Question

Determine the value of x, for a company’s two investment alternatives to be equivalent at an interest rate of 5\% per year:

Option 1. Invest $15 at the end of the first year. Starting from the end of 9th year, the company gets sales of $x for 3 years, followed by $2.0x for 8 years, and $x for 3 years.

Option 2. Invest $15 at year 0, and the company starts to get sales $15 at the end of year 8, increment by 1 cumulatively until at the end of year 14 (e.g., at end of year 9 is 16).

Answer 1

Present value of Option 1

= -15+ (x/1.05^9+x/1.05^10+x/1.05^11)+(2x/1.05^12+2x/1.05^13+.....+2x/1.05^19)+ (x/1.05^20+x/1.05^21+x/1.05^22)

= -15 + (x/1.05^9+x/1.05^10+......+x/1.05^22) + (x/1.05^12+x/1.05^13+.....+x/1.05^19)

=-15+ x/1.05^9*(1-1/1.05^14)/(1-1/1.05) + x/1.05^12*(1-1/1.05^8)/(1-1/1.05)

=-15 +10.47869646 * x

Present value of Option 2

= -15+(15/1.05^8+16/1.05^9+17/1.05^10+18/1.05^11+19/1.05^12+20/1.05^13+21/1.05^14)

=$58.21985

For the two investment alternatives to be equivalent

-15 +10.47869646 * x = 58.21985

=> x = 6.9875

So, value of x is $6.9875