Question

On a day of fluctuating market prices, the share price of XYZ Ltd. ends with a gain, i.e, it is higher at the close of the day compared to the opening value. Which trader got the maximum return on that day?

• A. Bikram
• B. Chetan
• C. cannot be determined
• D. Bikram or Chetan
• E. Abdul
• A. Abdul will not be one with the minimum return
• B. Return for Chetan will be higher than that of Bikram
• C. Return for Bikram will be higher than that of Chetan
• D. none of the above
• J. Return for Chetan cannot be higher than that of Abdul
• A. Chetan
• B. Bikram
• C. Abdul
• D. Abdul or Chetan
• O. cannot be determined

Answer 1

The Correct Option is C

Step 1: Set up.
Let the prices at 10 am, 11 am, 12 noon, 1 pm, 2 pm be \(p_1,p_2,p_3,p_4,p_5\) and the closing price be \(S\). The day ends with a gain, so \(S>p_1\).

Step 2: Return of each trader.
Return \(=\dfrac{\text{sale value}}{\text{investment}}-1=\dfrac{S}{\text{average cost per share}}-1\).

• Abdul buys everything at \(p_1\): average cost \(=p_1\).
• Bikram buys an equal number of shares each time: average cost \(=\dfrac{p_1+p_2+p_3+p_4+p_5}{5}\) (arithmetic mean, AM).
• Chetan invests an equal amount each time: average cost \(=\dfrac{5}{\frac1{p_1}+\frac1{p_2}+\frac1{p_3}+\frac1{p_4}+\frac1{p_5}}\) (harmonic mean, HM).

Step 3: Bikram versus Chetan.
Since HM \(\le\) AM, Chetan's average cost is never more than Bikram's, so Chetan's return is never less than Bikram's. So Bikram can never be the sole maximum; the maximum is Abdul or Chetan.

Step 4: Abdul versus Chetan depends on the prices.
Case 1: prices 100, 120, 120, 120, 120 and \(S=110\) (a gain). Abdul: \(+10\%\); Bikram (AM \(=116\)): about \(-5.2\%\); Chetan (HM \(\approx115.4\)): about \(-4.7\%\). Abdul is the best.
Case 2: prices 100, 50, 50, 50, 50 and \(S=105\) (a gain). Abdul: \(+5\%\); Bikram (AM \(=60\)): \(+75\%\); Chetan (HM \(\approx55.6\)): \(+89\%\). Chetan is the best.

The winner changes with the price pattern, so it cannot be determined from the given information.

Final Answer:
The correct option is (3): cannot be determined

Answer 2

Step 1: Returns in terms of average cost.
With prices \(p_1,\dots,p_5\) at the five buying times and closing price \(S\): Abdul's return \(=\dfrac{S}{p_1}-1\), Bikram's \(=\dfrac{S}{\text{AM}}-1\), Chetan's \(=\dfrac{S}{\text{HM}}-1\), where AM and HM are the arithmetic and harmonic means of \(p_1,\dots,p_5\).

Step 2: Bikram versus Chetan.
HM \(\le\) AM, with equality only if all five prices are equal. Hence Chetan's return \(\ge\) Bikram's return always, but they are equal when the price does not change.

Step 3: Test the statements.

• (1) Abdul not the minimum: false in general. Take prices 100, 50, 50, 50, 50 and \(S=105\): Abdul \(5\%\), Bikram \(75\%\), Chetan \(\approx89\%\). Abdul has the minimum return.
• (2) Chetan higher than Bikram: not always, since they are equal when all prices are equal.
• (3) Bikram higher than Chetan: never true (Chetan is at least as good).
• (5) Chetan cannot be higher than Abdul: false, in the example above Chetan (\(89\%\)) beats Abdul (\(5\%\)).

None of (1) to (5) is always true.

Final Answer:
The correct option is (4): none of the above

Answer 3

Step 1: Returns in terms of average cost.
Return \(=\dfrac{S}{\text{average cost}}-1\). Abdul's average cost is \(p_1\); Bikram's is the arithmetic mean (AM) of \(p_1,\dots,p_5\); Chetan's is their harmonic mean (HM).

Step 2: Use the rising prices.
The prices rise all day: \(p_1<p_2<p_3<p_4<p_5<S\). Since \(p_1\) is the smallest price, \(p_1<\text{HM}<\text{AM}\). The smallest average cost gives the largest return, so Abdul has the maximum return.

Step 3: Bikram versus Chetan.
The prices are not all equal, so HM \(<\) AM strictly. Chetan's average cost is lower than Bikram's, so Chetan's return is higher than Bikram's.

Check with prices 100, 110, 120, 130, 140 and \(S=150\): Abdul \(50\%\); Bikram (AM \(=120\)) \(25\%\); Chetan (HM \(\approx118.3\)) \(\approx26.8\%\). Bikram is the minimum.

Final Answer:
The correct option is (2): Bikram