Duration Duration is the weighted average time to maturity using the relative present values of the cash flows as weights For zero coupon bond duration maturity For all others Duration maturity Cash s Duration is always zero PVIF PV CF CF Coupon x PVIF Duration Features D increases with M but at a decreasing rate D decreases as YTM increases D decreases as coupon interest increases Everything else equal 1 When the maturity of a bond lengthens the duration rises as well 2 When interest rates rise the duration of a coupon bond falls 3 The higher is the coupon rate on the bond the shorter is the duration of the bond 4 Duration is additive the duration of a portfolio of securities is the weighted average of the durations of the individual securities with the weights equaling the proportion of the portfolio invested in each Convexity is desirable but greater convexity causes larger errors in the duration based estimate of price changes Duration gap a measure of overall interest rate risk exposure for a FI DGAP DA kDL where k a k a the Leverage Adjusted Duration Gap measured in years and reflects the degree of duration mismatch in a FI s balance sheet Specifically the larger this gap in absolute terms the more exposed the FI is to interest rate shocks If Duration gap is exposed to interest rates because liabilities duration assets duration Size of the Interest Rate Shock Expose of Net Worth of FI adjusted duration gap x Asset Size x Interest Rate Shock Interest rate shock is negative if rate decreased Size of FI A Unbias Expectancy Theory RN 1 R1 1 E r21 1 E rN1 1 N 1 calculate for each year yield curves reflect market s expectation of ST rates LT rates are geometric average of current and expected rates When Plotting on graph Vertical axis YTM and horizontal axis Term to maturity years Value At Risk VAR Largest potential loss at that Confidence Interval calculated under normal market conditions also used to measure risk of a portfolio Market VAR DEAR x N Cumulative DEARs over a specified period of time The higher the confidence level desired the larger the maximum expected loss that might occur for a given type of investment JP Morgan made its RiskMetrics system available free from charge over the Internet this system provided financial data methodology to calculate a portfolio s VaR Drawbacks Markets are NOT normal Portfolios are non linear Volatility is NOT constant Markets move together but no one knows how JPMorgan s Assumptions in measurement of VAR 1 Prices of financial instruments follow a stable random walk thus price changes are normally distributed 2 Price changes are serially uncorrelated there is no correlation between change today and changes in the past 3 Standard deviation volatility of price or rate changes is stable over time i e past movements may be used to characterize future movements 4 Interrelationships between 2 different price movements follow a joint normal distribution Daily Earnings at Risk DEAR DEAR dollar MV of position x price volatility DEAR Estimated potential loss of a portfolio s value over a one day unwind period as a result of adverse moves in market conditions such as changes in interest rates foreign exchange rates and market volatility Price Volatility MD x adverse daily yield move Modified Duration MD D 1 R Market Value FV 1 R t Potential adverse move in yields Confidence limit value x Standard Deviation 90 CI 1 65 98 CI 2 33 Aggregating DEAR Estimates Cannot simply sum up individual DEARs In order to aggregate the DEARs from individual exposures we require the correlation matrix Three asset case DEAR portfolio DEARa DEARb DEARa DEARc 2rbc DEARb DEARc 1 2 Chapter 11 Term Structure p 1 k y 1 p 1 k 1 i p 1 k 1 i 2 2 2rab DEARa DEARb 2rac 2 DEARc can be rewritten as 1 i 1 p y 1 k p 1 k p probability corp debt will be paid in full 1 p probability of default 1 i risk free rate p 1 k expected return on corporate securities y collateral in the event of default k 1 risk premium k contractually promised return on corp debt i contractually promised return on credit risk free one year treasury Treasury Strips and Zero Coupon Bonds bonds that are created or issued bearing no coupons and only a face value to paid on maturity They are issued at a large discount from face value aka deep discount bonds FDIC s 3 main goals of Resolution Process 1 keep promise to insured depositors 2 minimize disruption to bank customers and local community 3 resolve the bank at least cost to deposit insurance fund Leverage Ratio Least Cost Resolution January 1995 FDICIA required least cost resolution Purpose for implementing this strategy was to pass more of the failure resolution cost to uninsured depositors Method of failure resolution encourage uninsured depositors to more closely monitor the strategies of bank managers because uninsured depositors assume all of the losses they have a much stronger incentive to monitor and control the actions of bank owners A procedure requiring deposit insurer or other designated entity to implement the resolution method that is determined to be the less cost to the system than all other alternatives including liquidation of the failed bank Insured Deposit Payoff when a bank closes FDIC is appointed receiver and all depositors with insured funds are paid in full amount of their deposits Uninsured depositors and other general creditors are given receivership certificates entitling them to a share of the net proceeds from the sale and liquidation of the failed institutions assets Purchase and Assumption Loss Share closed bank transaction in which a health bank buys some or all assets of a failed bank or thrift and assumes some or all of the liabilities including all insured depositors Acquirer usually pays premium for assumed deposits decreasing FDIC s total resolution cost Insured Deposit Transfer IDT variation of deposit payoff another FI takes responsibility for paying insured depositors the amount they are owed Encourages depositor vigilance Uninsured depositors must file a claim against the receiver of the failed bank share with FDIC in any receivership distributions from the liquidation of the closed bank s assets This usually results in a loss for uninsured depositors a so called haircut All losses are born by shareholders followed by uninsured depositors Total Deposit Transfer all deposits are transferred to another FI Discount Window Central bank as lender of last resort through discount window Short term non permanent Requires
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