Dr. Krzysztof Ostaszewski, FSA, CFA. MAAA …math.illinoisstate.edu/krzysio/var-rbc.pdf · Dr....

Dr. Krzysztof Ostaszewski, FSA, CFA. MAAA Professor of Mathematics and Actuarial Program Director Illinois State University, Normal, IL 61790-4520, U.S.A. URL: http://www.math.ilstu.edu/krzysio/ E-mail: [email protected] Fax: 1-309-438-5866 Some comments about inadequacy of mathematical modeling revealed by the current credit crisis

© 2009 by Krzysztof Ostaszewski. All rights reserved. These notes are authored by Krzysztof Ostaszewski and are meant for educational use by Commission for Financial Supervision in Poland. No reproduction for any other purpose than educational use by the Commission for Financial Supervision is permitted.

1

Risk-Based Capital Throughout the 1990s, bank regulators worldwide and insurance regulators in the United States have developed a system of risk-based capital requirements. Furthermore, risk-based capital requirements are a part of the Solvency II directive under implementation in the European Union. Recent credit crisis has provided a challenge to the entire system of risk-based capital requirements, and changes to the system may be implemented, so this is a dynamic environment that should continued to be studied. This note will discuss several risk-based capital regimes, but the reader should be advised that additional references should be sought for more detailed analysis and implementation. The following will be outlined:

- Basel I and Basel II banking capital requirements. - Solvency II European Union directive for insurance. - United States Life Insurance Risk-Based Capital. - United States Property/Casualty Risk-Based Capital.

Banking capital requirements Banking capital requirement have been created for the purpose of aligning solvency requirements with risks undertaken by banks. In the regulation, categorization of assets and capital is standardized so that it can be risk weighted. Internationally, the Basel Committee on Banking Supervision housed at the Bank for International Settlements influence each country's banking capital requirements. In 1988, the Committee decided to introduce a capital measurement system commonly referred to as the Basel Accord, now known as Basel I. This framework is now being replaced by a new and significantly more complex capital adequacy framework commonly known as Basel II, created in June 2004. While Basel II significantly alters the calculation of the risk weights, it leaves alone the calculation of the capital. The capital ratio is the percentage of a bank's capital to its risk-weighted assets. Weights are defined by risk-sensitivity ratios whose calculation is dictated under the relevant Accord. Each national regulator has a very slightly different way of calculating bank capital, designed to meet the common requirements within their individual national legal framework. Most countries, and Basel I and II, impose lending limit as a multiple of a banks capital (also affected by the inflation rate). Traditionally, banks’ lending was determined by the 5 C's of Credit, Character, Cash Flow, Collateral, Conditions and Capital. This has been replaced by one single criterion. Examples of national regulators implementing Basel II include the FSA in the UK, BAFIN in Germany, and OSFI in Canada. An example of a national regulator implementing Basel I, but not Basel II, is in the United States. However, the Federal Reserve Bank supports implementation of Basel II in the United States, and is working on it. As of now, depository institutions in the United States are subject to risk-based capital guidelines issued by the Board of Governors of the Federal Reserve System


2

(FRB). These guidelines are used to evaluate capital adequacy based primarily on the perceived credit risk associated with balance sheet assets, as well as certain off-balance sheet exposures such as unfunded loan commitments, letters of credit, and derivatives and foreign exchange contracts. The risk-based capital guidelines are supplemented by a leverage ratio requirement. To be adequately capitalized under federal bank regulatory agency definitions, a bank holding company must have a Tier 1 capital ratio of at least 4%, a combined Tier 1 and Tier 2 capital ratio of at least 8%, and a leverage ratio of at least 4%, and not be subject to a directive, order, or written agreement to meet and maintain specific capital levels. To be well-capitalized under federal bank regulatory agency definitions, a bank holding company must have a Tier 1 capital ratio of at least 6%, a combined Tier 1 and Tier 2 capital ratio of at least 10%, and a leverage ratio of at least 5%, and not be subject to a directive, order, or written agreement to meet and maintain specific capital levels. The three pillars of Basel II Pillar 1 Minimum capital requirements: set of rules and methodologies for calculating the minimum capital to be held against key risks: credit, market, and operational. Pillar 2 Supervisory review process: four principles that outline the expectations about the role and responsibilities of banks, their boards, and their supervisors in identifying and assessing all the risks they face and in holding sufficient capital in line with their risk profile. Pillar 2 is a strong push for strengthening both risk management and bank supervision systems. Pillar 3 Market discipline: It requires banks to disclose sufficient information on their Pillar 1 risks to enable other stakeholders to monitor bank conditions. The first pillar The first pillar deals with maintenance of regulatory capital calculated for three major components of risk that a bank faces: credit risk, operational risk and market risk. Other risks are not considered fully quantifiable at this stage.

- The credit risk component can be calculated in three different ways of varying degree of sophistication, namely standardized approach, Foundation IRB and Advanced IRB. IRB stands for "Internal Rating-Based Approach".

- For operational risk, there are three different approaches - basic indicator approach or BIA, standardized approach or STA, and advanced measurement approach or AMA.

- For market risk the preferred approach is VaR (value at risk). Value-at-Risk In 1996 the Basel Committee published the 1996 BIS Amendment, which was implemented in 1998, and required banks to hold capital for market risk as well as for credit risk. The amendment makes a distinction between a bank’s trading book and banking book. The banking book consists of loans and is usually not revalued on a regular basis for managerial and accounting purposes. The trading book consists of different instruments that are traded by the bank: stocks, bonds, swaps, forwards,


3

contracts, options, etc., and is normally revalued daily. The amendment calculates the required risk-based capital using the Value-at-Risk (VaR) measure over 10 trading days at the 99% confidence level of probability. The capital requirement is k times the VaR (with a possible adjustment for specific risks), where the multiplier k is chosen on a bank-by-bank basis and must be at least 3. For a bank with an excellent VaR procedure the multiplier may be set at the minimum value of 3, for others it is typically higher. For a given portfolio, Value-at-Risk (VaR) is defined as the number VaR such that:

Pr Portfolio loses more than VaR within time period t( ) <α. all this given:

- amount of time t, and - probability level α (confidence level) - all this under normal market conditions!

Example:

Probability ($1 million in S&P 500 Index will decline by more than 20% within a year) < 10%

means that VaR = $200,000 (20% of $1,000,000) with α = 0.10, t = 1 year (typically time period is much shorter, expressed in days). VaR is typically a dollar amount, not %. Conditional VaR (C-VaR) is defined as the expected loss during an N-day period, conditional that we are in the (100 - X)% left tail of the distribution. This concept overcomes problems with distributions with two peaks, one of which is in the left tail, or other type of possibilities of an irregular tail of the distribution. But it is much less popular than VaR (after all, regulators require VaR). Basic VaR methodologies:

- Parametric; - Historical; - Simulation.


4

How is parametric done? - Estimate historical parameters: asset returns, variances and covariances, for all

asset classes, or assets comprising the portfolio; - Calculate portfolio expected return and standard deviation; - Estimate VaR assuming normal distribution of portfolio return. Typically assumes

normality and serial independence. - Wrong theoretically, but practitioners do not care. Problems with estimating

parameters, especially volatility. How is historical done?

- Assemble and maintain historical database; - Use historical data as the future distribution. - Also wrong: What if the future isn’t what it used to be? But … if generalized to

the nonparametric method of bootstrap (resampling), may be the best there is. Of course, bootstrap is not used in practice, because practitioners generally do not know what it is.

How is simulation done?

- Specify distributions of model input factors, - Use Monte Carlo simulation for factors, - Combine them into global outcome, get a probability distribution. - Assumptions on factors crucial. If one can get that distribution ideally, this may

be an ideal method. Outline of the parametric VaR calculation approach In the calculation, we generally assume normal distribution, and need to estimate the standard deviation of daily returns, i.e., daily volatility. Most estimates available are annual. Here is how we handle volatility per year versus volatility per day:

σ yr = σ day 252, σ day =σ yr

252.

There are 252 trading days, and studies indicate that volatility on non-trading days is minimal if nonexistent. But this is an issue of market efficiency, which is not fully settled. Let us now present a basic parametric VaR calculation. Consider $10 million in IBM stock, N = 10 days (two trading weeks), and X = 99% confidence level. Assume daily volatility of 2%, i.e., daily standard deviation (SD) of $200,000. Assume successive days’ returns are independent, then over 10 days SD is

10 ⋅$200,000 ≈ $632,456. It is customary to assume in VaR calculations that the expected return over period considered is 0% (because in practice calculations are done over very short periods). It is also customary to assume normal distribution of returns. Because the 1st percentile of the standard normal distribution is –2.33 (and the 99-th percentile is 2.33), VaR of this IBM stock portfolio is:

2.33 ⋅$632,456 = $1,473,621.


5

Now consider a $5 million in AT&T stock (symbol T). Assume its daily volatility is 1%. Then its 10-day SD is $50,000 10. Its VaR is

2.33 ⋅$50,000 10 ≈ $368,405. Now combine the two assets in a portfolio and assume that the correlation of their returns is 0.7. We have

σ X+Y = σ X2 +σY

2 + 2ρσ XσY , and we get 10-day volatility σ X+Y = $751,665. Thus 10-day 99% VaR of the combined IBM + T portfolio is

2.33 ⋅$751,665 = $1,751,379. The amount

$1,473,621+ $368,405( ) − $1,751,379 = $90,647. is the VaR benefit of diversification. A linear model Consider a portfolio of assets such that the changes in the values of those assets have a multivariate joint normal distribution. Let Δxi be the change in value of asset i in one day, and α i be the allocation to asset i, then for the change in value of the portfolio

ΔP = α ii=1

n

∑ Δxi

is normally distributed because of the multivariate joint normal distribution. Since E Δxi( ) = 0 is assumed for every i, E ΔP( ) = 0.We also have:

σ i = Var Δxi( ),ρij = Corr Δxi ,Δx j( )

σ P2 = ρij

j=1

n

∑i=1

n

∑ α iα jσ iσ j

and the 99% VAR for N days is: 2.33 σ P N . How bonds/interest rates are handled Duration estimate gives

ΔP = −D ⋅P ⋅ Δy. Let σ y be the yield volatility per day. What is it? One way to look at it: SD of Δy. Then

σ P = D ⋅P ⋅σ y .

Another way of looking at it: SD of Δyy, where y is the zero coupon bond yield for

maturity D. Then

ΔP = −D ⋅P ⋅ y ⋅Δyy,

so that σ P = D ⋅P ⋅ y ⋅σ y .


6

Can include convexity in this approach (in addition to duration), but this still does not account for nonparallel yield curve shifts. Cash flow mapping But yield and volatility data is not available for most bonds, which are infrequently traded. The only bonds that are frequently traded are U.S. Federal Government Bonds. Alternative approach has been developed in handling interest rates, dealing with the problem of not having data on volatility of most bonds. In this approach, we use prices and volatilities of Treasury zero-coupon bonds with standard maturities (e.g., 1/12 year, 0.25 year, 1, 2, 5, 7, 10, 30 year) as market variables. Note that any Treasury, or a bond in general, can be stripped into a packet of zeros. The mapping procedure is illustrated by an example. Consider a $1 million bond maturing in 0.8 years, with 10% semi-annual coupon. It can be viewed as a 0.3-year $50K zero-coupon bond plus 0.8-year $1050K zero-coupon bond. Suppose that the rates and zero-coupon Treasury Bills prices are as follows: 3 mos. 6 mos. 1 year zero yield 5.50% 6.00% 7.00% zero price volatillity 0.06% 0.10% 0.20% (% per day) Assume the following daily returns correlations for these T-Bills 3 mos. 6 mos. 1 year 3 mos. 1.00 0.90 0.60 6 mos. 0.90 1.00 0.70 1 year 0.60 0.70 1.00 What is the rate for discounting 0.80-year cash flows? We interpolate between 0.5 and 1.0 years and get yield of 6.60%. We also interpolate daily volatility for 0.8-year maturity and get 0.16%. We now try to replicate volatility of the 0.8-year payment with 0.5-year T-Bill and 1 year T-Bill, by a position of α in the 6 months T-Bill and 1−α in the 1 year T-Bill. Matching variances we get the following equation:

0.00162 = 0.0012α 2 + 0.0022 1−α( )2 + 2 ⋅0.7 ⋅0.001 ⋅0.002 ⋅α 1−α( ) This is a quadratic equation which gives α = 0.3203. The 0.8-year payment is worth

$1,050,0001.0660.8

= $997,662.

α is the portion of this amount, i.e., 0.3203 ⋅$997,662 = $319,589, which is allocated to the six-months T-Bill, and the rest, i.e., 0.6797 ⋅$997,662 = $678,073, is allocated to the one-year T-Bill. We can do the same calculations for the 0.3-year $50,000 payment. The 0.25-year and 0.50-year rates are 5.50% and 6.00%, respectively. Linear interpolation gives the 0.30-year rate as 5.60%. The present value of $50,000 received at time 0.30 years is


7

$50,0001.0560.3

= $49,189.32.

The volatilities of 0.25-year and 0.50-year T-Bills are 0.06% and 0.10% per day, respectively. Using linear interpolation we get the volatility of a 0.30-year payment as 0.068% per day. Assume that α is the value of the 0.30-year payment allocated to a three-month T-Bill, and 1−α is allocated to a six-month T-Bill. We match variances obtaining the equation:

0.000682 = 0.00062α 2 + 0.0012 1−α( )2 + 2 ⋅0.9 ⋅0.0006 ⋅0.001 ⋅α 1−α( ) which simplifies to

0.28α 2 − 0.92α + 0.5376 = 0. This is a quadratic equation, and its solution is:

α =−0.92 + 0.922 − 4 ⋅0.28 ⋅0.5376

2 ⋅0.28= 0.760259.

This means that a value of 0.760259 ⋅$49,189.32 = $37,397

is allocated to the three-month T-Bill and a value of 0.239741 ⋅$49,189.32 = $11,793

is allocated to the six-month T-Bill. This way the entire bond is “mapped” into positions in standard maturity zero Treasury Bills. Since we are given the volatilities and correlations of those bonds, and since we are assuming zero return in a short period of time, we just take the portfolio as mapped, and calculate its standard deviation. The portfolio consists of $678,073 in 1-year bond, $11,793 + $319,589 = $331,382 in 0.5-year bond, $37,397 in 0.25-year bond. The 10-day 99% VaR of the bond is then 2.33 times 10 times the SD calculated. When the linear model can be used The linear model starts with the equation

ΔP = α ii=1

n

∑ Δxi

which means that the change in the value of the portfolio is a linear function of the changes in the values of the underlying. This is not the case for many derivatives, especially options. Is there a case when derivatives can be handled with the linear model? Here are some examples. Assets denominated in foreign currency can be accommodated, by measuring them in domestic currency. Forward contract on a foreign currency can be regarded as an exchange of a foreign zero coupon bond maturing at contract maturity for a domestic zero maturing at the same time. Interest rate swap can be viewed as the exchange of a floating rate bond for a fixed rate bond. The floater can be regarded as a zero with maturity equal the next reset date. Thus this is a bond portfolio and can be handled by the linear model. When the portfolio contains options, the linear model can be used as an approximation. Consider a portfolio of options on a single stock with price S.

Suppose the delta of the portfolio is δ , so that: δ ≈ΔPΔS. Define Δx = ΔS

S. Then


8

ΔP ≈ SδΔx. If the portfolio consists of many such instruments, we get ΔP ≈ Sii=1

n

∑ δ iΔxi ,

which is essentially the linear model. Example. A portfolio consists of options on IBM, with delta of 1,000 and options on T, with delta of 20,000. You are given IBM share price of $120 and T share price of $30. Then

ΔP = 120 ⋅1000 ⋅ Δx1 + 30 ⋅20,000 ⋅ Δx2 = 120,000Δx1 + 600,000Δx2 . If daily volatility of IBM is 2% and daily volatility of T is 1%, with correlation 0.70, the standard deviation of ΔP is

1000 120 ⋅0.02( )2 + 600 ⋅0.01( )2 + 2 ⋅120 ⋅0.02 ⋅600 ⋅0.01 ⋅0.7 = 7,869. The 5th percentile of standard normal distribution is -1.65, and so the 5-day 95% VaR is

1.65 ⋅ 5 ⋅ 7,869 = $29,033. A quadratic model When a portfolio includes options, its gamma γ( ) should be included in the analysis. We have

ΔP = δΔS + 12γ ΔS( )2 ,

and withΔx = ΔSS, we can write this as

ΔP = SδΔx + 12S2γ Δx( )2 .

The problem is that ΔP is not normally distributed, although we assume Δx is, Δx ~ N 0,σ 2( ). The moments of ΔP are

E ΔP( ) = 12S2γσ 2 ,

E ΔP( )2( ) = S2δ 2σ 2 +34S4γ 2σ 4 ,

and

E ΔP( )3( ) = 92 S4δ 2γσ 4 +158S6γ 3σ 6 .

We can pretend that ΔP is normal and fit a normal distribution to the first two moments. The alternative is to use Cornish-Fischer expansion. For a portfolio in which each instrument depends on one market variable, we get

ΔP = Sii=1

n

∑ δ iΔxi +12i=1

n

∑ Si2γ i Δxi( )2 .

If pieces of the portfolio can depend on more than one variable then we get a more complicated picture:


9

ΔP = Sii=1

n

∑ δ iΔxi +12i=1

n

∑ SiSjγ ijΔxiΔx ji=1

n

∑ ,

where γ ij =∂2P∂SiSj

. This can be used to estimate moments of ΔP.

An alternative: Monte Carlo Simulation One day VaR calculation 1.Value the portfolio today in the usual way using the current values of market variables. 2. Sample once from the multivariate normal probability distribution of theΔxi ’s. 3. Use the values of theΔxi ’s that are sampled to determine the value of each market variable at the end of one day. 4. Revalue the portfolio at the end of the day in the usual way. 5. Subtract the value calculated in step one from the value in step four to determine a sample ΔP. 6. Repeat steps two to five many times to build up a probability distribution for ΔP. VaR is then calculated as the appropriate percentile of the probability distribution so obtained. An alternative approach, lowering the number of calculations is to assume that

ΔP = Sii=1

n

∑ δ iΔxi +12i=1

n

∑ SiSjγ ijΔxiΔx ji=1

n

∑ ,

and skip steps 3, 4 above. This is called a partial simulation. Historical Simulation Create a database of daily movements of all market variables for several years. Use the database as the probability distribution (the book says first day in database is first day in your simulation, and so on, but that is not the only way you could do this, you can also simulate from the empirical distribution given by the sample). ΔP is then calculated for each simulation trial, and the empirical distribution of ΔP has the percentiles determining VaR. The strength of this method lies in having no artificial assumption of normal distribution. Bu the method has weaknesses:

- Limited by data set available (but you can do bootstrap/resampling), both in length of time and data availability.

- Sensitivity analysis difficult. - Can’t use volatility-updating schemes (volatility changes over time, although

some methodologies have been designed to account for that). Stress testing and back testing

- Stress testing: Estimating how the portfolio would have performed under the most extreme market conditions. For example: five standard deviation move in a market variable in a day. This is next to impossible under normal distribution assumption (happens once in 7000 years) but in reality it happens about once


10

every 10 years (and some researchers say that one such move in some market happens every year).

- Back testing: checking how well our model did in predicting things in the past. For example: how often did a one day loss exceed 1-day 99% VAR. If this happens roughly one percent of the time, methodology appears appropriate.

An example of the use of historical simulation methodology: Spring 2007 Casualty Actuarial Society Course 8 Examination, Problem No. 33 You are given the following observations regarding the price of stocks A and B: Day Stock A Stock B 0 55.00 45.00 1 56.10 45.45 2 54.98 43.18 3 52.23 41.88 4 54.84 43.98 5 57.03 44.86 6 55.32 43.06 7 52.00 42.63 8 57.20 43.91 9 54.34 44.79 10 53.80 42.55 Assume that your portfolio consists of 1,000 shares of each stock. Estimate the one-day VaR for an 80% confidence level using the historical simulation methodology. Solution. We calculate the pairs of historical daily changes and apply them to current values: Stock A daily returns were

56.1055.00

−1 = 1.02 −1 = 2.00%,

54.9856.10

−1 = 0.980036 −1 = −1.9964%,

52.2354.98

−1 = 0.949982 −1 = −5.0018%,

54.8452.23

−1 = 1.049971−1 = 4.9971%,

57.0354.84

−1 = 1.039934 −1 = 3.9934%,

55.3257.03

−1 = 0.970016 −1 = −2.9984%,

52.0055.32

−1 = 0.939986 −1 = −6.0014%,


11

57.2052.00

−1 = 1.10 −1 = 10%,

54.3457.20

−1 = 0.95 −1 = −5%,

53.8054.34

−1 = 0.999063−1 = −0.9937%,

and the stock B daily returns were

45.4545.00

−1 = 1.01−1 = 1%,

43.1845.45

−1 = 0.950055 −1 = −4.9945%,

41.8843.18

−1 = 0.969893−1 = −3.0107%,

43.9841.88

−1 = 1.050143−1 = 5.0143%,

44.8643.98

−1 = 1.020009 −1 = 2.0009%,

43.0644.86

−1 = 0.959875 −1 = −4.0125%,

42.6343.06

−1 = 0.990014 −1 = −0.9986%,

43.9142.63

−1 = 1.030026 −1 = 3.0026%,

44.7943.91

−1 = 1.020041−1 = 2.0041%,

42.5544.79

−1 = 0.949989 −1 = −5.0011%.

We apply now these rates of return to last prices of stocks A and B: 1.02 ⋅53.80 +1.01 ⋅ 42.55 ≈ 97.85, 0.980036 ⋅53.80 + 0.950055 ⋅ 42.55 ≈ 93.15, 0.949982 ⋅53.80 + 0.969893 ⋅ 42.55 ≈ 92.38, 1.049971 ⋅53.80 +1.050143 ⋅ 42.55 ≈ 101.17, 1.039934 ⋅53.80 +1.020009 ⋅ 42.55 ≈ 99.35, 0.970016 ⋅53.80 + 0.959875 ⋅ 42.55 ≈ 93.03, 0.939986 ⋅53.80 + 0.990014 ⋅ 42.55 ≈ 92.70, 1.10 ⋅53.80 +1.030026 ⋅ 42.55 ≈ 103.01, 0.95 ⋅53.80 +1.020041 ⋅ 42.55 ≈ 94.51,

0.999063 ⋅53.80 + 0.949989 ⋅ 42.55 ≈ 93.69. The 80-th percentile of the loss corresponds to the 20-th percentile of the sample above, and since there are 10 values, that is the second lowest value, 92.70. Current value of the


12

portfolio of one share of each stock is 53.80 + 42.55 = 96.35. For the 1000 shares of each stock portfolio, the 80% VaR is 1000 ⋅ 96.35 − 92.70( ) = 3650. An example of a calculation of VaR for a foreign currency/option portfolio A bank owns a portfolio of options on the U.S. dollar – Pound Sterling exchange rate. The delta of the portfolio is given as 56.00. Current exchange rate is $1.50 per Pound Sterling. You are given that the daily volatility of the exchange rate is 0.70%. What is the approximate linear relationship between the change in the portfolio value and the proportional change in the exchange rate? Estimate the 10-day 99% Value at Risk. Solution. Given the value of delta, the approximate relationship between the daily change in the portfolio value, ΔP, and the daily change in the exchange rate, ΔS, is ΔP = 56ΔS. Let

Δx be the proportional daily change in the exchange rate. The Δx = ΔS1.5. Therefore

ΔP = 56 ⋅1.5Δx = 84Δx. The standard deviation of Δx equals the daily volatility of the exchange rate, i.e., 0.70%. The standard deviation of ΔP therefore is 84(0.70%)=0.588. The 10-day 99% Value at Risk is thus estimated as: 0.588 ⋅2.33 ⋅ 10 = 4.33. An example of a non-linear calculation We know that option portfolios cannot be easily represented by a linear model. Assume that the portfolio gamma for the previous problem is 16.2. How does this change the estimate of the relationship between the change in the portfolio value and the proportional change in the exchange rate? Calculate an update of the 10-day 99% Value at Risk based on estimate of the first two moments of the change in the portfolio value. Solution. Based on the Taylor series expansion

ΔP = 56 ⋅1.5 ⋅ Δx + 12⋅1.52 ⋅16.2 ⋅ Δx( )2 .

This simplifies to ΔP = 84Δx +18.225 Δx( )2 .

The first two moments of ΔP are

E ΔP( ) = E 12⋅1.52 ⋅16.2 ⋅ Δx( )2⎛

⎝⎜⎞⎠⎟=12⋅1.52 ⋅16.2 ⋅0.0072 = 0.000893

and

E ΔP( )2( ) = 1.52 ⋅562 ⋅0.0072 + 34 ⋅1.54 ⋅16.22 ⋅0.0074 = 0.346. The standard deviation of ΔP is

0.346 − 0.0008932 = 0.588.


13

We use the mean and standard deviation so calculated and pretend that ΔP has normal distribution, fitting a normal distribution with the same mean and variance. The ten-day 99% Value at Risk is calculated as:

10 ⋅2.33 ⋅0.588 −10 ⋅0.000893 = 4.3235. An example of a foreign currency swap A company has entered into a six-month forward contract to buy 1 million Pound Sterling for $1.5 million. The daily volatility of a six-month zero-coupon Pound Sterling bond (when its price is translated to U.S. dollars) is 0.06% and the daily volatility of the six-month zero-coupon dollar bond is 0.05%. The correlation between returns from the two bonds is 0.80. Current exchange rate is 1.53. Calculate the standard deviation of the change in the dollar value of forward contract in one day. What is the 10-day 99% Value at Risk? Assume that the six-month interest rate in both Pound Sterling and dollars is 5% per annum with continuous compounding. Solution. The contract if a long position in a Pound Sterling bond combined with a short position in a dollar bond. The value of the Pound Sterling bond is

$1.53 ⋅ e−0.05⋅0.5 million = $1.492 million. The value of the dollar bond is

$1.5 ⋅ e−0.05⋅0.5 million = $1.463 million. The variance of the change in the value of the contract in one day is, based on the formula forVar X −Y( ),

1.4922 ⋅0.00062 +1.4632 ⋅0.00052 − − 2 ⋅0.8 ⋅1.492 ⋅0.0006 ⋅1.463 ⋅0.0005 = 0.000000288.

The square root of this quantity, 0.000537, is the standard deviation, in millions of dollars. Therefore, the 10-day Value at Risk is 0.000537 ⋅ 10 ⋅2.33 = $0.00396 million. Solvency II Directive in the European Union Solvency II is the European Commission project which (quoting from the European Commission’s “Framework for Consultation”) aims to develop “a new solvency system to be applied to life assurance, non-life insurance and reinsurance undertakings, which Member States and supervised institutions are able to apply in a robust, consistent and harmonized way.” A three-pillar system is proposed, similar to Basel II: • Pillar 1: quantification of capital requirements; • Pillar 2: supervisory review process; and • Pillar 3: market analysis of published data. Pillar 1 encompasses two capital requirements (MCR and SCR) sitting on top of technical provisions made up of the best estimate of the liability plus a risk margin, as shown in the following diagram taken from Committee of European Insurance and Occupational Pensions Supervisors (CEIOPS):


14


15

Solvency II capital requirement approach Pillar 1 of the Solvency II framework will set out two levels of capital requirements. • The first of these is the Solvency Capital Requirement. This is a risk-based assessment of the level of capital that the business needs to operate. Should the capital fall below this level then the supervisor may intervene. The SCR is based on an assessment of insurance, credit, market and operational risk elements. The capital requirement is based on a 1 in 200 (99.5%) confidence level over a one-year time horizon. The capital requirement can be calculated using internal models (full or partial models are being considered) or a standard approach. When the SCR has been calculated, it is subject to supervisory review and an adjusted SCR will be issued by the regulator (i.e., reflecting their view of how well the SCR reflects the risks in the business). • The second capital requirement is the Minimum Capital Requirement. The exact role and nature of this is still under consultation. However, in simple terms it is the absolute minimum capital requirement, below which the regulator would ask the firm to cease writing business. The MCR will be based on a much simplified calculation basis, which is currently subject to consultation. It may be a simplified version of the SCR calculation or it may simply be a percentage of last year's SCR calculation. Pillar 2 of Solvency II sets out the internal risk and capital management standards that firms should follow. All firms will be expected to undertake an Individual Risk and Capital Assessment (IRCA) to identify the level of capital that the business requires. The use of an internal model is not expected if the standard model is being used under Pillar 1, but it is a quantitative as well as qualitative exercise. The second aspect of this Pillar is the supervisory review process. Solvency II allows the supervisors to undertake an assessment of the adequacy of the assessment of the risk-based capital requirements and risk management processes and to give an adjusted SCR requirement to the firm. Given that Pillar 2 gives the supervisors of the nation states review powers and the ability to adjust the capital requirements of a firm, it is a significant element of the framework. United Kingdom firms have experienced a similar regime but many European states have no equivalent regime. There are concerns that, notwithstanding the supervisory peer review process proposed, there may be inconsistencies in the application of the rules in the early stages of implementation. Given the scarcity of resources with experience of reviewing internal capital models this is not surprising. Pillar 3 of Solvency II concerns the disclosures that firms will be expected to make. These take two forms – disclosures to the supervisors and public disclosures. However, broadly the disclosures are expected to be in respect of the following areas:

- Business overview and performance, - Governance, - Valuation basis used for solvency purposes, - Risk and capital management.


16

One of the key topics under consultation is the disclosure around the SCR. Under discussion is whether the adjusted SCR (i.e. after supervisory review) should be disclosed and whether the disclosure should split the original estimate and the supervisory loading. The requirements will undoubtedly mean disclosure of more information than is currently disclosed and will increase the reporting burden on firms significantly. A key issue under debate currently is how far the reporting intrudes into sensitive commercial areas and what elements of information should be available only to the regulator or should be publicly showed. There are some similarities with the current IFRS reporting proposals, but the Solvency II regime goes further in its requirements. Risk-Based Capital Requirements for Insurance Companies in the United States The (United States) National Association of Insurance Commissioners (NAIC) instituted its RBC system for life insurance companies in 1993, followed by a property-casualty system in 1994 and a health system in 1998. The NAIC’s RBC system consists of two parts:

• A formula that is used to set a regulatory minimum capital level for each insurer, based on that insurer’s mix of assets, liabilities, and risk, and

• Definition of “financial impairment” and remedies to state insurance regulators in the event that an insurer meets that definition of impairment.

Formulas continuously evolve. NAIC publishes newsletters and guidelines for the calculation of Risk-Based Capital. The RBC system is meant to be a supplement, not a replacement, for the existing fixed minimum capital requirements that exist in each state. That is, the RBC formula requirements can be higher or lower than the fixed minimum capital requirements (which are typically $1 to $2 million), but each insurance company must meet both sets of standards. Many small insurance companies generate RBC requirements that are lower than the fixed dollar minimums, but for virtually all medium-sized and large insurers, the capital requirements generated by the RBC formula are higher than the state fixed minimums. The RBC requirement (level of capital required in view of risk undertaken) is calculated by multiplying risk factors by statement values, adding the results together, and then adjusting for covariance between major risk categories. The formula results are compared to the risk-adjusted capital of the insurer to develop the RBC ratio, which is the ratio of risk-adjusted capital to RBC. The ratio results are used to determine the degree to which an insurance company’s surplus is impaired. The model act specifies a series of increasingly stringent regulatory responses, as the RBC ratio decreases below 200%. A trend test is included to test whether insurers that were between the 200% breakpoint and 250% level were trending downward, which will trigger regulatory action, but an RBC ratio over 250% for a life company is sufficient to receive a passing grade on this pass/fail test. There are four “action levels” under the NAIC RBC system.


17

• Company Action Level (CAL). If this level is reached, insurer is required to automatically submit a written, detailed business plan within 45 days that details the causes and actions that have led up to the capital impairment as well as a plan for the restructuring of the insurer’s business to rebuild capital to acceptable levels. Alternatively, the company can detail plans to reduce its risk to a level commensurate with its actual capital level.

• Regulatory Action Level (RAL). In this case, insurer must conform to the requirements stated in the Company Action Level, and in addition is subject to an immediate regulatory audit. The regulator can then issue protective orders to force the insurer to either lower its risk profile or increase its capital to a level commensurate with its risk. A company that has reached the Company Action Level and that does not conform to the statutory requirements spelled out in the statute is also automatically deemed to have triggered the Regulatory Action Level.

• Authorized Control Level (ACL) is triggered by having statutory capital that is less than the Authorized Control Level RBC, as computed by the RBC formula or by failing to meet regulatory requirements imposed by the Regulatory Action Level. The Authorized Control Level is the capital level at which the state insurance commissioner is authorized, although not required, to place the insurance company under regulatory supervision.

• Mandatory Control Level. When that happens, the state regulator is required by statute to take steps to place the insurer under regulatory supervision.

Risk categories in the Life RBC formula Originally, the major risk categories in the Life RBC formula were C1 – Asset Risk, C2 – Insurance Risk, C3 – Interest Rate Risk and C4 – Business Risk. These generic categories have been later refined and currently they are:

C0 Affiliates Risk C1cs Asset Risk – Unaffiliated Common Stock C1o Asset Risk – Other Assets Risk C2 Insurance Risk C3a Interest Rate Risk C3b Health Credit Risk C4a General Business Risk C4b Administrative Expense Risk

The values calculated for each category are then combined in what is commonly called the covariance formula. The results of the covariance formula produce the Company Action Level RBC capital requirement. The Company Action Level requirement is twice the Authorized Control Level requirement.


18

If the insurer’s Total Adjusted Capital is less than the Authorized Control Level RBC requirement, the regulator is authorized to seize control of the company. The ACL RBC and the Total Adjusted Capital are both reported in the Five-Year History page of the annual statement. The RBC formula inputs and calculations are not made public. Total Adjusted Capital = Statutory Capital & Surplus + Asset Valuation Reserve (AVR) including AVR in separate accounts + Half of company’s liability for dividends + company’s ownership share of AVR of subsidiaries + Half of company’s ownership share of subsidiaries’ dividend liability Separate risk-based capital models apply to life companies, property/casualty companies and health organizations. The common risks identified in the NAIC models for all types of companies include Asset Risk-Affiliates, Asset Risk-Other, Credit Risk, Underwriting Risk, and Business Risk. Steps in RBC calculation:

• Apply risk factors against annual statement values, • Sum risk amounts and adjust for statistical independence (using the covariance

formula), • Calculate Authorized Control Level Risk-Based Capital amount, • Compare ACL RBC to Adjusted Capital.

Total Adjusted Capital (Actual Capital) is divided by Authorized Control Level RBC (Hypothetical Minimum Capital) to get the RBC Ratio • No Action (98% of companies) -- TAC/RBC over 200% • Company Action Level -- TAC/RBC is 150% to 200% • Authorized Control Level -- TAC/RBC is 70% to 100% • Mandatory Control Level -- TAC/RBC is less than 70% After the calculation of RBC, the company is also expected to perform sensitivity tests to indicate how sensitive the results are to certain risk factors’ changes. Major categories in life RBC formula:

C0 – Subsidiary Insurers Risk C1 – Asset Risk C2 – Insurance Risk C3 – Interest Rate Risk C4 – Business Risk

Major categories in property/casualty RBC formula:

R0 – Subsidiary Insurers Risk R1 – Fixed Income Asset Risk R2 – Equity Asset Risk


19

R3 – Credit Risk R4 – Insurance Risk – Reserve Development R5 – Insurance Risk – Written Premiums

Major categories in health RBC formula:

H0 - Insurance Subsidiaries Risk H1 –Asset Risk H2 – Insurance Risk H3 – Credit Risk H4 – Business and Admin Expense Risk

Asset risk (C1) calculation example: Note: Risk factors are developed by an NAIC Advisory Group. Based on simulation testing for portfolios of bonds. They are intended to account for default risk only. In what follows we use factors that may not be exactly the ones currently used. Asset portfolio given: Factor RBC NAIC Class 1, U.S. Government $1000 0.000 $0.00 NAIC Class 1, non-U.S. Government $1000 0.003 $3.00 NAIC Class 2 $1000 0.010 $10.00 NAIC Class 3 $1000 0.020 $20.00 NAIC Class 4 $1000 0.045 $45.00 NAIC Class 5 $1000 0.100 $100.00 NAIC Class 6 $1000 0.300 $300.00 Total $7000 $478.00 There is then a bond size adjustment factor, i.e., the resulting RBC is multiplied by a number fSIZE, which is determined by the number of bonds in the portfolio. Note: there are no adjustments for portfolio size for stocks and mortgages. For stocks, the C1 RBC is derived by multiplying the total value of all stocks by a factor provided by NAIC (30% for life companies, 15% for property/casualty companies, but note that factors do change over time). Life mortgages formula is based on Asset Valuation Reserve (AVR)

- Mortgage Experience Adjustment (MEA) - Separate Risk Factors By Mortgage Type

o Farm o Residential o Commercial o Restructured

- Separate Risk Factors By Quality o In Good Standing


20

o Overdue o In Foreclosure

Life real estate formula:

- Separate calculations for each property, then add up to get the total. - Large number of separate RBC factors depending on type and quality of property,

very complex. - Questionable accuracy.

Accounting for insurance risk C2:

• Life formula uses tiered factors to adjust for size differences. • Life insurance - Apply factors against net amount at risk. • Health insurance - Apply factors against earned premiums. • Flat factor is applied against health insurance reserves (e.g., not size-based). • Credit allowed for premium stabilization reserves.

Example of tiered premium insurance risk RBC calculation: A company has $10,000,000 in net amount at risk. A factor of 3.5% is applied to the first $5 million, and 2.0% for the next $5 million. Then the insurance risk RBC for this company is: 3.5% of $5,000,000 plus 2.0% of $5,000,000, for a total of $175,000 + $100,000 = $275,000. The C3 risk RBC calculation uses the asset-liability model used for year-end Asset Adequacy Analysis cash flow testing, or a consistent model. You start by running the scenarios (12 or 50) produced from the interest-rate scenario generator. The statutory capital and surplus, S(t), should be captured for every scenario for each calendar year-end of the testing horizon. For each scenario, the C-3 measure is the most negative of the series of present values S(t), using 105 percent of the after-tax one-year Treasury rates for that scenario. Then one should rank the scenario-specific C-3 measures in descending order, with scenario number 1’s measure being the positive capital amount needed to equal the very worst present value measure. Taking the weighted average of a subset (currently, scenarios ranked 17th through 5th are used with weights 0.02, 0.04, …, 0.16, …, 0.04, 0.02) of the scenario specific C-3 scores derives the final C-3 factor for the 50 scenario set. For the 12 scenario set, the charge is calculated as the average of the C-3 scores ranked 2 and 3, but cannot be less than half the worst scenario score. There are also cases when single scenario testing is allowed. Finally, the C4 RBC is calculated generally as a small percentage of premiums, in the range of 2%. The original life RBC covariance formula was:

C1+ C3( )2 + C22 + C4.


21

It has been since then changed to:

C0 + C1+ C3( )2 + C22 + C4. The property/casualty covariance formula is:

R0 + R1+ R2 + R3+ R4 + R5( )2 . The result of this formula is used in the RBC ratio to compare to the Company Action Level RBC (200% of Authorized Control Level). If you would rather compare to the Authorized Control Level, the life covariance formula is:

12C0 + C1+ C3( )2 + C22 + C4( ).

Risk-Based Capital Standards for U.S. Property/Casualty Companies The property/casualty covariance formula is:

R0 + R1+ R2 + R3+ R4 + R5( )2 . R0: Investments in insurance affiliates

Non-controlled assets Guarantees for affiliates Contingent liabilities

R1: Fixed income securities Cash: 0.3% charge Bonds: NAIC scale Bond size adjustment factor: 2.5 for less than 50, 1.3 for next 50, then 1, but for

issues above 400 the factor is 0.9 Mortgage loans Short-term investments Collateral loans Asset concentration adjustment for fixed income securities

R2: Equity investments Common stocks: 15% charge Preferred stocks: NAIC scale Real estate Other invested assets Aggregate write-ins for invested assets Asset concentration adjustment for equity investments

R3: Credit risk Reinsurance recoverables: 10% charge Other receivables: 5% charge

R4: Reserving risk Basic reserving risk charge: Quantified separately by line of business, using

statutory Schedule P data for the past ten years. Start with adverse


22

development ratios, and then apply interest adjustment (i.e., discount). Offset for loss-sensitive business Adjustment for claims-made business Loss concentration factor Growth charge for reserving risk

R5: Written premium risk Basic premium risk charge: charge quantifies the amount of capital needed to

guard against unexpectedly poor underwriting results during the coming year. Offset for loss-sensitive business Adjustment for claims-made business Premium concentration factor Growth charge for premium risk

Example Regulated Insurance is a company domiciled in Freedonia, a country that implemented a risk-based capital requirement system for regulation of its insurance companies. Risk-based capital requirement is the amount equal to the 95-th percentile of random amount of the loss that the company experiences in one year. The company has liabilities of 1,000,000 freebies (the currency of Freedonia) in life annuities reserves, and assets of $1,200,000 freebies, invested 40% in stocks and 60% in bonds. The liabilities of the company will increase by 3% over the next year. The effective annual rate of return on bonds follows a normal distribution with mean 4% and standard deviation of 2%, while the effective annual rate of return on stocks follows a normal distribution with mean 9% and standard deviation of 12%. The correlation of returns of stocks and bonds is 0.6. Find the ratio of the capital held by Regulated Insurance to the capital required by regulation in Freedonia. The 95-th percentile of the standard normal distribution is 1.645. Solution. The capital held by Regulated Insurance is equal to the excess of its assets over its liabilities, i.e., 1,200,000 – 1,000,000 = 200,000 freebies. Let X be the random rate of return of stocks, and Y be the random rate of return on bonds. The loss of Regulated Insurance in one year is

0.03 ⋅1000000 − X ⋅0.4 ⋅1200000 −Y ⋅0.6 ⋅1200000 = = 30000 − 480000X − 720000Y .

This is a normal random variable with mean E 30000 − 480000X − 720000Y( ) = 30000 −19200 − 64800 = −54000, and variance

Var 30000 − 480000X − 720000Y( ) = Var 480000X + 720000Y( ) = = 4800002 ⋅0.122 + 7200002 ⋅0.022 + 480000 ⋅ 720000 ⋅0.12 ⋅0.02 ⋅0.7 = = 4105728000.

The 95-th percentile of the distribution of the loss is −54000 +1.645 ⋅ 4105728000 ≈ 51404.9459. The ratio sought is


23

20000051404.9459

≈ 3.8907.

Example Honorable Life Insurance Co. (HLIC) is headquartered in Freedonia, a country that just established new Risk-Based Capital regulation. Required capital equals

CR = 1.25 C1 + C3( )2 + C22 ,

where: • C1 = capital requirement for asset portfolio, equal to the sum of 20% of the value of stock portfolio held and 2% of the value of the bond portfolio held. Companies in Freedonia only invest in stocks or bonds. • C2 = capital requirements for liabilities, equal to 1% of amount at risk, defined as the difference between the amount of insurance and the liability established for it. • C3 = capital requirement for asset-liabilities interaction, set as the change in the value of the surplus as a result in the increase of interest rate of 100 basis points, calculated with the standard duration-convexity approximation. You are given the following: • HICS portfolio is always invested 20% in stocks and 80% in bonds, and its asset portfolio is worth 1,100,000 freebies, the currency in Freedonia. If assets are sold, stocks and bonds are sold proportionately, so that the allocation of 20% in stocks and 80% in bonds is retained. • The liabilities are one million freebies and the net amount at risk is also one million freebies. • The company’s surplus has the duration of 20 years and convexity of 100 years squared. HLIC can purchase a proportional reinsurance, in which it receive credit for a fraction w of its liabilities, and covering the same fraction of insurance amount, costing 105% of the amount of liabilities reinsured. Assume that the duration and convexity of the surplus is unchanged by the reinsurance contract. Calculate the smallest w such that the reinsurance the company purchases causes it to have the capital that is greater than or equal to the capital required. A. 48.56% B. 33.33% C. 12.40% D. 5.96%

E. 0% (no reinsurance is needed to satisfy the capital requirement) Solution. Before reinsurance, the company has Assets of 1,100,000 consisting of 220,000 in stocks and 880,000 in bonds, Liabilities of 1,000,000, and amount of insurance of 2,000,000, Surplus of 100,000. Also before reinsurance, risk-based capital calculation is as follows C1 = 0.2 ⋅220,000 + 0.02 ⋅880,000 = 61600, C2 = 0.01 ⋅1,000,000 = 10000,


24

C3 = − −100,000 ⋅0.01 ⋅20 + 12⋅100,000 ⋅0.012 ⋅100⎛

⎝⎜⎞⎠⎟= 19,500,

CR = 1.25 61600 +19500( )2 +100002 ≈ 102,142.746. But the company only has the capital of 100,000 freebies, so it is failing the regulatory requirements. If the company purchases reinsurance for a fraction w of its liabilities, it will have Assets of 1100000 −1050000w = 1100000 1− w( ) + 50000w invested 20% in

stocks and 80% in bonds, Liabilities of 1000000 1− w( ) and amount of insurance of 2000000 1− w( ),

Surplus of 1100000 1− w( ) + 50000w( ) − 1000000 1− w( )( ) = = 100000 1− w( ) + 50000w = 100000 − 50000w.

Risk-based capital calculation is as follows

C1 = 0.2 ⋅220,000 1− w( ) + 0.2 ⋅0.2 ⋅50000w + + 0.02 ⋅880,000 1− w( ) + 0.02 ⋅0.8 ⋅50000w = = 61600 1− w( ) + 2800w = 61600 − 58800w,

C2 = 0.01 ⋅1,000,000 1− w( ) = 10000 1− w( ), C3 =

= − − 100000 1− w( ) + 50000w( ) ⋅0.01 ⋅20 + 100000 1− w( ) + 50000w( ) ⋅ 12⋅0.012 ⋅100⎛

⎝⎜⎞⎠⎟=

= 19,500 1− w( ) +10250w = 19500 − 9250w,and

CR = 1.25 61600 − 58800w +19500 − 9250w( )2 + 10000 1− w( )( )2 =

= 1.25 81100 − 68050w( )2 + 10000 1− w( )( )2 .

The capital held and capital required will be equal if

100000 − 50000w = 1.25 81100 − 68050w( )2 + 10000 1− w( )( )2 , or 80000 − 40000w( )2 = 81100 − 68050w( )2 + 10000 1− w( )( )2 . We solve this equation

0 = 10000 1− w( )( )2 + 81100 − 68050w( )2 − 80000 − 40000w( )2 == 100002 ⋅ 1− 2w + w2( ) + 1100 − 28050w( ) 161100 −108050w( ) == 3130802500w2 − 4837710000w + 27721000,

as


25

w =4837710000 ± 48377100002 − 4 ⋅ 3130802500 ⋅27721000

2 ⋅ 3130802500≈ 1.4856,

0.0596.⎧⎨⎪

⎩⎪

Since it is not possible to reinsure more than 100% of policies issued, only the second solution is feasible, and the answer is w = 0.0596 = 5.96%. Answer D.


26

References: Barth, Michael, “Life Risk-Based Capital: The U.S. Experience”, presented at the World

Bank’s Contractual Savings Conference, April 29-May 3, 2002, Washington, D.C.

Caruana, Jaime, and Aditya Narain, “Banking on More Capital,” Finance and Development, June 2008, volume 42, number 2, International Monetary Fund, publication available online at: http://www.imf.org/external/pubs/ft/fandd/2008/06/caruana.htm

Committee of European Insurance and Occupational Pensions Supervisors (CEIOPS), Quantitative Studies in the Framework of the Solvency II project, available online at: http://www.ceiops.eu/content/view/118/124/

Feldblum, Sholom, “NAIC Property/Casualty Insurance Company Risk-Based Capital Requirements,” Proceedings of the Casualty Actuarial Society 83(1996), pp. 297- 435. Available online at: http://www.casact.org/pubs/proceed/proceed96/96297.pdf

Hull, John, Options, Futures and Other Derivatives, Seventh Edition, Pearson Education, Inc., Upper Saddle River, New Jersey, 2009.

Dr. Krzysztof Ostaszewski, FSA, CFA. MAAA …math.illinoisstate.edu/krzysio/var-rbc.pdf · Dr....

Documents

Transcript of Dr. Krzysztof Ostaszewski, FSA, CFA. MAAA …math.illinoisstate.edu/krzysio/var-rbc.pdf · Dr....