>Top 2. The theme of financial engineering is risk:
 Risk: potential of gaining or losing something of value caused by uncertain action.
 our life if full of various risks; such as disease, accident, or disaster
 agriculture by weather, fishery, hunting, and mineral exploration.
 MTS (maketostock) to BTO (builttoorder)
 from hunting to stockfarming; from fishery to aqua farming cultivation
 >Top Risk Management:
 Minimax principle
 Risk: uncertainty having probability distribution
 True uncertainty having no probability distribution
 Expected value, variance and standard deviation (SD) (or volatility in finance):
 law of large numbers; normal distribution
 >Top Risk premium:
 minimum amount by which the expected return on a risky asset must exceed the known return on a riskfree asset.
 compensation for investor who tolerate the extra risk, compared to that of a riskfree asset.
 hedger:
 speculator:
 to maximise the expected utility; convex curve to the top
 >Top the marginal utility is digressive (law of diminishing marginal utility):
 the first unit of consumption of a good or service yields more utility than the second and subsequent units.
 Variance ($\sigma^2$): measures how far a data set is spread out.
 Variance ($\sigma^2$): $u(X)=(X\mu)^2$, where $\mu$ is mean.
 Standard Deviation (SD) ($\sigma$)
 St. Petersburg Paradox: (proposed by Nicolaus Bernoulli)
 a particular lottery game that leads to a random variable with infinite expected value, but nevertheless seems to be worth only a very small amount to the participants.
 People want to maximize 'expected utility' rather than 'expected profit.'
 >Top 'Indifference curve': and equal utility curve
 showing combination of two goods that gives equal satisfaction and utility. Each point on the curve indicates that the consumer is indifferent between the two.
 Risk management: (>Fig.)
 Diversified investments
 Insurance contract
 Sales of risky assets, or forward contract
 Decrease of SD by combination of investment: (>Fig.)
 the risk of combination is smaller than the weighted average.
 To minimize risk (or SD) is not the object of asset, but to maximize the return of investment. (Portfolio)
 Investment Opportunities Curve: (>Fig.)
 Efficient frontier:
 An investment portfolio which occupies the efficient parts of the riskreturn spectrum.; formulated by Harry Markowits in 1952.
 Tobin's Separation Theorem:
 Fund operation is available by adding norisk asset (like government bond, cash, etc.) to the best riskreturn portfolio, depending on each investor's risk allowance.
 If the risk allowance is low, the investor prefers to have more nonrisk asset or risk allowance is high, the investor prefers not to have norisk assets, or raise return ration by leverage.
 Fig:

 Capital Market Line (CML); gradient of the line is risk premium.
 >Top Modern portfolio theory:
Harry Markowitz
theory had not been attention by economists. But James Tobin extended his theory, establishing the separation theorem and market value of risk.
 B point of the above figure is called safe asset, like government bond.
 A certain point as a mix of safety asset and risk asset is shown by the blue line (risk premium, or market risk premium).
 The indifference curve, an equally acceptable risk curve, which is chosen by a consumer, is shown here as green curve.
 To gain the optimized combination, this line should contact the outmost point (D) of the investment opportunities curve (red curve).
 This D point is determined irrelevantly by investors' preferences, i.e., mix of risk assets is determined independently from riskreturn preferences of investors.(Tobin theorem, or Portfolio separation theorem)
 Thus, the optimized portfolio for any investor is plotted on the above blue line (Capital Market Line).
 The gradient of the blue line shows: (Risk asset return  safe asset return)/(SD of asset return); here the numerator shows risk premium of risk asset.

2.
金融工学のテーマはリスク:
 risk: uncertain: likely turn out well or badly
 expected value: 期待値
 probability distribution: 確率分布
 marginal utility: 限界効用
 indifference curve: 無差別曲線
 investment opportunities curve: 投資機会曲線
 capital market line: 資本市場線

 期待値・分散・標準偏差
 リスクプレミアム (期待収益率):
 限界効用逓減の法則
 St. Peterburgの逆理
 人々が最大化するのは期待効用であって期待利得ではない。
 無差別曲線:
期待効用が一定となる期待値と分散; 同じ効用が得られる財の組合わせ
 Market risk & unique risk:
 モダンポートフォリオ理論:
 低リスク銘柄と高リスク銘柄による組合せパターン
 各人のリスク許容度に応じて株式と国債の割合を決める (効率的ポートフォリオ)
 Index fund vs. Active fund
 トービンのPortfolio分離定理:
