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Answer:
Equal probability = 0.5
If you invest your own = $1,000
Expected value = 0.5(800) + 0.5(1,400)
= 400 + 700
= 1,100 or 10%
Standard deviation = 300
If you borrowed an additional $1,000 and invested a total of $2,000,
Expected value = 0.5(1,600-1,000) + 0.5(2,800-1,000)
= 300 + 900
= 1,200 or 20%
You have doubled the expected return.
[tex]SD=[0.5(600-1,200)^{2}+0.5(1,800-1,200)^{2}] ^{\frac{1}{2} }[/tex]
SD = 600
Therefore, standard deviation has doubled.
If you borrowed $2,000 to invest a total of $3,000, then,
Expected value = 0.5(2,400-2,000) + 0.5(4,200-2,000)
= 200 + 1,100
= 1,300 or 30%
You have tripled the expected return versus the unleveraged investment.
[tex]SD=[0.5(400-1,300)^{2}+0.5(2,200-1,300)^{2}] ^{\frac{1}{2} }[/tex]
SD = 900
Therefore, standard deviation has tripled versus the unleveraged investment.
The expected value and standard deviation would double in the case where an additional $ 1,000 is borrowed and it will triple where an additional $2,000 is borrowed.
What is an Expected value?
In Statistics and probability analysis, the expected value is calculated by multiplying each possible outcome with the probability that each outcome will appear and summarizing all those values.
In the given information, equal probability is 0.5
Own investment is $1,000
[tex]\rm\,Expected\,value= 0.5(800) + 0.5(1,400)\\ \\ = 400 + 700\\ \\= 1,100\\\\Standard\,deviation = \sqrt{(800-1,100)^{2} \times 0.5 + (1,400-1,100)^{2}\times 0.5}\\\\= 300[/tex]
Case A: Borrowed an additional $1,000 and invested a total of $2,000
[tex]\rm\, Expected\,value = 0.5(1,600-1,000) + 0.5(2,800-1,000) \\ \\= 300 + 900 \\\\= 1,200\,or\,20\%\\\\\\Standard\,deviation = \sqrt{(600 - 1200)^{2} \times 0.5 + (1,800 - 1,200)^{2}\times 0.5}\\\\= 600[/tex]
The expected value and Standard deviation has doubled.
Case B: Borrowed $2,000 to invest a total of $3,000:
[tex]\rm\, Expected\,value = 0.5(2,400 -2,000) + 0.5(4,200-2,000) \\ \\\\= 200+ 1,100 \\\\= 1,300\,or\,30\%\\\\\\\\Standard\,deviation = \sqrt{(400 - 1300)^{2} \times 0.5 + (2,200 - 1,300)^{2}\times 0.5}\\\\= 900[/tex]
Hence, as we can see the value of expected value and standard deviation has tripled due to the additional amount of borrowing. It is proved that with every amount of increased borrowing the returns keep increasing.
To learn more about expected value, refer to the link:
https://brainly.com/question/17080700
