<style> body:has(.aqs-scr-article) .blog-template-wrapper{width:100%;max-width:960px!important;padding:0!important;margin:auto} body:has(.aqs-scr-article) .blog-template-wrapper .w-richtext{width:100%!important;max-width:none!important;padding:0!important;margin:0!important} body:has(.aqs-scr-article) .banner-section.blog-post{background:linear-gradient(125deg,#10091f,#2d1851)} body:has(.aqs-scr-article) .div-block-223 h2{font-size:32px!important;line-height:1.3!important} .aqs-scr-article h2{font-size:30px!important;line-height:1.3!important;margin-top:44px!important;margin-bottom:18px!important} .aqs-scr-article p,.aqs-scr-article li{font-size:18px!important;line-height:1.8!important} .aqs-scr-article p{margin-bottom:22px!important} .aqs-scr-article li{margin-bottom:12px!important} .aqs-scr-article .math-block{max-width:100%;overflow-x:auto;padding:6px 0;margin:24px 0;text-align:center} .aqs-scr-article mjx-container{max-width:100%} .aqs-scr-article .aqs-scr-note{font-size:15px!important;color:#6b6575;border-left:3px solid #c8bfdf;padding-left:18px} .aqs-scr-article .aqs-scr-callout{background:#f6f3fb;border-left:3px solid #7459b4;padding:22px 24px;margin:28px 0;border-radius:0 8px 8px 0} .aqs-scr-article .aqs-scr-callout p{margin:0!important;font-size:16px!important} .aqs-scr-article .aqs-scr-figure{margin:34px auto;width:100%!important;max-width:640px!important;display:block} .aqs-scr-article .aqs-scr-figure svg{display:block;width:100%;height:auto} .aqs-scr-article figcaption{font-size:13px;line-height:1.7;color:#756e7f;margin-top:12px;text-align:left} .aqs-scr-article .aqs-scr-references li{font-size:15px!important;line-height:1.7!important} @media(max-width:767px){body:has(.aqs-scr-article) .section.articles .base-container{padding-left:22px!important;padding-right:22px!important}.aqs-scr-article h2,body:has(.aqs-scr-article) .div-block-223 h2{font-size:25px!important}.aqs-scr-article p,.aqs-scr-article li{font-size:17px!important}.aqs-scr-article .aqs-scr-callout{padding:18px}} .aqs-scr-article .aqs-scr-figure img{display:block;width:100%;height:auto}</style><div class="aqs-scr-article w-richtext"> <h2>Core Problem</h2> <p>Suppose you are an insurer in an insurance company that insures thousands of policyholders. On a typical day, you may expect to accept claims of a few policyholders. But now imagine a catastrophic event such as an earthquake which resulted in many claims of policyholders all at once. If all these claims are accepted, then the company would have to bear the loss in millions which would overwhelm their financials.</p> <p>As an insurer, how would you make sure that companies are able to absorb these catastrophic losses with a high probability?</p> <h2>Risk Measures</h2> <p>When it comes to dealing with risks, it is easy for you to identify potential sources of it. For example, if an insurance policy covers only natural disasters such as earthquakes, then that coverage is already a source of risk. But, the hardest part is when you’re trying to look at risk from the quantitative side. So how would you actually quantify risks? This is where the idea of risk measures comes in.</p> <p>Risk measures are there to help you translate numbers to potential risks. It helps provide a numerical intuition on how likely a certain event may happen or how large an impact is going to be given a ‘risky’ event. There is no fixed approach on how you can measure risk, but typically you want to make sure that your risk measure is built to capture the properties you wanted. However, not all properties need to be captured which can be seen in how a risk measure that does not exhibit all expected properties remains one of the most commonly used risk measures in the financial world which is the Value-at-Risk or VaR in short.</p> <h2>Value-at-Risk</h2> <p>To give a picture of what Value-at-Risk does, we want to first introduce a random variable $X$ which represents loss alongside $x$ as the loss bound. Then, $F_X(x)$ would be defined as the probability that the loss bound is not exceeded by $X$.</p> <p>What the Value-at-Risk does is provide a threshold of how much you could lose at a given confidence level denoted by $1-\alpha$.</p> <div class="math-block"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="383.28pt" height="31.8675pt" viewBox="0 0 383.28 31.8675" xmlns="http://www.w3.org/2000/svg" version="1.1" class="aqs-var-equation" role="img" aria-labelledby="aqs-var-equation-title" style="display:block;width:100%;max-width:620px;height:auto;margin:0 auto"><title id="aqs-var-equation-title">VaR at confidence level 1 minus alpha: infimum of x in the real numbers such that F subscript X of x is at least 1 minus alpha.</title> <defs> </defs> <g id="figure_1"> <g id="patch_1"> <path d="M 0 31.8675 L 383.28 31.8675 L 383.28 0 L 0 0 L 0 31.8675 z " style="fill: none"/> </g> <g id="text_1"> <!-- $\mathrm{VaR}_{1-\alpha}(X)=\inf\left\{x\in\mathbb{R}\mid F_X(x)\geq 1-\alpha\right\}$ 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</defs> <use xlink:href="#STIXGeneral-Regular-56"/> <use xlink:href="#STIXGeneral-Regular-61" transform="translate(72.199982 0)"/> <use xlink:href="#STIXGeneral-Regular-52" transform="translate(116.599976 0)"/> <use xlink:href="#STIXGeneral-Regular-31" transform="translate(184.843254 -13.228125) scale(0.7)"/> <use xlink:href="#STIXGeneral-Regular-2212" transform="translate(229.951241 -13.228125) scale(0.7)"/> <use xlink:href="#STIXGeneral-Italic-3b1" transform="translate(288.009227 -13.228125) scale(0.7)"/> <use xlink:href="#STIXGeneral-Regular-28" transform="translate(331.0586 0)"/> <use xlink:href="#STIXGeneral-Italic-58" transform="translate(364.358588 0)"/> <use xlink:href="#STIXGeneral-Regular-29" transform="translate(425.458579 0)"/> <use xlink:href="#STIXGeneral-Regular-3d" transform="translate(473.198563 0)"/> <use xlink:href="#STIXGeneral-Regular-69" transform="translate(556.138544 0)"/> <use xlink:href="#STIXGeneral-Regular-6e" transform="translate(583.938532 0)"/> <use 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xlink:href="#STIXGeneral-Regular-31" transform="translate(1309.891303 0)"/> <use xlink:href="#STIXGeneral-Regular-2212" transform="translate(1374.331284 0)"/> <use xlink:href="#STIXGeneral-Italic-3b1" transform="translate(1457.271265 0)"/> <use xlink:href="#STIXGeneral-Regular-7d" transform="translate(1512.471262 0)"/> </g> </g> </g> </svg></div> <p>In other words, the VaR provides the smallest loss bound $x$ such that the probability of the actual loss staying below the loss bound is at least $1-\alpha$.</p> <p>For example, at a 99% confidence level, the probability that your loss does not exceed the loss bound is 99%, whereas there is still a 1% chance that your loss can still exceed the loss bound.</p> <p>One thing to keep in mind is that the VaR is just a quantile of your loss distribution. It doesn’t give you exactly on how large your loss would actually be if it exceeds the loss bound. Despite that, VaR remains important, especially with its use in regulatory frameworks such as Solvency II.</p> <h2>VaR in Solvency II</h2> <p>Under Solvency II, the VaR is used to determine the Solvency Capital Requirements (SCR) which is a capital requirement for insurance and reinsurance companies in the European Economic Area to make sure they have enough capital to fulfill their obligations to policyholders. Under the framework, the SCR must be calibrated to a 1-in-200 year event over a one-year horizon or in other words, the SCR is the 99.5% Value-at-Risk of the basic own funds over a period of one year. So what does this actually mean in practice? Let’s take a simple example to illustrate how the VaR works in the context of Solvency II.</p> <h2>Situation</h2> <p>You are an insurer in an insurance company which handles earthquake insurance. The company’s balance sheet is as follows:</p> <ul><li>Assets : 250 million euros</li><li>Liabilities : 120 million euros</li></ul> <p>Each year, there is a 5% probability that an earthquake occurs in a year and in an event that an earthquake occurs, the claim size is lognormal with a logmean of log 80 million and the log standard deviation of 30%. Assuming that the only risk is an earthquake risk, will the insurance company have sufficient capital to absorb the loss?</p> <p>We will perform a simulation which estimates the Solvency Capital Requirement by generating possible outcomes for next year and use the resulting distribution to get the SCR.</p> <p>For $n = 10000$, we will simulate $n$ years and for each year, a number between 0 and 1 is drawn to decide whether an earthquake occurs. An earthquake occurs if the drawn number is lower than the probability of an earthquake occurring.</p> <p>Next, we will draw the claim size from the lognormal distribution and keep only the ones where an earthquake occurs. In the case where no earthquake occurs, the claim size remains 0.</p> <p>Then we will identify the change in the basic own fund which is only represented by the negative value of the claim size. For a higher claim size, it means more money is out of the insurance firm’s pockets, so the change in its basic own fund will be more negative.</p> <p>The graph below represents the distribution of the change in basic own funds and the red line represents the 0.005 quantile of the distribution where we obtain the 99.5% VaR of the loss in basic own funds. The SCR is just the positive counterpart of the VaR.</p> <figure class="aqs-scr-figure"><img src="https://cdn.prod.website-files.com/6913793ee068962f268ed83c/6ae5f372a3eb9f62179c7828_solvency-ii-original-simulation.png" alt="Distribution of Change in own Funds" loading="lazy"></figure> <p>Our scenario shows an SCR of around 116 million euros, while the insurance company’s basic own funds amount to 130 million euros. Since the basic own funds are higher than the SCR, the insurer has sufficient capital to absorb the catastrophic loss due to the earthquake.</p> <p>Our findings point out how risk measures are used to support insurance companies in assessing and managing potential risks. In particular, the Value-at-Risk provides a loss threshold at a given confidence level that allows the company to determine the minimum capital it needs to absorb catastrophic losses and solvent according to Solvency II.</p> <h2>References:</h2> <ul class="aqs-scr-references"> <li><a href="https://www.solvencyiisolutions.com/solvency-ii-pillar-1/">https://www.solvencyiisolutions.com/solvency-ii-pillar-1/</a></li> <li><a href="https://eur-lex.europa.eu/eli/dir/2009/138/oj">https://eur-lex.europa.eu/eli/dir/2009/138/oj</a></li> <li>Jon Danielsson, Financial Risk forecasting: The Theory and Practice of Forecasting Market Risk with Implementation in R and Matlab, Wiley, 2011</li> </ul> </div>
