買指數型ETF的槓桿方式 - 投資

Blog post:
https://daze68.blogspot.com/2021/03/higher-than-risk-free-rate.html

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不太確定這個想法數學上是否robust，姑且聽聽看

Merton's solution 的理想股債比是 (1/η)*x/(s^2)
其中η是相對風險趨避係數，x是股票溢酬，s是標準差
(如何估計自己的η可參考 https://tinyurl.com/j68dzmv6 )

當借錢投資的時候
如果借款利率高於無風險利率
則槓桿部分的x會減少，但s不變

If η=3, x=0.05, s=0.16，理想股債比是 65%
若借款利率是無風險利率+1%，理想股債比變成 52%
若借款利率是無風險利率+2%，理想股債比變成 39%

如果現有資產遠遠少於未來人力資本
則只要借款利差低於股票溢酬
仍然應該借好借滿

但如果現有資產已占人生總資產的相當比例
則隨著借款利率增加，理想股債比降低
達到理想股票配置的槓桿需求也會下降

再考慮到股票溢酬的不確定性、破產的disutility等等
某些狀況下也許持有100%股票配置會比較理想
使用槓桿並不會增加utility

另外，Currency risk 通常認為長期來說是中立的
但Currency risk會增加Variance，s^2的部分會增加
同樣會降低理想股債比
但作用的方式不太一樣

假設USDTWD滿足常態分佈，標準差0.05
variance的上下界是 (0.16-0.05)^2 ~ (0.16+0.05)^2
(上下界應該可以再縮窄，但以我的數學能力有點困難)
理想股債比要乘以 2.11~0.58
但似乎沒有很好的理由相信增加currency risk反而會減少變異度
或許在 1~0.58之間猜一個你覺得合理的數字

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也有些paper用數值方法測試不同借款限制下的理想配置
Take it with some grain of salt.

譬如 Steven J. Davis & Felix Kubler & Paul Willen, 2006. "Borrowing Costs and
the Demand for Equity over the Life Cycle," The Review of Economics and
Statistics, MIT Press, vol. 88(2), pages 348-362, May.

--
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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