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Agenda Some duration formulas (CT1, Unit 13, Sec. 5.3) An aside on annuity bonds (Lando & Poulsen Sec. 3.3 – attached to hand-out) Convexity (CT1, Unit 13, Sec. 5.4) Immunisation (CT1, Unit 13, Sec. 5.5) 1 October 26, 2010 MATH 2510: Fin. Math. 2 Duration Measures the sensitivity of present values/prices to changes in the interest rate. It has ”dual” meaning: A derivative wrt. the interest rate A value-weighted discounted average of payment times (so: its unit is ”years”) 2 October 26, 2010 MATH 2510: Fin. Math. 2 Set-up: Cash-flows C at tktk Yield curve flat at i (or continuously compounded/on force form: ) Present value of cash-flows: A k Ct k (1 i)tk k Ct k P(tk ) 3 October 26, 2010 MATH 2510: Fin. Math. 2 Macauley Duration The Macauley duration (or: discounted mean term) is defined by k tk Ctk (1 i) tk : A Clearly a weighted average of payment dates. But also: Sensitivity to changes in the force of interest. Or put differently: To parallel shifts in the (continuously compounded) yield curve. 4 October 26, 2010 MATH 2510: Fin. Math. 2 Duration of an Annuitiy The duration of an n-year annuity making payments D is (independent of D and) equal to ( Ia) n| , an| (Note: On the Oct. 21 slides there was either a D too much or a D too little) where as usual a (1 v n ) / i with v (1 i ) 1 n| 5 October 26, 2010 MATH 2510: Fin. Math. 2 and (IA) is the value of an increasing annuity ( IA) n| : v 2v 2 3v 3 ... nv n v v 2 v 3 ... v n v(v 2v 2 3v 3 ... (n 1)v n 1 ) an| (1 i ) 1 ((IA) n| nv n ) (1 i )an| nv n ( IA) n| i 6 October 26, 2010 MATH 2510: Fin. Math. 2 Annuity Bonds The remaining principal (outstanding notional) of an annuity bond with (nominal) coupon rate r and (annual) payment D satisfies pt (1 r ) pt 1 D Repeated substitution gives pn (1 r ) p0 D j 0 (1 r ) j n n 1 7 October 26, 2010 MATH 2510: Fin. Math. 2 Suppose (wlog) p0=100. For the loan to be paid off after n periods we must have pn=0, i.e. 100 D j 0 (1 r ) ( j n ) 0 n 1 8 October 26, 2010 MATH 2510: Fin. Math. 2 This we can rewrite to solve for the yearly payment r 100 D 100 n . 1 (1 r ) an| In finance people will often refer to this as the annuity formula. The yearly payments (or: instalments) consist of interest payments and repayment of principal. 9 October 26, 2010 MATH 2510: Fin. Math. 2 Duration of a Bullet Bond Using similar reasoning, the duration of a bullet bond w/ coupon payments D and notional R is D ( Ia ) n| Rnv n Da n| Rv n (Note: On the Oct. 21 slides D was missing in the denominator.) 10 October 26, 2010 MATH 2510: Fin. Math. 2 Convexity The convexity of Ct k is defined as k tk (tk 1)Ctk (1 i) (tk 2) 1 A 2 c(i ) : A i 2 A This is the ”effective” (or: ”volatility” version) – could also do Macauley or Fisher-Weil style. 11 October 26, 2010 MATH 2510: Fin. Math. 2 Convexity and (effective) duration give a 2nd order accurate (Taylor expansion) approximation to changes in present value for (small) interest rate changes A(i ) A(i) 1 A(i) 1 1 2 A(i) 2 A(i) A(i) i 2 A(i) i 2 1 v(i) c(i) 2 2 12 October 26, 2010 MATH 2510: Fin. Math. 2 A classical ”picture of” duration and convexity Approximating present value changes w/ duration and convexity 0.5 0.4 Relative change in present value 0.3 0.2 0.1 Exact Dur. & conv. (2nd order) 0 Dur. (1st order) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -0.1 -0.2 -0.3 -0.4 Interest rate 13 October 26, 2010 MATH 2510: Fin. Math. 2 Convexity is a measure of dispersion around the duration We can write Macauley duration as t twt and Macauley-style convexity as t t 2 wt A measure of dispersion around the duration is : t (t ) 2 wt t (t 2 2 2t ) wt t t 2 wt 2 t twt 2 t wt 2 14 October 26, 2010 MATH 2510: Fin. Math. 2 Immunisation Consider a pension fund that has assets At and k liabilities Lt . k We say that the fund is immunised against movements in the interest rate around i0 if PVA (i0 ) PVL (i0 ) and PV A (i0 ) PVL (i0 ) 15 October 26, 2010 MATH 2510: Fin. Math. 2 By Taylor expanding the difference between assets and liabilities (known as the surplus) we get that the ”inequality condition” is fulfilled if - the assets and the liabilities have the same duration, and - convexity of the assets is higher than the convexity of the liabilities 16 October 26, 2010 MATH 2510: Fin. Math. 2 These are known as Redington’s conditions. It doesn’t matter whether we use effective or Macaluey duration. Typical exercise approach: The ”equality conditions” give two linear equations in two unknows; the convexity condition follows from a dispersion consideration. 17 October 26, 2010 MATH 2510: Fin. Math. 2 Immunisation isn’t the be-all-and-end- all of interest rate risk management Note that an immunised portfolio is looks very much like an arbitrage. That tells us that considering only parallel shifts to flat yield curves isn’t the perfect way to model interest rate uncertainty. And models with genuinely random behaviour is the next topic. 18 October 26, 2010 MATH 2510: Fin. Math. 2