Question

3. A certain entrepreneur opened a fashion boutique which experienced the cash flows shown in the table below. She closed the boutique after four years due to falling revenues. (a) How many possible rates of return are there for these cash flows? (b) Find all the actual rate of return values between 0% and 100%. (c) If the entrepreneur's minimum acceptable rate of return is 20% per year, was this a good investment? د Year Costs ($) Revenues ($) 0 75,000 49,500 42,000 67,500 210,000 90,000 57,000 45,000 60,000 م درا هم
include cash flow diagram

Answer 1

ANSWER:

A certain entrepreneur opened a fashion boutique which experienced the cash flows shown in the table below. She closed the boutique after four years due to falling revenues.

Year 0 1 wn Cash Flow Costs ($) Revenues ($) Revenues-Costs 75,000 -75,000 49,500 42,000 -7,500 67,500 2,10,000 1,42,500 90,000 57,000 -33,000 45,000 60,000 15,000 Year Positive 4 3 cash flow diagram 1,42,500 15,000 Negative -75,000 -7,500 -33,000

Net present value of cash flows=NPV=-75000-7500/(1+r) + 1,42,500/(1+r)2 - 33,000/(1+r)3 + 15,000/(1+r)4

r is the possible rates of return are there for these cash flows then NPV=0.

Let 1/(1+r) =x therefore

-75000-7500x + 1,42,500x2 - 33,000x3 + 15,000x4=0

=>15 x4 - 33 x3 + 142.5 x2 - 7.5x -75=0=f(x)

a)Since there are three sign changes for the equation there are max 3 +ve roots possible for the equation.Therefore either there are 3 +ve roots or 1 +ve root.In all over all 4 rates of returns are possible for these cash flows as there are 4 possible solutions of x and thus 4 possible corresponding rates of returns.

f(-x)=15 x4 +33 x3 + 142.5 x2 + 7.5x -75 has 1 change in sign so there are max 1 possible -ve root possible for x.

We shall find value of x for which the f(x)=P=0 using excel.And then find the corresponding values of r.

b) 24.54% -255.60% x=1/(1+r) 80% -64% P=15*x^4 -33*x^3+142.5*x^2-7.5*x-75 0.00 0.00 run the sensitivity analysis using the values of rand P above -500% -63.6445 -400% -55.2593 -300% -30.5625 -200% 123 -120% 17025 -20% 110.449 80% -39.4147 180% -60.7618 280% -67.6347 380% -70.6477 480% -72.213 -300.00% -290.00% -280.00% -270.00% -260.00% -255.60% -245.60% -235.60% -225.60% -215.60% -205.60% -30.56 -25.62 -19.76 -12.77 -4.30 0.00 11.40 25.70 43.98 67.88 99.98 20.00% 5.84491 24.54% 0.00416 29.54% -5.72444 34.54% -10.8222 39.54% -15.3798 44.54% -19.4719 49.54% -23.1605 54.54% -26.4975

We find that there are two solutions 1 positive of 80% and other negative of -64% of x. The correspondin rates of returns are 24.54% and -255.6%.

We find no other real solutions therefore it suggests that the other two roots are imaginary. of x.Thus actual rate of return values between 0% and 100% is 24.54%

There are two real and two imaginary solutions.
c)Since the rate of return .24.54%>the entrepreneur's minimum acceptable rate of return 20% per year This was a good investment.

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