{"id":2162,"date":"2021-12-02T11:05:45","date_gmt":"2021-12-02T03:05:45","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=2162"},"modified":"2022-03-02T16:51:27","modified_gmt":"2022-03-02T08:51:27","slug":"risk-aversion","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=2162","title":{"rendered":"Risk Aversion"},"content":{"rendered":"\n<p><strong>The Arrow-Pratt coefficient of absolute risk aversion<\/strong><\/p>\n\n\n\n<p><strong>Definition <\/strong>(Arrow-Pratt coefficient of absolute risk aversion). Given a twice differentiable Bernoullio utility function \\(u(\\cdot)\\),<\/p>\n\n\n\n<p>$$ A_u(x):=-\\frac{u&#8221;(x)}{u'(x)} $$<\/p>\n\n\n\n<ul><li>Risk-aversion is related to concavity of \\(u(\\cdot)\\); a &#8220;more concave&#8221; function has a smaller (more negative) second derivative hence a larger \\(u&#8221;(x)\\).<\/li><\/ul>\n\n\n\n<ul><li>Normalisation by \\(u'(x)\\) takes care of the fact that \\(au(\\cdot)+b\\) represents the same preferences as \\(u(\\cdot)\\).<\/li><\/ul>\n\n\n\n<ul><li>In probability premium<\/li><\/ul>\n\n\n\n<p>Consider a risk-averse consumer: <\/p>\n\n\n\n<p><strong>1<\/strong>. prefers \\(x\\) for certain to a 50-50 gamble between \\(x+\\epsilon\\) and \\(x-\\epsilon\\). <\/p>\n\n\n\n<p><strong>2<\/strong>. If we want to convince the agent to take the gamble, it could not be 50-50 &#8211; we need to make the \\(x+\\epsilon\\) payout more likely. <\/p>\n\n\n\n<p><strong>3<\/strong>. Consider the gamble G such that the agent is indifferent between G and receiving x for certain, where <\/p>\n\n\n\n<p>$$G= \\begin{cases}     x+\\epsilon, &amp; \\text{with probability $\\frac{1}{2}+\\pi$}.\\\\     x-\\epsilon, &amp; \\text{with probability  $\\frac{1}{2}-\\pi$ }  \\end{cases}$$<\/p>\n\n\n\n<p><strong>4<\/strong>. It turns out that \\(A_u(x)\\) is <em>proportional to <\/em>\\(\\pi\/\\epsilon\\) as \\(\\epsilon \\rightarrow 0\\); i.e., \\(A_u(x)\\) tells us the &#8220;premium&#8221; measured in probability that the decision-maker demands per unit of spread \\(\\epsilon\\).<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code> <\/code><\/pre>\n\n\n\n<p><strong>ARA<\/strong>.<\/p>\n\n\n\n<p>Decreasing Absolute Risk Aversion. The Bernoulli function \\(u\\cdot)\\) has decreasing absolute risk aversion iff \\(A_u(\\cdot)\\) is a decreasing function of \\(x\\). Increasing Absolute Risk Aversion&#8230; Constant Absolute Risk Aversion &#8211; Bernoulli utility function has constant absolute risk aversion iff \\(A_u(\\cdot)\\) is a constant function of \\(x\\).<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Relative Risk Aversion<\/strong><\/h4>\n\n\n\n<p>Definition (coefficient of relative risk aversion). Given a twice differentiable Bernoulli utility function \\(u(\\cdot)\\),<\/p>\n\n\n\n<p>$$ R_u(x):=-x\\frac{u&#8221;(x)}{u'(x)}=xA_u(x) $$<\/p>\n\n\n\n<p>There could be decreasing\/increasing\/constant relative risk aversion as above.<\/p>\n\n\n\n<p><strong>Implication:<\/strong> DARA means that if I take a <span class=\"katex math inline\">10 gamble when poor, I will take a<\/span>10 gamble when risk. DRRA means that if I gamble 10% of my wealth when poor, I will gamble 10% when rich.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Arrow-Pratt coefficient of absolute risk aversion Definition (Arrow-Pratt coefficient of absolute risk aversion). Given a twice differentiable Bernoullio utility function \\(u(\\cdot)\\), $$ A_u(x):=-\\frac{u&#8221;(x)}{u'(x)} $$ Risk-aversion is related to concavity of \\(u(\\cdot)\\); a &#8220;more concave&#8221; function has a smaller (more negative) second derivative hence a larger \\(u&#8221;(x)\\). Normalisation by \\(u'(x)\\) takes care of the fact &hellip; <a href=\"https:\/\/fanyuzhao.com\/?p=2162\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Risk Aversion<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,6],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/2162"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2162"}],"version-history":[{"count":29,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/2162\/revisions"}],"predecessor-version":[{"id":2194,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/2162\/revisions\/2194"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2162"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2162"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2162"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}