A retail firm\'s quarterly profits have following annual pattern due to seasonal

Question

A retail firm's quarterly profits have following annual pattern due to seasonal change in demand. where x is the first quarter's profit. The quarterly required return is 1%. Calculate the present value (evaluated at the beginning of a year) of the annual profit. The result should contain x. If x = $300,000, what is the present value of the annual profit? Assuming x = $300,000 in the first year. It is expected that the economy will grow steadily so that x will grow at the rate of 2% every year forever. What is the present value of all future profits? A financial crisis hits the economy. Now it is expected that x = $200,000 in the first year and there will be zero growth over the first five years. After that x will resume the 2% growth rate. What is the present value of the losses for the firm due to the financial crisis?

Explanation / Answer

a) PV of annual profit = x/(1.01)1 + 1.5x/(1.01)2 + 0.7x/(1.01)3 + 2x/(1.01)4
= 5.0619x (ans)

b) x = $300,000
PV of annual profit = 5.0619 * 300,000
= $ 1,518,575.06 (ans)

c) Profit of first year = $300,000 (given)
Quarterly Return = 1% (given)
Annualised Return (Re) = (1+0.01)4-1
= 0.0406 = 4.06%
g= 0.02 p.a forever (given)
PV of future profits (perpetuity) = Profit of year 1 / (Re - g)

= 300,000 / (0.0406 - 0.02)
= $14,563,106.80 (ans)

d) x = $200,000
r = 4.06% annualised.
g = 2% after 5 years

PV of future profits with financial crisis:
No growth period-
PV of first five years = 200,000 * PVIFA (4.06%,5yrs) = 200,000 * 4.4443 = $888,865.71
Growth Period-
PV at the end of year 5 = x / (re-g)
= 200,000/(0.0406-0.02) = $9,708,737.86
PV today = 9,708,737.86 / (1+ 0.0406)5 = $7,956,895.74

PV of future profits during financial crisis = PV of no growth period + PV of growth period
= $888,865.71 + $7,956,895.74
= $8,845,761.45

PV of losses for the firm due to financial crisis = PV of future profits without crisis - PV of future profits with crisis
= $14,563,106.80 - $8,845,761.45
= $57,17,345.35 (loss) (ans)

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