An annuity consists of $n$ payments of $K,K(1+r)dots,K(1+r{)}^{n-1}$ starting one period from now and $rj.$ Determine which of the following is the present value of this annuity.

$($a$)])/(jr[u(1+r)$

$($b$)])/(j[u(1+r)$

$($c$)])/(j-r[u(1+r)$

$($d$)])/(j[u(1+r)$

$($e$)$ none of these

A perpetuity pays $100$ per year. Immediately after the fifth payment, the perpetuity is exchanged for a $25$ year annuity$-$immediate that will pay $x$ at the beginning of the first year. Each subsequent payment will be $10\%$ greater than the preceeding payment. The annual interest rate is $10\%.$ Find $x.$ Hint: The PV of the perpetuity and the annuity$-$immediate are the same at the time of exchange.

$($a$)25$

$($b$)40$

$($c$)50$

$($c$)20$

$($d$)33$

$($e$)$ none of these

If the interest rate on $OB$ is $j,$ and is paid off by $n$ payments with $P{R}_1=$dots$=P{R}_n,$ then $I_t=$ : $($Hint: Find $O{B}_{t-1}$ first$)$

$($a$)O{B}_0(n+1-t)\frac{j}{n}$

$($b$)O{B}_{0}t\frac{j}{n}$

$($c$)O{B}_0(n-t)j$

$($d$)O{B}_0(n+1-t)\frac{j}{t}$

$($e$)$ none of these

A loan of $1000$ is repaid with $20$ payments at the end of each year. The first ten payments equals $200\%$ of the amount of interest due. Determine which of following is false, where $1t10.$

$($a$)O{B}_1=L(1-i)$

$($b$)K_t=2iO{B}_{t-1}$

$($c$)O{B}_2=L(1-i{)}^2$

$($d$)O{B}_t=L(1-i{)}^t$

$($e$)$ none

of these

Continue from above. Each of the last ten payments is $x$ and $i=10\%.$ Then

$($a$)|)$

$($b$)1000({\mathrm{.}95}^{10})=x{a}_{\frac{?}{\text{b}\text{a}\text{r}}(10)*10}$

$($c$)|)$

$($d$)1000({\mathrm{.}95}^{10})=|)$

$($e$)$ none of these

Let $x$ be the present values of an n$-$payment annuity$-$immediate that pays $n,n-1,$dots,$1$ at time $1,2,$dots,$n.$ Find an expression for $x.$

$($a$)|)-n$

$($b$)|))/(i$

$($c$)\frac{n}{i}$

$()n|)$

$($d$)|)-n))/(i(()n+1$

$($e$)$ none of these