Exercise $5(13$ MARKS$)$

Throughout this question, the annual interest rate is assumed to be $i.$

a$)(3$ marks$)$ Alice makes a payment of $1$ at the end of each of year for $n$ years. Show that the present value of the payments is given by

$\frac{1-(1+i{)}^{-n}}{i}$

b$)(3$ marks$)$ What is the accumulated value of Alice's payments after $n$ years?

$2.(3$ marks$)$ Bob pays back a loan with $n$ payments made at the end of each year for $n$ years. He makes a payment of $1$ at the end of the first year. At the end of the second year, he makes a payment of $0\mathrm{.}98.$ At the end of the third year, he makes a payment of ${0\mathrm{.}98}^2.$ At the end of the $m-th$ year, he makes a payment of ${0\mathrm{.}98}^{m-1}$ for every integer min$[1,n].$ Show that the present value of Bob's payment is given by

$\frac{1-{0\mathrm{.}98}^n(1+i{)}^{-n}}{i+0\mathrm{.}02}$

$(4$ marks$)$ At the end of the first year, Carl makes a payment of $1.$ At the end of second year, he makes a payment of $2.$ At the end of third year, he makes a payment of $3.$ In other words, at the end of the $m-$th year, Carl makes a payment of $m,$ for every integer min$[1,n].$ Show that present value of Carl's payment is

$\frac{1+i}{i^2}\{1-(1+i{)}^{-n}\}-n\frac{(1+i{)}^{-n}}{i}$