference notionals, recovery rates and defaulted entities

Suppose in Example 6, the reference entity 4 defaulted sometime after the CDO started trading. Suppose also the contract dates are given as in Example 4. Then the reference table needs a default status column as is shown below:

Reference Notional and Recovery Rate Table

Notional	Recovery Rate	Default Status (0=no, 1 =yes)
1000000	0.4	0
1000000	0.4	0
1000000	0.4	0
2000000	0.4	1
2000000	0.4	0
2000000	0.3	0
2000000	0.3	0
3000000	0.3	0
3000000	0.3	0
3000000	0.3	0

Results

Output Statistics	Description	Tranche 1	Tranche 2
1	Fair value	298361.62	-530327.37
2	Value of payoff	400815.95	-650501.47
3	Value of premium	-102454.34	120174.10
4	Accrued interest of premium	-8000.00	8000.00
5	Fair value minus accrued interest	306361.62	-538327.37
6	Par spread	0.09	0.08
7	Ratio of par spread and reference credit spread aggregate	0.37	0.35
8	Attachment point	0.00	2000000.00
9	Detachment point	2000000.00	4000000.00
10	Tranche size	2000000.00	2000000.00
11	Tranche notional	2000000.00	2000000.00
12	Tranche loss up to the valuation date, inclusive	1200000.00	0.00
13	Outstanding tranche notional	800000.00	2000000.00
14	DVOX of credit default spreads	927.48	-1958.66
15	Rho of recovery rate	10250.25	4449.53
16	Rho of correlations or factor loading	-3571.70	3679.82
17	BPV of risk-free rate	-45.21	92.46
18	Theta	-216.62	382.34
19	Number of days that premium accrues	72.00	72.00
20	Next coupon date	39071.00	39071.00
21	Previous coupon date	38980.00	38980.00
22	Number of remaining cash flows	16.00	16.00
23	Implied upfront fee given premium coupon	0.38295202	0.269163684

Example 8: A CDO with factor loadings

Suppose in example 6 neither correlation nor base correlation is available, but the betas of the reference names can be obtained. To value the CDO we can scale the betas so that they are between -1 and 1 and use them as factor loadings. Suppose the factor loadings are shown as in the following table:

Model Parameter Table

Name	1	2	3	4	5	6	7	8	9	10
Factor Loading	0.5	0.6	0.4	0.7	0.8	0.2	0.6	0.5	0.3	0.1

To value the CDO call the function aaCDO_ST_DS_fl_p with the following input table:

aaCDO_ST_DS_fl_p

Argument	Description	Example Data	Switch
d_v	Valuation date	1-Dec-2006
contra_d	CDS date table	see below
ref_bskt	Reference notional and recovery rate table	see example 7
tranche_type	Type of a tranche (or loss layer) definition	1	Proportional
tranche_corr	Tranche table	see below
corr_type	Correlation type	1	tranche correlation
freq_pr	Premium payment frequency	3	Quarterly
acc	Accrual method	2	act./360
d_rul	Business day convention, a switch.	2	next business day
modl_type	Model type	1	normal copula
modl_para	Model parameter table	see above
dp_type	Default curve type	1	par CDS spread curve
dp_bskt	Par CDS spread curve	see example 6
intrp_tb	Default curve parameter table	see example 1
hl	A holiday list	0
dfstd	The discount factor curve
intrp	Interpolation method of the discount factor curve, a switch.	1
accu_level	Level of accuracy	2	accuracy level 2
calc_para	Calculation parameter table	see above
stat	Stat list	{1, 2,…,23}
tranche_list	Tranche list	1, 2

Tranche Table

Attachment	Detachment	Notional	Coupon	Position	Notional adjustment method	Upfront fee
0	0.1	0	0.05	1	1	0.3
0.1	0.2	0	0.02	2	1	0.1
0.2	0.3	0	0	1	1	0

Results

Output Statistics	Description	Tranche 1	Tranche 2
1	Fair value	782977.281	-352400.7328
2	Value of payoff	1048989.81	-488908.6835
3	Value of premium	-266012.53	136507.9507
4	Accrued interest of premium	-20000	8000
5	Fair value minus accrued interest	802977.281	-360400.7328
6	Par spread	0.09125344	0.04496355
7	Ratio of par spread and reference credit spread aggregate	0.35097478	0.172936731
8	Attachment point	0	2000000
9	Detachment point	2000000	4000000
10	Tranche size	2000000	2000000
11	Tranche notional	2000000	2000000
12	Tranche loss up to the valuation date, inclusive	0	0
13	Outstanding tranche notional	2000000	2000000
14	DVOX of credit default spreads	2675.09	-2109.10
15	Rho of recovery rate	6342.18	2393.29
16	Rho of correlations or factor loading	-92586.93	520.43
17	BPV of risk-free rate	-130.48	81.37
18	Theta	-548.46	379.61
19	Number of days that premium accrues	72	72
20	Next coupon date	20-Dec-06	20-Dec-06
21	Previous coupon date	20-Sep-06	20-Sep-06
22	Number of remaining cash flows	17	17
23	Implied upfront fee given premium coupon	0.40148864	0.180200366

Example 9: Deltas with respect to the CDS spreads of individual entities

Suppose in example 8 we want to calculate the deltas of name 1 and name 2. Then we should use the function aaCDO_ST_DS_risk. Its input table is similar to that of aaCDO_ST_DS_p except that it has an output type parameter and a name list table and has no statistics list. For this example, the output type is 1 (delta) and the entity list is {1, 2}. The following are the outputs:

Output Table Results

Name	Tranche 1	Tranche 2
1	-0.42511	0.290643661
2	-0.38811	0.300931327

From the above table we see that to hedge the equity tranche the trader should sell CDS of both name 1 and name 2, and to hedge tranche 2 the holder of the tranche portfolio should buy CDSs of the names. The net hedge will be to sell CDSs of name 1 and name 2 with notionals of:

and

respectively.

Example 10: Bespoke tranches

To value bespoke tranches of synthetic credit risk, one often makes the standardized tranches, such as tranches of iTraxx or Dow Jones CDX, benchmarks for calibrating credit models. Function aaCDO_std_baseCorr_map can be used to calibrate base correlations of bespoke tranches. Suppose that the reference pool and base correlations of a benchmark CDO are given as follows:

Benchmark references table

notional of an entity	recovery rate	number of entities
100	40.000%	3

Benchmark basket default probability curve

time in years	default probability of entity 1	default probability of entity 2	default probability of entity 3
1	0.022032	0.0317	0.0219
2	0.046242	0.0655	0.0461
3	0.07266	0.1022	0.0725
4	0.101233	0.142	0.101
5	0.131885	0.1752	0.1315

Benchmark base correlation table

detachment point	base correlation
0.03	0.07
0.07	0.11
0.15	0.15

Also suppose that reference pool and tranche detachment points of the bespoke CDO are described in the following tables:

Bespoke reference table

notional amount	recovery rate
100	0.3
200	0.4
300	0.4

Bespoke basket default probability curve

time in years	default probability of entity 1	default probability of entity 2	default probability of entity 3
1	0.022032	0.0317	0.0219
2	0.046242	0.0655	0.0461
3	0.07266	0.1022	0.0725
4	0.101233	0.142	0.101
5	0.131885	0.1752	0.1315

Detachment points of bespoke tranches

detachment point

0.05

0.1

0.2

Input details are described in the following table:

aaCDO_ST_baseCorr_map

Argument	Description	Example Data	Switch
base_corr	benchmark base correlation table	See above
intrp_gen	base correlation interpolation method	1	linear
ref_bMark	benchmark references table	See above
ref_custom	bespoke references table	See above
prob_bMark	benchmark basket default probability curve	See above
prob_custom	bespoke basket default probability curve	See above
intrp_prob	interpolation method of probability curve	1	linear
time_m	time to maturity of CDO	2.5
detach_custom	detachment points of bespoke tranches	See above
map_type	mapping method of base correlations	1	match loss profiles
rescale	scaling factor (used when mapping type = 2 only)	100%
accu_level	level of accuracy	1	accuracy level 1
calc_method	calculation method	2	recursion method

Then the base correlations of the bespoke tranches can be estimated by calling aaCDO_baseCorr_map:

Output table

detachment point	base correlation
0.05	0.088824283
0.1	0.12196818
0.2	0.15

Example 11: Tranche loss distribution

The tranche value of a synthetic CDO is based on the tranche loss distribution. Function aaCredit_tranche_loss_prob_cum can be used to calculate the tranche loss distribution. Suppose, in Example 10, one likes to know the loss distribution of the bespoke equity tranche. To use this function, we specify the following inputs:

aaCredit_tranche_loss_prob_cum

Argument	Description	Example Data	Switch
ref_bskt	notional and recovery rate table	see above
prob_bskt	default probability curve	see below
intrp_prob	interpolation method of probability curve	1	linear
tranche_bound	tranche and loss range table	see below
tranche_type	type of tranche definition	1	tranche boundaries are proportional to the total notional
modl_type	model type	1	normal copula
modl_para	model parameters	see below
time_points	time points (in years) at which loss probabilities are to be calculated	see below
accu_level	level of accuracy	1	accuracy level 1
calc_method	calculation method	2	recursion method
output_type	output type	1	tranche loss distribution

Tranche and loss range table

attachment point	detachment point	lower loss bound	upper loss bound
0	0.05	0	0.05

Time points (in years) at which loss probabilities are to be calculated

time length
(in years)

0.5

1

5

Note that the factor loading is the square root of the correlation.

Model parameters

factor loading

0.298034029690451

Calling aaCredit_tranche_loss_prob_cum, we get the following loss distributions for the equity tranche:

Output table

attachment point	detachment point	probability of loss from time 0 to time point 1	probability of loss from time 0 to time point 2	probability of loss from time 0 to time point N
0	0.05	0.0370100730996259	0.0727404397902457	0.297664746815289

References

[1] Andersen, L. Sidenius, J. and Basu, S. (November 2003), ‘All your hedges in one basket,’ Risk, 67-72.

[2] Andersen L., Sidenius, J. et al, (2004), The Bank of America Guide to Advanced Correlation Products, Incisive Financial Publishing.

[3] Brasch, H. (2004), ‘A note on efficient pricing and risk calculation of credit basket products’, TD Securities.

[4] Burtschell, X., Gregory, J. and Laurrent, J., (2005), A Comparative Analysis of CDO Pricing models, preprint.

[5] Debuysscher, A. and Szego, M., (2003), The Fourier Transform Method – Technical document, Moody’s Investors Service.

[6] Fabozzi, F. and Goodman, L., (2001), Investing in Collateralized Debt Obligations, New Hope, J. Fabozzi Associates.

[7] Luo L., (May 2006), ‘CDOs and Search for Simplicity’, Wilmott Magazine,

[8] Merrill Lynch, (June 2004), ‘Standardized Tranches’, Credit Derivatives Research.

[9] McGinty, L., Beinstern, E. and Watts, M., (2004), Credit Correlation, A Guide, JP Morgan publication.

[10] Pugachevsky, D. (2005), ‘How to value bespoke synthetic CDO tranches consistent with standard ones’, preprint.

[11] Schonbucher, P., (2003), Credit Derivatives Pricing, Models, Pricing and Implementation, Wiley Finance.

Disclaimer

With respect to this document, FinancialCAD Corporation (“FINCAD”) makes no warranty either express or implied, including, but not limited to, any implied warranty of merchantability or fitness for a particular purpose. In no event shall FINCAD be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of this document or the information contained in it. This document should not be relied on as a substitute for your own independent research or the advice of your professional financial, accounting or other advisors.

This information is subject to change without notice. FINCAD assumes no responsibility for any errors in this document or their consequences and reserves the right to make changes to this document without notice.

Copyright

Copyright © FinancialCAD Corporation 2008. All rights reserved.