There is a loan obligation to pay $$1,000$ a year from now, $$1,500$ two years from

now, and $$2,000$ three years from now. Find the general expression for

$(1)$ Amount of the loan

$(2)$ Macaulay duration

$(3)$ Modified duration

$(4)$ Convexity

Evaluate those expression for an annual effective rate of interest of $6\%$