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<HTML> <HEAD> <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=windows-1252"> <META NAME="Generator" CONTENT="Microsoft Word 97"> <TITLE>Interest</TITLE> <META NAME="Template" CONTENT="C:\PROGRAM FILES\MICROSOFT OFFICE\OFFICE\html.dot"> </HEAD> <BODY LINK="#0000ff" VLINK="#800080"> <B><FONT FACE="Arial"><P ALIGN="CENTER">Interest Rate Risk Management</P> </B></FONT><FONT FACE="Arial" SIZE=1><P ALIGN="CENTER">Copyright 1997 by FinSoft, Inc.</P><DIR> <P ALIGN="CENTER"> </P> <P>As appropriate for the instrument, ValueCalc provides the following types of valuation and interest rate risk measures for use in portfolio risk management.</P></DIR> <OL> <B><LI>Value </B>is the model value of the instrument.</LI> <B><LI>Delta </B>is the rate of change in Value as an underlying variable changes. In the case of interest rate instruments, the underlying variable is the term structure of continuously compounded spot interest rates.</LI> <B><LI>Gamma</B> is the rate of change in Delta as the underlying variable changes.</LI> <B><LI>Theta</B> is the rate of change in Value as time changes.</LI> <B><LI>Vega</B> is the rate of change in Value as Volatility changes.</LI></OL> <P> </P><DIR> <B><P>Use of Delta</P> </B><P>The Delta related change in the value of an interest rate instrument resulting from a small shift in the term structure of continuously compounded spot interest rates is given by</P> <P> </P> </FONT><P ALIGN="CENTER"><IMG SRC="binary.gif" WIDTH=57 HEIGHT=18><FONT FACE="Arial" SIZE=1>.</P> <P ALIGN="CENTER"> </P> <P>Delta is thus equal to the negative of the instrument's <B>Dollar Duration </B>(see Fabozzi 1993, pp. 170-171) calculated with respect to shifts in spot rates. That is,</P> <P> </P> <I><P ALIGN="CENTER">Delta = - Dollar Duration</I>.</P> <P> </P> <P>This estimate of Dollar Duration is moderately larger than the one which could be calculated with respect to changes in semi-annually compounded yields to maturity. Also an instrument's <B>Effective Duration</B><I> </I>(see Fabozzi 1993, pp. 170-171), which gives a measure of relative price volatility, can be calculated by dividing its Dollar Duration (i.e. - Delta) by its Value. That is,</P> <P> </P> <I><P ALIGN="CENTER">Effective Duration = -Delta/Value</I>.</P> <I><P ALIGN="CENTER"> </P> </I><P>The change in value resulting from a one basis point change in spot rates is equal to Delta times .0001. That is,</P> <P> </P> <I><P ALIGN="CENTER">Price Value of a Basis Point</I> =<I> .0001Delta</I>.</P> <P> </P> <P>Portfolio deltas with respect to various underlying variables can be calculated as the sum of the deltas for individual instruments included in the portfolio. That is, for an <I>n</I> asset portfolio </P> <P> </P> <I><P ALIGN="CENTER">Portfolio Delta</I> =<IMG SRC="Image124.gif" WIDTH=78 HEIGHT=45>.</P> <P ALIGN="CENTER"> </P> <P>The Delta related change in the value of the portfolio resulting from a small change in spot interest rates is given by</P> <P> </P> <I><P ALIGN="CENTER">(Portfolio Delta)<IMG SRC="Image125.gif" WIDTH=22 HEIGHT=17></I>.</P> <P ALIGN="CENTER"> </P> <P>The Portfolio Delta Hedge Ratio can be calculated by dividing the portfolio's delta by the delta of the hedging vehicle. In cases where the asset to be hedged and the hedging vehicle are priced relative to different term structures, this hedge ratio should be multiplied by a <B>yield beta</B> (see Fabozzi 1993, p.174) which reflects the relative volatility of the different term structures. Thus,</P> <P> </P> <I><P ALIGN="CENTER">Portfolio Delta Hedge Ratio = (Portfolio Delta)(Yield Beta)/(Delta of Hedging Vehicle)</I>.</P> <B><P> </P> <P>Use of Gamma</P> </B><P>The Value of many instruments has a non-linear relationship to an underlying variable. Gamma is the rate of change in Delta as the underlying variable changes. An estimate of the Gamma related change in the value of an interest rate instrument resulting from a small shift in spot interest rates is given by</P> <P> </P> </FONT><P ALIGN="CENTER"><IMG SRC="Image126.gif" WIDTH=100 HEIGHT=24><FONT FACE="Arial" SIZE=1>.</P> <P ALIGN="CENTER"> </P> <P>Gamma is thus equal to the instrument's <B>Dollar Convexity</B><I> </I>(see Fabozzi 1993, p. 198) calculated with respect to shifts in spot rates. That is,</P> <I><P ALIGN="CENTER">Gamma = Dollar Convexity</I>.</P> <I><P ALIGN="CENTER"> </P> </I><P>This estimate of Dollar Convexity is moderately larger than the one which could be calculated with respect to changes in semi-annually compounded yields to maturity. Also an instrument's <B>Effective Convexity</B><I> </I>(see Fabozzi 1993, p. 212), which gives a measure of relative price volatility, can be calculated by dividing its Dollar Convexity (i.e. Gamma) by its Value. That is,</P> <P> </P> <I><P ALIGN="CENTER">Effective Convexity = Gamma/Value</I>.</P> <P ALIGN="CENTER"> </P> <P>Portfolio gammas with respect to various underlying variables can be calculated as the sum of the gammas for individual instruments included in the portfolio. That is, for an <I>n</I> asset portfolio</P> <P> </P> <I><P ALIGN="CENTER">Portfolio Gamma</I> =<IMG SRC="Image127.gif" WIDTH=93 HEIGHT=45>.</P> <P ALIGN="CENTER"> </P> <P>The Gamma related change in the value of the portfolio with respect to a small change in spot interest rates is</P> <P> </P> <P ALIGN="CENTER">.<I>5(Portfolio Gamma)<IMG SRC="Image128.gif" WIDTH=40 HEIGHT=24></I>.</P> <P ALIGN="CENTER"> </P> <P>Portfolio gamma hedge ratios can be calculated by dividing the portfolio's gamma by the gamma of the hedging vehicle. Thus,</P> <P> </P> <I><P ALIGN="CENTER">Portfolio Gamma Hedge Ratio = (Portfolio Gamma)/(Gamma of Hedging Vehicle)</I>.</P> <P ALIGN="CENTER"> </P> <P> <B>Use of Theta</P> </B><P>An estimate of the change in the value of an instrument resulting from a small shift in time is given by</P> <P> </P> </FONT><P ALIGN="CENTER"><IMG SRC="Image129.gif" WIDTH=56 HEIGHT=18><FONT FACE="Arial" SIZE=1>.</P> <P ALIGN="CENTER"> </P> <P>Portfolio thetas can be calculated as the sum of the thetas for individual instruments included in the portfolio. That is, for an <I>n</I> asset portfolio</P> <I><P ALIGN="CENTER">Portfolio Theta</I> =<IMG SRC="Image130.gif" WIDTH=80 HEIGHT=45>.</P> <B><P> </P> <P>Use of Vega</P> </B><P>An estimate of the change in the value of an option resulting from a small shift in the volatility (vol) of an underlying variable is given by</P> <P> </P> </FONT><P ALIGN="CENTER"><IMG SRC="Image131.gif" WIDTH=66 HEIGHT=21><FONT FACE="Arial" SIZE=1>.</P> <P ALIGN="CENTER"> </P> <P>Portfolio Vegas with respect to underlying variables can be calculated as the sum of the vegas for individual instruments included in the portfolio. That is, for an <I>n</I> asset portfolio </P></DIR> </FONT><I><FONT FACE="Arial" SIZE=2><P ALIGN="CENTER">Portfolio Vega</I> </FONT><FONT FACE="Arial">=<IMG SRC="Image132.gif" WIDTH=73 HEIGHT=45>.</P><DIR> </FONT><FONT FACE="Arial" SIZE=1><P ALIGN="CENTER"> </P> <P>The change in the value of the portfolio resulting from volatility changes in an underlying variable is </P> <P> </P> <I><P ALIGN="CENTER">(Portfolio Vega)<IMG SRC="Image133.gif" WIDTH=34 HEIGHT=17></I>.</P> <P ALIGN="CENTER"> </P> <P>Portfolio vega hedge ratios can be calculated by dividing the portfolio's vega by the vega of the hedging vehicle. Thus,</P> <P> </P> <I><P ALIGN="CENTER">Portfolio Vega Hedge ratio =(Portfolio Vega)/(Vega of Hedging Vehicle)</I>.</P> </FONT><FONT FACE="Arial"><P> </P></DIR> </FONT></BODY> </HTML>