[心得] 效用函數-3-應用:保險及其他

Blog post:
https://daze68.blogspot.com/2021/03/3-utility-function-3-insurance-and.html

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相對風險趨避係數η在資產配置的應用
數學上相對比較複雜
也許不是每位朋友都會認同
讓我們嘗試一些比較直觀的應用

比如說，保險:

假設總資產1000萬，其中汽車價值100萬
預期一年內平安無事機率99%，發生車禍完全撞毀機率1%
是否該買要價x萬元的車體險?

If η=1，u(c)=ln(c)
不購買車體險 utility = ln(1000/1000)*0.99 + ln( (1000-100)/1000)*0.01
購買車體險 utility = ln((1000-x)/1000)*0.99 + ln((1000-x)/1000)*0.01
車體險要價小於1.05萬元時，購買車體險的utility就會大於不購買

If η=3，u(c)=(c^(1-3)-1)/(1-3)
不購買車體險 utility =
((1000/1000)^(-2)-1)/(-2)*0.99 + (((1000-100)/1000)^(-2)-1)/(-2)*0.01
購買車體險 utility =
(((1000-x)/1000)^(-2)-1)/(-2)*0.99 + (((1000-x)/1000)^(-2)-1)/(-2)*0.01
車體險要價小於1.17萬元時，購買車體險的utility就會大於不購買

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utility function也不只能用在財務規劃

舉例來說
假設核四發生核災機率為x
核四商轉不發生核災可讓資產增加1%
發生核災會讓資產減少90%
是否該支持核四商轉?

If η=1，u(c)=ln(c)
(1-x)ln(1.01)+x*ln(0.1)==0, solve x
=> x=0.0043
如果核災機率大於0.43%，utility就變負

If η=3，u(c)=(c^(1-3)-1)/(1-3)
(1-x)*(1.01^(-2)-1)/(-2)+x*(0.1^(-2)-1)/(-2)==0, solve x
=> x=0.00020
如果核災機率大於0.02%，utility就變負

即使大家對於財產收益與損失的預期及核災的發生率相同
在不同的η之下
決定也可能是不同的

--
You got to know when to hold 'em, know when to fold 'em, Know when to walk away and know when to run.
You never count your money when you're sittin' at the table. There'll be time enough for countin' when the dealin's done.

'Cause ev'ry hand's a winner and ev'ry hand's a loser, And the best that you can hope for is to die in your sleep."
now Ev'ry gambler knows that the secret to survivin' Is knowin' what to throw away and knowing what to keep.

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