問7
頻出度★★☆
いずれも時価100円の四つの株式があり，そのうちの一つに投資したい。経済の成長が高成長，中成長，低成長の場合，それぞれの株式の予想値上がり幅が表のとおりであるとき，値上がり幅の期待値が最も高い株式はどれか。ここで，高成長，中成長，低成長になる確率はそれぞれ0.4，0.4，0.2であり，経済が成長しない場合は考えないものとする。 単位 円
| 株式 | 高成長 | 中成長 | 低成長 |
|---|---|---|---|
| A | 20 | 10 | 15 |
| B | 30 | 20 | 5 |
| C | 25 | 5 | 20 |
| D | 40 | 10 | −10 |
平成21年度 春期 ITパスポート試験 問7
テクノロジ系基礎理論 › 応用数学
This question's explanation and your answers so far.
解説
問 7
Incorrectア
A has an expected value of 20×0.4+10×0.4+15×0.2 = 8+4+3 = 15 yen, the lowest of the four. A's 15 yen under low growth is higher than B (5 yen) and D (−10 yen), but it does not reach C (20 yen) in the same low growth column, and low growth occurs with a probability of only 0.2, so its contribution to the total stops at 15×0.2 = 3 yen. Because the price increases under high growth and medium growth (0.8 in total), which carry the larger probabilities, are only 20 yen and 10 yen, the sum still falls short of B's 21 yen.
- イ
正解 All four stocks share the same market price of 100 yen, so you can simply compare the expected values of the price increase. An expected value is the weighted average obtained by adding up “the value in each scenario × the probability that the scenario occurs” over all scenarios. The probabilities are 0.4 for high growth, 0.4 for medium growth and 0.2 for low growth (1.0 in total), so multiply each row of the table by these weights and add them up. A is 20×0.4+10×0.4+15×0.2 = 8+4+3 = 15 yen, B is 30×0.4+20×0.4+5×0.2 = 12+8+1 = 21 yen, C is 25×0.4+5×0.4+20×0.2 = 10+2+4 = 16 yen, and D is 40×0.4+10×0.4+(−10)×0.2 = 16+4-2 = 18 yen. Comparing the four, B with 21 yen is the highest, so the correct answer is イ (B). B has 30 yen under high growth and 20 yen under medium growth (the largest in the medium growth column), giving it a large price increase in both scenarios, which together account for a probability of 0.8, so its weighted average also exceeds the others.
Incorrectウ
C has an expected value of 25×0.4+5×0.4+20×0.2 = 10+2+4 = 16 yen. In the low growth column, C's 20 yen is the largest of the four, but that low growth occurs with a probability of only 0.2, so the contribution stops at 20×0.2 = 4 yen. In addition, medium growth is as low as 5 yen, so it gains only 5×0.4 = 2 yen over that section with probability 0.4; the total of 16 yen therefore does not reach B's 21 yen.
Incorrectエ
D has an expected value of 40×0.4+10×0.4+(−10)×0.2 = 16+4-2 = 18 yen. The 40 yen under high growth is the largest price increase in the table and looks advantageous at first glance, but under low growth it becomes a negative −10 yen, which pushes the expected value down by 2 yen. You have to compare the totals weighted by probability, not the maximum value of a single scenario, and 18 yen is lower than B's 21 yen.
ここがポイント
- Expected value = Σ (the value of each outcome × the probability that the outcome occurs). In this question, compute a weighted average for each stock using 0.4 / 0.4 / 0.2 as the weights and compare the results (always confirm that the probabilities used as weights add up to 1). Since every market price is equally 100 yen, you may compare the expected values of the price increase directly.
- Do not judge from one particular scenario alone. The scenario that produces the maximum value (D's 40 yen under high growth), or the stock that is strongest in the worst scenario of low growth (C's 20 yen), is not necessarily the one selected; the result is decided by the sum weighted by probability. A negative value (D's −10 yen) must be subtracted according to its sign in the calculation.
用語