Question

The table below shows the one-year return distribution of Startup Inc.

Probability	35%	20%	20%	10%	?%
Return	-90%	-75%	-50%	-25%	1000%

a. Calculate the expected return.

b. Calculate the standard deviation of the return.

c. Replace the expected return of 1000% in the last column in the table above with the expected return value that minimizes the standard deviation of the returns.

a. The expected return is %. (round to one decimal)

b. The standard deviation of the return is %. (round to one decimal)

c. The value of expected return in last column which minimizes the standard deviation of the returns is %. (round to one decimal. If negative, enter a minus sign "-".)

Answer 1

Solution a) Calculation of Expected Return

Probabilty (p)	Return (x)	Expected Return = (p) * (x)
0.35	-90	-31.5
0.2	-75	-15
0.2	-50	-10
0.1	-25	-2.5
0.15	1000	150
		91

Therefore, Expected Return = ∑ (p) * (x)   = 91%

Solution 2) Calculation of Standard Deviation of the return

Probabilty (p)	Return (x)	Expected Return = (p) * (x)	(x) - Expected Return	((x) - Expected Return) ^ 2	((x) - Expected Return ^ 2) * (p)
0.35	-90	-31.5	-181	32761	11466.35
0.2	-75	-15	-166	27556	5511.2
0.2	-50	-10	-141	19881	3976.2
0.1	-25	-2.5	-116	13456	1345.6
0.15	1000	150	909	826281	123942.15
		91			146241.5

Standard Deviation of the return = Variance^(1/2)

                                                            = 146241.5 ^ (1/2)

                                                            = 382.42%

   Solution c) Calculation of Expected Return and Standard Deviation after replacing 1000% in the last column by Expected Return value.

Probabilty (p)	Return (x)	Expected Return = (p) * (x)	(x) - Expected Return	((x) - Expected Return) ^ 2	((x) - Expected Return ^ 2) * (p)
0.35	-90	-31.5	-44.65	1993.62	697.77
0.2	-75	-15	-29.65	879.12	175.82
0.2	-50	-10	-4.65	21.62	4.32
0.1	-25	-2.5	20.35	414.12	41.41
0.15	91	13.65	136.35	18591.32	2788.70
		-45.35			3708.03

Standard Deviation of the return = Variance^(1/2)

                                                            = 3708.03 ^ (1/2)

                                                            = 60.89%