Answer the following time value of money questions assuming the interest rate is $12$ percent.

a$.$ What is the present value of $$1,200$ to be received in four years?

b$.$ What is the present value of $$1,200$ to be received in eight years?

c$.$ What will be the value in seven years of $$11,700$ invested today?

d$.$ How much would you pay for the right to receive $$4,700$ at the end of year $1,$$$3,700$ at the end of year $2,$ and $$7,700$ at the end of

year $5?$

e$.$ How long will it take for a $$1,700$ investment to double in value?

f$.$ What will be the value in $20$ years of $$1,500$ invested at the end of each year for the next $20$ years?