FRA

A broker offers you to agree today with your client on an FRA with which you will provide a loan of $$5,000,000\mathrm{.}00$ that will begin within $9$ months and will be settled two years after it begins. The Rk rate that you will be paid is $9\%$ nominal quarterly.

a$)$ Is it convenient for you to sign the contract? Show why yes or why not.

b$)$ If you decide to agree to the FRA with that Rk$=9\%$ nominal quarterly, how much is the FRA contract worth to you today?

c$)$ How much is the FRA contract worth to its counterparty today?

d$)$ Suppose that you agreed to the FRA with the fair rate and within a month the interest rates have changed: the $1$ and $2$ year rates rose $40$ bp and the $3$ year and later rates rose $20$ bp$.$ What is the value of the forward contract for you?

e$)$ Suppose that the FRA was agreed upon with the fair rate and $3$ after the contract was agreed upon, all interest rates have fallen by $8$ b$.$p$.$ What is the value of the forward contract to its counterparty?

f$)$ Suppose that you decided to agree to the contract with Rk$=9\%$ nominal quarterly and that when it reaches T$1=8$ months it turns out that RT$1,$T$2=7\mathrm{.}8\%$ effective semiannual, what is the pay$-$off of your FRA seen as a speculative asset?

g$)$ Now, if you reach T$1=8$ months and it turns out that RT$1,$T$2=4\mathrm{.}82\%$ bimonthly nominal, what is the pay$-$off for your counterparty if the FRA guarantees a truly realizable loan?

Consider the following picture$)$:

$(1+\frac{10\mathrm{.}8\%}{2}{)}^4=(1+\frac{10\mathrm{.}2\%}{2}{)}^2(1+\frac{f_{1,2}}{2}{)}^2=>f_{1,2}=2\{[(1+5\mathrm{.}4\%{)}^4(1+5\mathrm{.}1\%{)}^{-2}{]}^{\frac{1}{2}}-1\}$

$=>f_{1,2}=2[(1\mathrm{.}11727{)}^{\frac{1}{2}}-1]=2(0\mathrm{.}0570086)=11\mathrm{.}4017\%$ nom. semestral

$(1+\frac{11\mathrm{.}4\%}{2}{)}^6=(1+\frac{10\mathrm{.}8\%}{2}{)}^4(1+\frac{f_{2,3}}{12}{)}^{12}=>f_{2,3}=12\{[(1\mathrm{.}057{)}^6(1\mathrm{.}054{)}^{\mathrm{.}4}{]}^{\frac{1}{12}}-1\}$

$=>f_{2,3}=12(1\mathrm{.}01024-1)=12\mathrm{.}2863\%$ nom. mensual

$(1+\frac{11\mathrm{.}6\%}{2}{)}^8=(1+\frac{11\mathrm{.}4\%}{2}{)}^6(1+\frac{f_{3,4}}{6}{)}^6=>f_{3,4}=6\{[(1\mathrm{.}058{)}^6(1\mathrm{.}057{)}^{-4}{]}^{\frac{1}{6}}-1\}$

$=>f_{3,4}=6(0\mathrm{.}0199352)=11\mathrm{.}9611\%$ nom. bimestral

$(1+\frac{12\mathrm{.}1\%}{2}{)}^{14}=(1+\frac{11\mathrm{.}6\%}{2}{)}^{10}{e}^{f_{5,7}(2)}=>f_{5,7}=\frac{1}{2}ln[(1\mathrm{.}0605{)}^{14}(1\mathrm{.}058{)}^{-10}]$

$=>f_{9,7}=\frac{1}{2}(0\mathrm{.}258563596)=12\mathrm{.}9281798\%$ continua