Should have $4$ answers to the $4$ parts

The following case is similar to the case covered in Session 3. Using general notations, the case is as follows. Consider a do-it-yourself pension fund based on regular savings invested in a bank account attracting an annual interest rate of r. Today is t = 0, and you are planning to retire after n years. You want to receive a pension equal to e (0, 1) of your final salary and payable for the next T years. In other words, the pension will be delivered between n and n + T years. Your earnings are assumed to grow at an annual rate of g, and you want the pension payments to grow at the same rate. The objective is to determine today a fraction of salary denoted by x in order to satisfy the retirement goal. 1. Provide a closed-form solution for the value of T. Note that the solution should be expressed as a function of o, r, g, n, and T 2. To make sure you get the right analytical form, you need to validate your answer based on the numbers from the special case we covered in class - see table below. Given these values, what is T? 0.50 0.01 0.02 40.00 20.00 3. Consider the problem now from a continuous time. In this regard, the growth of the salary increases continuously while the interest is compounded continuously as well. The idea here is you continuously keep contributing to your pension. Similar to part 1, your task here is derive a closed-form solution for in the continuous case. Note: Suppose that a single year can be cut into m increments. In this ease, discount factor over one increment is The key is repeat the previous analysis while taking the limit of m i x. Hint: Refer to the same parameters from the above table. The value of T should be close to the one you computed for the discrete time. 4. Given the continuous time closed-form solution, demonstrate the sensitivity of with respect to each one of the five parameters. For instance, what happens to T when g increases, while holding everything else equal?