Fixed-income security - duration

4 important questions on Fixed-income security - duration

Duration of fixed-income instrument

weighted average of the times that payments (cash flows) are made. 
The weighting coefficients are the present values of the individual cash flows

D = PV(t0)t0 + ... + PV(tn)tn / PV

Macualay duration formula

all coupon payments are identical

The macualay duration for a bond with a coupon rate c per period, yield y per period, m periods per year, and exactly n periods remaining is

D= 1+y  -  1+y+n(c-y)                    
       my       mc((1+y)^n - 1)) + my

Price sensitivity formula

the derivative of price P with respect to yield λ of a fixed-income security is:

dP/dλ = -DmP

where Dm=D/(1+(λ/m)) is the modified duration


Note that DmD for large values of m or small values of λ
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Duration of a portfolio

Suppose there are m fixed-income securities with prices and durations of Pi and Di, respectively i=1,2...,m, all computed at a common yield. The portfolio consisting of the aggregate of the securities has price P and duration D, given by

P=P1 + P2 + ... + Pm
D= w1D1 + w2D2 + ... + wmDm

where wi=Pi/P i=1,2...m

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