Tutorial $3$b

Question $1$:

Assume an amount $x$ is invested at times $1,2,3,$dots,$n$ and interest is payable in arrears at a rate of i per period. We know that the accumulated value of the payments at time $t=n$ is given by: $FV=x[\frac{(1+i{)}^n-1}{i}]$

Prove that the present value of the payments at time $t=0$ is given by: $PV=x[\frac{1-(1+i{)}^{-n}}{i}]$

Question $2$:

On $1$ July $2018$ an investor opens a savings account and starts to invest a constant amount of R$1,200$ at the start of every quarter into the account. On $1$ January $2022$ the investor decides that for the next three years an additional constant amount will be invested into the savings account at the end of every month. The interest rate is $2\mathrm{.}5\%$ effective per quarter.

If the investor wants R$125,000$ in the savings account on $30$ June $2028,$ calculate the value of the additional monthly investments that must be deposited into the savings account.

Question $3$:

On $1$ January $2020$ an investor opens a savings account and plans on making the following investments in the account:

R$15,000$ on $30$ April $2020.$

R$18,500$ on $1$ September $2021.$

From $1$ July $2022$ up to $31$ December $2024$ an amount of $Rx$ at the start of each month.

From $1$ January $2020$ up to $31$ December $2021$ the savings account earns interest at an effective rate of $7\mathrm{.}8\%$ per year. Thereafter the interest rate changes to $8\mathrm{.}64\%$ per year, compounded monthly.

If the total accumulated value of all the investments is $R79,658\mathrm{.}41$ on $31$ December $2024,$ calculate the value of $x.$

Question $4$:

An investor opens a savings account and plans on making the following investments into the account:

During the first two years an amount of R$1,250$ at the end of every month.

During the following three years an amount of R$3,875$ at the start of every quarter.

During the last five years an amount of $R8,860$ at the end of each half$-$year.

The savings account earns interest at $8\mathrm{.}6\%$ effective per year.

Calculate the total present value of the investments at the start of the $10-$year period.

Question $5$:

John has extra cash available and decides to invest it in a bank account. On $1$ June $2007$ he starts to deposit R$800$ at the start of each quarter for six years. The bank account earns interest at $3\mathrm{.}75\%$ effective per quarter, paid in arrears.

During $2010,$ John realises that he has an additional R$150$ per month at his disposal. He decides to invest this additional amount into the account from $31$ March $2010$ to $30$ May $2011$ at the end of each month.

Calculate the final value in the bank account on $1$ June $2013.$