Question

XYZ has an investment worth $56,000. The investment will make a special, extra payment of X to XYZ in 3 years from today. The investment also will make regular, fixed annual payments of $12,000 to XYZ with the first of these payments made to XYZ in 1 year from today and the last of these annual payments made to XYZ in 5 years from today. The expected return for the investment is 13.2 percent per year. What is X, the amount of the special payment that will be made to XYZ in 3 years?

	a.	An amount equal to or greater than $10,000 but less than $17,000
	b.	An amount equal to or greater than $17,000 but less than $22,000
	c.	An amount equal to or greater than $22,000 but less than $26,000
	d.	An amount equal to or greater than $26,000 but less than $30,000
	e.	An amount less than $10,000 or an amount equal to or greater than $30,000

Please show work.

Answer 1

13.2% is the IRR or the rate of return, such that the total cash inflows = 56,000
Total inflows = 33,729.02 + (12,000 + x) * 0.6894
Solve the linear equation :
33,729.02+(12,000+x) * 0.6894 = 56000 (33,729.02 = 10,600.71 + 9,364.58 + 7,307.95 + 6,455.79)
Solving for x, we get x = 20,305.17
Answer : b. An amount equal to or greater than $17,000 but less than $22,000

Approach :

The return on investment is 13.2%. In other words, the IRR of the investment is 13.2%. The investment is generating 13.2%, and also the annual cash flows are re-invested at 13.2% every year.

Hence, as per concept of IRR : $56,000 = All the cash inflows generated annually, discounted at 13.2%

Therefore, the annual cash flows are discounted at :"13.2% Discounting factor", and by this we derive the present value

DF = 1/(1+r)^time where, r = rate of interest or discounting rate and time is the year in which the cash flow is generated

0.6894 is the DF of year 3 in which the X is generated. 1/(1.132)^3

We know that 56,000 is the sum of present value of all the cash inflows. Hence, we can assume the cash inflow of year 3 as 12,000 + x and determine the value of x using linear equation

We know the cash inflow of all the other years and hence we can add those figures to derive 33,729.02. But, we also need to have (12,000 + x) * 0.6894 on the LHS of the linear equation. We can't straight away take 12,000 + x because we need to have the Present value, and thus it needs to be multipled by year 3 DF = 0.6894.

And by solving the linear equation, we can arrive at x.

What is the significance of Present value?

The investment is generating 13.2% every year. Hence, if the future value is 12,000 every year, the present value (value as of now) is much lesser. Hence, by multiplying with the discounting factor, we are eliminating the interest or the return earned every year at a compounding rate.

For instance, 12,000 at year 2, has a PV of 9,364.58 today (since, 9.364.58 generating 13.2% for two years with compounding every year will be worth 12,000 by end of year 2)