So a question was raised on another thread a couple of weeks ago about whether it is necessary to have each liability preceded and succeeded by assets. I feel like you can achieve full immunization for a series of liabilities using just two assets on either end of the series.
Correct me if I'm wrong: but given liabilities L_i occuring between times a and b, then you can get full immunity by bookending with cash flows of x and y at times t < a and t' > b using the amounts
x = sum(i=a...b, L_i * (t'-i) * e^{(t-i ) * delta} )/(t'-t)
y = sum(i=a...b, L_i * (i -t) * e^{(t'-i) * delta} )/(t'-t)
It satisfies conditions 1 and 2, and while I haven't worked out a proof of the last case, by doing some tests and looking at the surplus graphs, it seems to work: Here is the graph given initial liabilities at times 3 and 4 using assets at times 1 and 5 and delta = 0.02:
Surplus graph
Graph centered around delta=0.02, showing the convexity
Surplus graph with assets at 1 and 8 with liabilities of 1,4 and 5 at times 2,3 and 5
Correct me if I'm wrong: but given liabilities L_i occuring between times a and b, then you can get full immunity by bookending with cash flows of x and y at times t < a and t' > b using the amounts
x = sum(i=a...b, L_i * (t'-i) * e^{(t-i ) * delta} )/(t'-t)
y = sum(i=a...b, L_i * (i -t) * e^{(t'-i) * delta} )/(t'-t)
It satisfies conditions 1 and 2, and while I haven't worked out a proof of the last case, by doing some tests and looking at the surplus graphs, it seems to work: Here is the graph given initial liabilities at times 3 and 4 using assets at times 1 and 5 and delta = 0.02:
Surplus graph
Graph centered around delta=0.02, showing the convexity
Surplus graph with assets at 1 and 8 with liabilities of 1,4 and 5 at times 2,3 and 5
Full immunization
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