Calculate the present values at t $=0($now$)$ of the following cash flows:

a$.$ $$100$ every year forever, with the first payment at t $=1($t counts years$),$ where the effective

annual rate is $\mathrm{.}05($i$.$e$.,5\%).$

b$.$ $$100$ every year forever, with the first payment at t $=11($t counts years$),$ where the effective

annual rate is $\mathrm{.}05($i$.$e$.,5\%).$ Hint: What will be the value of this stream at t $=10($ten years

from now$)$ if the discount rate remains $5\%?$ To get the value at t $=0,$ discount this single

value back $10$ years at $5\%.$

c$.$ $$100$ every year for $10$ years, with the first payment at t$=1($t counts years$),$ where the

effective annual rate is $\mathrm{.}05(5\%).$ Calculate this value using the present value of an annuity

formula. Compare this value to the value you get by subtracting your answer to b above from

your answer to a above. Why are they related as they are?

d$.$ $$100$ every $3$ years forever, with the first payment at t $=3($t counts years$),$ where the effective

annual rate is $\mathrm{.}05($i$.$e$.,5\%).$

e$.$ $$1000$ every $3$ years forever, with the first payment at t $=3($t counts years$),$ where the

effective annual rate is $\mathrm{.}05($i$.$e$.,5\%).$

f$.$ $$1000$ every $3$ years forever, with the first payment at t $=3($t counts years$),$ where the

effective annual rate is $\mathrm{.}10($i$.$e$.,10\%).$

g$.$ The first cash flow at t $=1$ is $$100.$ Every year thereafter, the payment increases by $3\%$ over

the previous year$$s payment. This continues on forever. What is the present value of this

growing perpetuity if the effective annual discount rate is $\mathrm{.}1(10\%)?$

h$.$ The first cash flow at t $=1$ is $$100.$ Every year thereafter, the payment increases by $3\%$ over

the previous year$$s payment. This continues on forever. What is the present value of this

growing perpetuity if the effective annual discount rate is $\mathrm{.}05(5\%)?$

i$.$ The first cash flow at t $=1$ is $$100.$ Every year thereafter, the payment increases by $3\%$ over

the previous year$$s payment. This continues on forever. What is the present value of this

growing perpetuity if the effective annual discount rate is $\mathrm{.}035(3\mathrm{.}5\%)?$

j$.$ The first cash flow at t $=1$ is $$100.$ Every year thereafter, the payment increases by $3\%$ over

the previous year$$s payment. This continues for $9$ years past the first payment $($for a total of

$10$ payments$).$ What is the present value of this growing annuity if the effective annual

discount rate is $\mathrm{.}02(2\%)?$

k$.$ $$50$ every year and a half forever, with the first payment after $1\mathrm{.}5$ years, where the effective

annual rate is $\mathrm{.}05($i$.$e$.,5\%)$