Insurance Market
SOA and CAS Exam P, Probability
Here, I provide my summary notes of the SOA/CAS P Exam. The purpose of the use is non-commercial research and/or private study. Please do not copy or distribute without permission. If you find any errors, please email me at: liyifinhub@outlook.com
Note: Ch1: Sets; Ch2: Combinatorics; Ch3: Conditional Probability; Ch4: Bayes’ Theorem; Ch5: Random Variable;
Ch6: Conditional Probability for Random Variable; Ch7: Mean; Ch8: Variance and Other Moments; Ch9: Percentiles;
Ch10: Mode; Ch11 and CH15: Joint Distribution; Ch12 and Ch27: Marginal Distribution; Ch13 and Ch28: Joint Moments;
Ch14 and Ch29: Covariance; Ch15: Conditional Moments; Ch16 and CH26: Unifrom Distribution;
Ch17: Double Expectation Formulas; Ch18 and Ch19: Binominal and Negative Binominal Distributions;
Ch20: Poisson Distribution; Ch21: Exponential Distribution; Ch22: Normal Distribution; Ch23: Central Limit Theorem;
Ch24: Order Statistics
SOA and CAS Exam FM
Here, I provide written solution to the SOA/CAS FM Exam. The purpose of the use is non-commercial research and/or private study. Please do not copy or distribute without permission. If you find any errors, please email me at: liyifinhub@outlook.com
Written Solutions: Question 1-84; Question 86-117; Question 119-170; Question 171-260; Question 261-309; Question 311-317;
Question 319-325; Question 327-385
Claim Reserve, IBNR, and Chain Ladder Method
Claim Reserves consist of “Reported but not settled (RBNS)” and “Incurred but Not Reported (IBNR)”. There are long-term reserve and claim reserve. For example: long-term reserve (direct) consists of long-term life reserve and long-term medical reserve. IBNR is a reserve account used for the claims that happened but not yet reported due to processing lag. Here we focus on using Chain Ladder Method (CLM) to estimate the IBNR claim reserve.
Chain ladder method is a widely used ultimate claim estimation method (Bornhuetter Ferguson Method is an alternative method used to calculate claim reserve.). The chain ladder method is based on the assumption that the past development pattern can represent the future development pattern. We use this pattern to project the future ultimate claim amount. IBNR is the difference between the future cumulative amount and the already reported claim amount. This method often appears in the property and casualty insurance filed. Non-life insurance policies typically have shorter policy term comparing to life.
First, the chain ladder method requires the use of cumulative data. If you are given incremental data, please accumulate first. Second, calculate the Loss Development Factors (LDF) and Cumulative Deveopment Factors (CDF). One can use the simple average LDF or weighted average LDF as the final seleted CDF. The CDF is calculated based on the selected CDF (multiplication). Finally, we can get the projected ultimate calim amount and IBNR claim reserve.
- Please find my note here.
Duration Mismatch
In theory, asset duration and liability duration should match each other. If a company has zero duration gap (The duration gap is the difference between asset duration and liability duration. It measures the timing matching of asset inflows and liability outflows.), the company is not affected by (immunized against) interest rate risk. For insurance companies, the duration of liabilities is provided by actuary and the duration of assets is determined by investment department.
Duration mismatch, also known as liquidity mismatched, is one of the major asset-liability mismatches besides currency mismatch and Interest rate mismatch in Asset and Liability Management (ALM).
The impacts of duration mismatch: (1) Case 1: Duration of assets larger than duration of liabilities. When interest rates rise, the value of both assets and liabilities falls, but the value of the asset side falls more due to its longer duration, further reducing the value of equity. This is the main reason for the collapse of SVB Bank. (2) Case 2: Duration of liabilities larger than duration of asset. When interest rates fall, the value of assets and liabilities both increase, but the value of the asset side increases less. It is a decrease in the value of the asset side compared to the liability side, which leads to a decrease in firm’s equity.
We know that liquidity mismatch caused SVB bank to fall. In 2019, the average liability duration of life insurance companies
operating in China is more than 12 years, while the average asset duration is around 6 years. Wang, Wang, and Chen (2021) reveal that Chinese listed companies often use short-term debt to finance long-term investments. Macaulay Duration is a measure of the sensitivity of bond prices to the changes in interest rates. It is related to, but is not, the slope of the price yield curve. We know that by applying Taylor’s theorem to ΔP=P(r+Δr)-P(r), we obtain ΔP≈(-P*D_mac/1+r)*Δr+P*(Δr^2)*C/2≈(-D_mod*P)*Δr+P*(Δr^2)*C/2
The tangent line underestimates prices rise when yields fall and overestimates the prices fall when yields increase. The last part of the formula is simply a convexity adjustment used for error correction. Line duration prediction produces an error, which is the difference between tangent line and curve. This means that the slope is -D_mac/(1+r)*P = -D_mod*P (known as the dollar duration). For example, Dmac = 1, 5% simple rate, 100 par, then a 100bps increase in rate corresponds to a (-1/(1+5%)*100)*0.01 = 0.95 decrease in price.
Convexity, also known as Modified Convexity, is related to the second derivative of the price of bond w.r.t. yield 
Recall that modified duration is related to the first derivative of the price of bond w.r.t. yield defined D_mod = – (dP_i/di)/P = D_mac/1+r. In SOA and CAS FM exam, you will be asked to calculate the Macaulay Duration and the Modified Duration. The Macaulay Duration is further used in the first-order Macaulay approximation and the first-order modified approximation.
There are two types of immunization in FM exam: Redington Immunization and Full Immunization. Redington Immunization: discount the CFs to time 0. Full Immunization: discount the CFs to the time of single liability CF. The rest is the same, take the first and second derivate w.r.t interest rate respectively. Then, you will find the answer. Please check the above “SOA and CAS Exam FM” section for more information.
- Please find my note: Duration Mismatch here.