Consider the following MBS pass through with principal $$300$ million. The original

mortgage pool has a WAM $=360$ months $(30$ years$)$ and a WAC $=7\mathrm{.}00\%.$ The

pass through security pays a coupon equal to $6\mathrm{.}5\%.$ Instead of the yield curve,

you are given the following parameters from the extended Nelson Siegel model $($see

Chapter $2)$: ${}_0=6,278\mathrm{.}30,{}_1=-6,278\mathrm{.}25,{}_2=-6,292\mathrm{.}47,{}_3=0\mathrm{.}04387,$

${}_1=27,056\mathrm{.}49,$ and ${}_2=30\mathrm{.}48.$ That is$,$ to compute the continuously compounded

zero coupon yield with maturity $T,$ the formula is

$r(0,T)={}_0+({}_1+{}_2)\frac{1-e^{-\frac{T}{{}_1}}}{\frac{T}{{}_1}}-{}_2{e}^{-\frac{T}{{}_1}}+{}_3(\frac{1-e^{-\frac{T}{{}_2}}}{\frac{T}{{}_2}}-e^{-\frac{T}{{}_2}})$

The discount with maturity $T$ is then $Z(0,T)=e^{-r(0,T)T}.$

$($a$)$ What is the price of the pass through? Assume a constant PSA $=150\%.$

$($b$)$ Compute the duration of this security assuming that the PSA remains constant

at $150\%.$

$($c$)$ Compute the effective duration of this security assuming that the PSA increases

to $200\%$ if the term structure shifts down by $50$ basis points, while it decreases

to $120\%$ if the term structure shifts up by $50$ basis points. Comment on any

difference compared to your result in part $($b$).$

$($d$)$ Compute the effective convexity of this security under the same PSA assump$-$

tions as in part $($c$).$ Interpret your results.