{"id":387,"date":"2015-01-05T18:59:34","date_gmt":"2015-01-05T18:59:34","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=387"},"modified":"2015-02-20T18:51:31","modified_gmt":"2015-02-21T00:51:31","slug":"1-financial-math-concepts","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/interest-rate-risks-and-simulation\/1-financial-math-concepts\/","title":{"rendered":"1. Financial Math Concepts"},"content":{"rendered":"<p>Recall a few concepts that you have seen an in earlier financial mathematics course:<br \/>\n\\(v(t)\\) &#8211; the <em>current market price<\/em> of a \\(t\\)-year coupon bond<br \/>\n\\(y_t\\) &#8211; the \\(t\\)<em>-year spot rate of interest<\/em>. This is defined through the expression<br \/>\n\\begin{eqnarray*}<br \/>\nv(t)(1+y_t)^t = 1 \\textrm{ which is the same thing as } v(t)=\\frac{1}{(1+y_t)^t}<br \/>\n\\end{eqnarray*}<br \/>\n<em>yield curve<\/em> &#8211; a plot of \\({y_t}\\) versus \\(t\\)<br \/>\n<em>term structure of interest rates<\/em> describes this relationship (between \\({y_t}\\) and \\(t\\)).<br \/>\n<em>flat<\/em> term structure &#8211; \\(v(t) =v^t = e^{-\\delta t}\\), or \\(y_t \\equiv y\\).<br \/>\n<em>forward rates of interest<\/em> &#8211; \\(f(t,t+k)\\), the (annual) interest rate contracted at time 0 earned on an investment made at time \\(t\\) that matures at time \\(t+k\\)<br \/>\n\\begin{eqnarray*}<br \/>\n(1+f(t,t+k))^k= \\frac{(1+y_{t+k})^{t+k}}{(1+y_t)^t}= \\frac{v(t)}{v(t+k)}<br \/>\n\\end{eqnarray*}<br \/>\n<em>no-arbitrage argument<\/em> &#8211; we should not be able to make money from nothing in risk free bonds by disinvesting and then reinvesting. An arbitrage opportunity exists if an investor can construct a portfolio that costs zero at inception and generates positive profits with a non-zero probability in the future, with no possibility of incurring a loss at any future time.<\/p>\n<p><div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/interest-rate-risks-and-simulation\/\" title=\"Interest Rate Risks and Simulation\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/interest-rate-risks-and-simulation\/2-valuation-of-insurances-and-life-annuities\/\" title=\"2. Valuation of Insurances and Life Annuities\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Recall a few concepts that you have seen an in earlier financial mathematics course: \\(v(t)\\) &#8211; the current market price of a \\(t\\)-year coupon bond \\(y_t\\) &#8211; the \\(t\\)-year spot rate of interest. This is &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":383,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-6f","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/387"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=387"}],"version-history":[{"count":6,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/387\/revisions"}],"predecessor-version":[{"id":794,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/387\/revisions\/794"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/383"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=387"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}