AlphaLess wrote: ↑Tue Oct 09, 2018 11:16 pm

nisiprius wrote: ↑Tue Oct 09, 2018 8:51 pm

c) The coin-flip adds risk without adding any return, and reduces the risk-adjusted return for both of you.

P: random variable (portfolio)

B: random variable (bet).

Var (P + B) = Var ( P ) + 2 * Cov ( P,B) + Var ( B)

By the design of the problem, Cov ( P,B) = 0 {let's say we can muster up an iPhone app to throw a random coin, and it is truly random and does not use any known asset returns for randomness generation},

Var(B) > 0

So Var (P + B) > Var (P).

Also, notice that Std (P+B) = Sqrt ( Var (P+B)).

Thus, the risk of the (P+B) combo is higher, in variance and standard deviation terms.

E [ P + B ] = E [ P ] + E [ B ] {as expectation is additive}.

By design, E [ B ] = 0.

Thus, E [ P + B] = E [ P ]

If we use a simplest form of risk-adjusted return, ignoring interest rates:

Sharpe (P+B) = E [ P + B ] / Std (P + B) < E [ P ] / Std ( P) = Sharpe (P) , because

: numerators are equal,

: denominator Std ( P + B) is larger.

However, let me ask you this problem:

- you and a friend (let's call your friend Warren 'GEICO' Buffet) decided to meet up once a month,

- you give your friend $100 / M, for twenty years, maybe more,

- your friend Warren cashes that check, with a smirk,

- occasionally, when you have trouble with your car, of accident variety, Warren reluctantly writes a check to you, covering some part of the expenses resulting from the accident,

- it is known that the total dollar value of the checks that Warren might write to you is strictly lower than the total value of checks that you write to Warren.

Do you play this game with Warren?