3 conditions:
(1) PV of liabilities = PV of assets, at market rate of interest
V(L)=V(A)

(2) Discounted mean term of assets equals the discounted mean term of liabilities
(Σt*V^t*L(t)) / (ΣV^t*L(t)) = (Σt*V^t*A(t)) / (ΣV^t*A(t))

(3) The spread of the liability proceeds about its mean term is less than that of the asset proceeds
(Σt^2*V^t*L(t))/(ΣV^t*L(t)) < (Σt^2*V^t*A(t))/(ΣV^t*A(t))

or if you go with derivative: Let  f (i) =V(L) −V(A)
Apply Taylor’s Expansion:
f (i + ε) = f(i) +εf ′(i) + (ε^2/2) f ′′ (i) + ..... o(i)
then 3 conditions come to:
(1) f(i) = 0
(2) f ′(i) = 0
(3) f ′′ (i) >0
Then, for small ε , f (i +ε ) > f (i) = 0. Hence portfolio is ‘immunised’ against a small change in the ruling interest rate.
i.e., present value of assets will not be lower than present value of the liabilities at the new ruling interest rate i + ε.

Thank you very much

tks~~~~~~~~~~~~