$1.$ You work for a pharmaceutical company that has developed a new drug. The patent on the drug will last $19$ years. You expect that the$$ drug's profits will be $$$$2\mathrm{.}5$ million in its first year and that this amount will grow at a rate of $6\%$ per year for the next $19$ years. Once the patent$$ expires, other pharmaceutical companies will be able to produce the same drug and competition will likely drive profits to zero. What is the present value of the new drug if the interest rate is $12\%$ per$$ year?

The present value of the new drug is $$$ $\_\_\_\_\_\_?($Round to the nearest$$ dollar.$)$

$2.$Suppose you invest $$$$2000$ today and receive $$$$9750$ in $5$ years.

a$.$ What is the internal rate of return$($IRR$)$ of this$$ opportunity?

b$.$ Suppose another investment opportunity also requires $$$$2000$upfront$,$ but pays an equal amount at the end of each year for the next $5$ years. If this investment has the same IRR as the first$$ one, what is the amount you will receive each$$ year?The periodic payment that gives the same IRR is$\_\_\_\_\_?$