Given the following bonds, calculate the continuously$-$compounded spot rates.

$($Total: $6$ marks$)$

$\backslash$table$[[$Par $($$$),$Maturity,Coupon,Price $($$$)],[100,0\mathrm{.}50,0\mathrm{.}00\%,97\mathrm{.}91],[100,1\mathrm{.}00,0\mathrm{.}00\%,95\mathrm{.}80],[100,1\mathrm{.}50,6\mathrm{.}00\%,102\mathrm{.}29]]$

The coupon rate is per year, and coupons are paid semi$-$annually.

a$)$ What is the spot rate for $6$ months? $$

$(1$ mark$)$

$97\mathrm{.}91=100*e^{-R_{1}0\mathrm{.}5}=>R_1=-ln\frac{\frac{97\mathrm{.}91}{100}}{0\mathrm{.}5}$

b$)$ What is the spot rate for $12$ months?

$95\mathrm{.}80=100*e^{-R_{1}1\mathrm{.}0}=>R_1=\frac{-{?}^{ln(\frac{95\mathrm{.}80}{100})}}{1\mathrm{.}0}$

c$)$ What is the discount factor for $6$ months $($DF$1)?(1$ mark$)$

d$)$ What is the discount factor for $12$ months $($DF$2)?(1$ mark$)$

e$)$ What is the discount factor for $18$ months $($DF$3)?(1$ mark$)$

$102\mathrm{.}29=3*DF13*DF2103*DF3=>DF3=\frac{102\mathrm{.}29-3*DF1-3*DF2}{103}$

f$)$ What is the spot rate for $18$ months? $(1$ mark$)$

$DF3=e^{-R_{3}1\mathrm{.}5}=>R_1=-ln\frac{DF3}{1\mathrm{.}5}$

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