Independent solution

How to solve this Redington Immunization question

Setup

Setup

Measure times in years. The liability time is 0.75, midway between asset times 0.50 and 1.00.

tX=0.50,tL=0.75,tY=1.00t_X=0.50,\quad t_L=0.75,\quad t_Y=1.00

Model

Model

Equal present value and duration require the asset present-value weights to average to 0.75. Hence each asset carries half the liability present value.

0.50wX+1.00wY=0.75(wX+wY)0.50w_X+1.00w_Y=0.75(w_X+w_Y)
wX=wY=5,000v0.75w_X=w_Y=5{,}000v^{0.75}

Compute

Compute

Convert the time-zero present-value allocation for X into its nominal payment at time 0.50.

Xv0.50=5,000v0.75Xv^{0.50}=5{,}000v^{0.75}
X=5,000v0.25=4,975.308X=5{,}000v^{0.25}=4{,}975.308

Answer

Answer

The six-month cash flow rounds to 4975, choice B.

X≈4,975(B)\boxed{X\approx4{,}975\quad\text{(B)}}