I am trying to figure out the calculation steps of the derivative of Duration equation that turns into Convexity. I feel stupid asking this as I haven’t touch calculus in so long. Anyone want to take a stab? Duration = (V- - V+)/ (2V0 (chng Y)) Convexity = (V+ - V- - 2V0)/(2V0(chng Y)^2)
Pointless with just 18 or so days to go until June 6. Just memorize the formulas, the derivative is uninteresting for now, you can derive it on June 7.
studymyrearoff Wrote: ------------------------------------------------------- > I am trying to figure out the calculation steps of > the derivative of Duration equation that turns > into Convexity. I feel stupid asking this as I > haven’t touch calculus in so long. Anyone want to > take a stab? > > Duration = (V- - V+)/ (2V0 (chng Y)) > > Convexity = (V+ - V- - 2V0)/(2V0(chng Y)^2) the v-, v+ isn’t the real mathematical formula–it’s just a simplification for the exams.
figure I give it a try. I’ll just memorize it as suggested.
You need the price-yield equations. Convexity is the second term of the Taylor expansion of price-yield. You can take the derivative of Duration formula but it will be tricky since its in a fraction form. Generally d/dx (y/x) = yx + yx or something like that. Its been a while.
you want to take the derivative of the macaulay duration formula which is simply the time weighted PV of cashflows divided by the PV of cashflows. the formula you are trying to use is effective duration, and is simply an estimate.
Derivative = V’ = dV/dy approximate V’(+)=[V(+)-V(0)]/dy and V’(-)=[V(0)-V(-)]/dy, then V’(0)=1/2*[V’(+)+V’(-)] = [V(+)-V(-)]/(2*dy) from there you have Duration = -V’/V = [V(-)-V(+)]/(2*V(0)*dy) Second derivative = V’’ = dV’/dy = [V’(+)-V’(-)]/(2*dy) = [V(+)+V(-)-2*V(0)]/(2*dy^2) Convexity = V’’/V = [V(+)+V(-)-2*V(0)]/(2*V(0)*dy^2)