Fixed Income portfolio management: - quantifying & measuring interest rate risk Finance 30233, Fall 2010 S. Mann Interest rate risk measures: Duration Convexity PVBP Interest Rate Risk Management

Zero-coupon bond prices for various yields & maturities

Duration Bond price (B c ) as a function of yield (y): Small change in y,  y, changes bond price by how much? Classical duration weights each cash flow by the time until receipt, then divides by the bond price:

Define D M = D c /(1+y) (annual coupon) = D c /(1+y/2) (semi-annual coupon) ( modified duration) approximate % change in Price:  P/P = - D M x  y Modified Duration example: D M = 4.5  y= + 30 bp expected % price change= -4.5 (.0030) = -1.35% linear approximation. Convexity matters.

Modified duration Percentage change in bond price: Change in bond price: Modified Duration (D M ): D M = D c /(1+y) (annual coupon) D M = D c /(1+y/2) (semiannual coupon) Duration is linear approximation

Duration for an annual coupon bond

Duration for a semi-annual coupon bond

Example: portfolio value = $100,000; D M = 4.62 PVBP = (4.62) x 100,000 x.0001 = $46.20 Exercise: estimate value of portfolio above if yield curve rises by 25 bp (in parallel shift). Food for thought: what about non-parallel shifts? Price Value of Basis Point (PVBP) PVBP = D M x Value x.0001

Predicted % price change using duration:  P/P = -D m  y Duration is FIRST derivative of bond price. (slope of curve) convexity is SECOND derivative of bond price (curvature: change in slope) Using duration AND convexity, we can estimate bond percentage price change as:  P/P = - D m  y + (1/2) Convexity (  y) 2 (a 2 nd order Taylor series expansion) (the convexity adjustment is always POSITIVE) (We will not hand-calculate convexity) Convexity: adjusting for non-linearity

example: 30 year, 8% coupon bond with y-t-m of 8%. Modified duration = 11.26, Convexity = What is predicted % price change for increase of yield to 10%? Duration prediction:  P/P = - D m  y = x 2.0% = % Duration & convexity prediction:  P/P = - D m  y + (1/2) Convexity (  y) 2 = x 2.0% + (1/2) (.02) 2 = % % = % Actual % price change: price at 8% yield = 100; price at 10% yield = % change = % Example using Convexity

Asset-Liability Interest Rate Rrisk Management Example: The BillyBob Bank Simplified balance sheet risk analysis: AmountDurationPVBP Assets$100 mm6.0100,000,000 x 6.0 x = $60,000 Liabilities 90 mm2.0 90,000,000 x 2.0 x = 18,000 Equity 10 mm???PVBP(E) = PVBP(A) – PVBP(L) = 60,000 – 18,000 = $42,000 Q: What is effective duration of equity? PVBP(E) = D E x V E x $42,000= D E x ($10,000,000) x D E = $42,000/$1000 = 42.0

The BillyBob Bank, continued Simplified balance sheet risk analysis: AmountDurationPVBP Assets$100 mm ,000,000 x 6.0 x = $60,000 Liabilities 90 mm ,000,000 x 2.0 x = 18,000 Equity 10 mm 42.0PVBP(E) = PVBP(A) – PVBP(L) = 60,000 – 18,000 = $42,000 Assume that the bank has minimum capital requirements of 8% of assets (bank equity must be at least 8% of assets) Q: What is the largest increase in rates that the bank can survive with the current asset/liability mix? A: Set 8% = E / A = ($10mm - $42,000  y) / (100mm – 60,000  y) and solve for  y: 0.08 (100mm – 60,000  y ) = 10mm - 42,000  y $8 mm – 4800  y = 10mm - 42,000  y (42,000 – 4800)  y = $2,000,000  y = $2,000,000/$37,200 = basis points