Fixed Income portfolio management: - quantifying & measuring interest rate risk

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

2. Zero-coupon bond prices for various yields & maturities

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

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

7. Modified duration Percentage change in bond price: Modified Duration (DM): DM = Dc/(1+y) (annual coupon) DM = Dc/(1+y/2) (semiannual coupon) Change in bond price: Duration is linear approximation

8. Duration for an annual coupon bond

9. Duration for a semi-annual coupon bond

10. Duration for a semi-annual coupon bond

11. Price Value of Basis Point (PVBP) PVBP = DM x Value x .0001 Example: portfolio value = $100,000; DM = 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?

13. Convexity: adjusting for non-linearity Predicted % price change using duration: DP/P = -DmDy 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: DP/P = - DmDy + (1/2) Convexity (Dy)2 (a 2nd order Taylor series expansion) (the convexity adjustment is always POSITIVE) (We will not hand-calculate convexity)

14. Example using Convexity example: 30 year, 8% coupon bond with y-t-m of 8%. Modified duration = 11.26, Convexity = 212.4 What is predicted % price change for increase of yield to 10%? Duration prediction: DP/P = - DmDy = -11.26 x 2.0% = -22.52% Duration & convexity prediction: DP/P = - DmDy + (1/2) Convexity (Dy)2 = -11.26 x 2.0% + (1/2) 212.4 (.02)2 = -22.52% + 4.25% = -18.27% Actual % price change: price at 8% yield = 100; price at 10% yield = 81.15. % change = -18.85%

15. Asset-Liability Interest Rate Rrisk Management Example: The BillyBob Bank Simplified balance sheet risk analysis: Amount Duration PVBP Assets $100 mm 6.0 100,000,000 x 6.0 x 0.0001 = $60,000 Liabilities 90 mm 2.0 90,000,000 x 2.0 x 0.0001 = 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) = DE x VE x 0.0001 $42,000 = DE x ($10,000,000) x 0.0001 DE = $42,000/$1000 = 42.0

16. The BillyBob Bank, continued Simplified balance sheet risk analysis: Amount Duration PVBP Assets $100 mm 6.0 100,000,000 x 6.0 x 0.0001 = $60,000 Liabilities 90 mm 2.0 90,000,000 x 2.0 x 0.0001 = 18,000 Equity 10 mm 42.0 PVBP(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 Dy) / (100mm – 60,000 Dy) and solve for Dy: 0.08 (100mm – 60,000 Dy ) = 10mm - 42,000 Dy $8 mm – 4800 Dy = 10mm - 42,000 Dy (42,000 – 4800) Dy = $2,000,000 Dy = $2,000,000/$37,200 = 53.76 basis points