***Markowitz's theory
2. A Number of Small Bets vs. a Single Large Bet
2-1. Variance & Standard Deviation: measurement of the riskiness
*negative correlation or very low positive correlations
*independent trials: a special case of zero correlations => as you add more assets, the volatility diminishes
2-2. Moody's diversity index
- Creating & measuring (the degree of diversification) ABSs (asset-based securities)
- Diversity score is based on the concept of low or zero correlation. It assumes that firms in the same industry tend to be correlated and that firms in different industries are less correlated (zero). The scoring system for diversity takes account of the number of different risks in the population and their degree of independence based on industry membership.
- The greater a portfolio's diversity score, the greater its diversification
- The diversity score is used in association with the BET(Binomial Expansion Technique) to model the portfolio as consisting of identical, uncorrelated assets with the same default probability.
**32 industries in Moody's classification system
e.g.) 2% max per name & 5% max per industry
=> at least 100 / 2 = 50 companies in the portfolio
if you place three companies from the same industry, the industry exposure 2% x 3 = 6%.
so, the diversity score will be 1.5 x (50 / 2) = 37.5
@ least 100/5 = 20 industries in the portfolio
there is a limit of 2% per name, at least three companies must be included from each industry.
so, the diversity score will be 2 x (100 / 5) = 40
==> The filnal diversity score is the lower of 37.5 and 40, that is, 37.5
2-3. Regional Diversification Scoring System
Coopers & Lybrand measured regional diversification based on the geographical distribution of the mortgage loans in the portfolio => regional diversification does not necessrily reduce portfolio risk. => a diversity index scoring system may oversimplify the issue of portfolio risk and may generate misleading results.
3. Standard Portfolio Approach to Credit Portfolio
There are practical difficulties in applying the standard portfolio model to investments, and to fixed income or loan asset portfolio in particular.
3-1. Correlation Estimates
Several implementation issues associated with the correlation matrix
- Size of correlation matrix: as assets increase, the size of the correlation matrix becomes quite large
- Correlation variable: e.g.) fixed income portfolio: the appropriate correlation variable is not clear => factors that affect returns (interest rate volatility, default probabilities, rating categories, spreads, etc) are subject to forecast error => the correlation matrix may not be very meaningful if the forecast errors is high
- Unstable correlations: most correlations are calculated without holding other structural variables constant (unconditional) => unstable correlation matrix => a portfolio strategy based on a given correlation matrix may not be suitable if the matrix changes
**correlation = cov(Ri,Rm) / [std. dev(Ri) x std. dev(Rm)]
**forecast error: provides a useful estimate of the average forecast error of the model
3-2. Distributions of Returns
- fixed income investments and loans: skewed toward losses
- fat-tail problem: the probability density of extreme value outcomes is greater than implied by the normal distribution
3-3. Holding Period
Single-period problem: Intra-period trading is not considered => transaction costs are not adequately considered especially when the assets are very illiquid, such as loans & many bonds.
3-4. Lack of Price Discovery
A continuum of prices is not as easy to obtain for debt securities as it is for equity securities. Debt security market is quite thin market.
3-5. Lack of Good Data
- Reliable data on the variables that affect value are difficult to obtain and analysts must frequently transform available data into a more meaningful form.
- Problem with the proper identification of the geographic location and industry classification for companies.
- one SIC(standard industry code) for a given firm => problem with the firm that have a significant portion of its operations in other industries
4. Current Portfolio Approaches
At the simplest level, institutions apply a slicing and dicing approach, whereby they set limits on the various types of concentration and monitor their exposure accordingly.
The limites are generally developed based on one or more of the following approaches:
- Historical or recent loss experience
- Standards based on maximum loss tolerance relative to capital
- Risk-adjusted return on capital
***recently introduced
- Diversity index
- Expected & unexpected losses
- RAROC
- RAROC2020
- Credit VaR
- Variants of the Sharp Ratio
4-1. Expected vs. Unexpected Losses
- Expected Losses: long-run average losses & reflected in pricing
associated with the mean of the loss distribution of a loan or a portfolio
*Vojta's expected loss
Expected Loss (%) = Probability of default x Default severity
where:
default severity = a percent of the loan that is lost
**Moody's diversity score and expected & unexpected losses
The higher the diversity score, the lower are the unexpected loss coverage requirments.
- Unexpected Losses: maximum potential loss or maixmum loss at a given level of confidence
unexpected losses are derived based on default probabilities and recovery rates => this approach is driven more by intrinsic value (model value) than by market value.
The concept of unexpected loss attempts to takes account of the uncertainty associated with the ex ante probability of default. The standard deviation of this default rate may be used as a proxy for the unexpected aspect of default => two standard deviation is used to derive the unexpected value of the loss to arrive the capital to be set aside.
5. Optimization of Capital Usage
Economic capital is the resource needed to cushion unexpected losses. It is important to optimize the amount of capital so that the firm can operate with more efficiency.
Optimization problem = to minimize expected losses in the portfolio, subject to the two constraints:
- The sum of the unexpected losses is less than or equal to the risk capital (the total amount of capital)
- The assets are allocated from among only those available.
**minimization function
where:wi: weight in each bond rating
pi: probability of default
ri: recovery rate, 1 - ri= loss given default
C: capital rtio = 8% (per Basel II)
The sum of this quantity taken over all the bond ratings and divided by the capital ratio gives the total value for the unexpected from the portfolio (=economic capital).
Three constraints for the objective function are:
***Expected Shortfall
http://www.youtube.com/watch?v=eHGJFOjyzr4
** Economic capital is impacted directly by the width of the loss distribution. In turn, the loss distribution is impacted by the amount of portfolio diversification, the type of risk measured, the type of loss being measured, the fundamental assumptions made, and the time horizon
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