1/

🧵How much skill (alpha) a concentrated stock picker needs to beat a fully diversified benchmark index?

Variance lowers expected geometric return. To compensate for the idiosyncratic variance drag, stock picker needs skill to catch up with and to exceed the index return. 2/

We:
- Consider geometric (instead of arithmetic) returns
- Assume continuously compounded returns, equally weighted portfolios & similar volatility for portfolios of equal size
- Completely ignore the risk reduction aspect of diversification and focus only on expected returns

1/23

🧵 The importance of diversification increases with time

In the absence of stock picking skill, compound wealth of a typical poorly diversified portfolio loses to compound wealth of a fully diversified benchmark the more the longer the time horizon. 2/

Probably the best known phrase in finance is ”diversification is a free lunch” by Markowitz.

This is true for annualized arithmetic (single period, no compounding) returns, for which mean remains constant while spread of returns narrows with diversification.

1/15

🧵 Consider a parameter-clairvoyant Kelly investor

The clairvoyant can borrow at risk-free rate, is immune to margin calls and can foresee two parameters for the next ten years: Sharpe ratio and volatility.

Having this gift, he can optimize leverage to maximize his CAGR. 2/

CAGR is maximized at full Kelly allocation, i.e. when portfolio’s volatility (fs) = Sharpe ratio (SR). f = leverage multiplier, s = volatility at 100% allocation.

[Note that Kelly use continuously compounded growth rate g = ln(1 + CAGR). Both g and CAGR max at full Kelly]

https://twitter.com/markku_kurtti/status/1541415451426693121

1/6

🧵 The Shannon limit

Digital communications:

Information channel capacity =
The max achievable average error-free data transfer rate over a noisy channel

Finance:

Compounding process capacity =
The max achievable average compound excess growth rate for a risky investment 2/6

Given a sufficient frequency bandwidth, information channel capacity depends only on signal-to-noise(per Hz) ratio (SNR).

Given a sufficient compounding frequency, compounding process capacity depends only on square of Sharpe ratio, the SNR of arithmetic excess returns.

1/5

Your expected growth rate and optimal leverage are lower than you estimate

Why?

Because you not only need to account for volatility drag, but also the drag from volatility (uncertainty) of your volatility estimate.

Unlike realized past, future always entails uncertainty. 2/5

We can derive

Kelly criterion:
f* = (m-r)/(Mean(SDest)^2+SD(SDest)^2)

and

Geometric risk premium:
g-r = f(m-r) - f^2(Mean(SDest)^2+SD(SDest)^2)/2

in the presence of uncertainty about risk SD(SDest)

f = leverage multiplier
m = arith. mean
g = geom. mean
r = riskless rate

1/5

🧵 Too little empirical data downplays long-term equity risk

Empirical long-term equity risk is lower than theory predicts based on shot-term risk

All 19 year periods have returned more than riskless rate

Loss statistics appear lower than predicted at horizons > 10 years 2/5

Lower than expected empirical long-term risk is usually explained by return mean reversion

Mean reversion may be true, but we don’t need it — we only need too little data

Simple i.i.d. simulation replicates empirical result (i.i.d. = no mean reversion)

1/5

Practical investors live their lives through time and care about both the growth rate and the path their wealth takes

We can describe:
- reward
- risk
- reward/risk

solely by two metrics elementary to path dependent finance:
- Kelly fraction
- square of Sharpe ratio 2/5

Long term investors take on equity risk to earn a REWARD: geometric return in excess of riskless return (G - RF)

Deviation from riskless portfolio exposes investors to RISK: drawdown from riskless portfolio value (DD_RF)

Let’s define
REWARD = E(G - RF)
RISK = E[max(DD_RF)]

1/11

Kansainvälinen hajauttaminen toimii – kun sille antaa aikaa.

Korrelaatiot määrittävät hajautushyödyn ja markkinoiden väliset korrelaatiot ovat tunnetusti korkeita etenkin kriiseissä.

Totta lyhyellä aikavälillä, mutta korrelaatiot pienenevät sijoitushorisontin pidentyessä. 2/11

”Global EW” portfolio koostuu viidestä maantieteellisestä alueesta tasapainoin välillä Jul/1990 - May/2022.

Alueiden keskinäiset korrelaatiot pienenevät ajan myötä poislukien “Emerging Markets” (EM) ja “Asia Pacific excluding Japan”, jotka korreloivat lähes täydellisesti.

1/7

For geometric returns, idiosyncratic risk is not only uncompensated but costly. The cost is lower expected growth rate (g).

Portfolio g - single stock g = Diversification Premium (DP). DP ≈ f^2*IVar_diff/2. f = leverage & IVar_diff = idiosyncratic variance difference. 2/7

Diversification not only lower risk, but increase geometric expected return particularly with leverage.

Essentially diversification increase Sharpe ratio (SR). Higher SR imply higher g-capacity and expanded opportunity set for rational compounder.

https://twitter.com/markku_kurtti/status/1541415451426693121

1/
Miten indeksiin sijoittava selvännäkijä määrittäisi sijoitusasteensa?

Selvännäkijällä on kyky nähdä tulevaisuuteen 10 vuoden periodille kaksi parametria: Sharpe ratio ja volatiliteetti, ei muuta. Näillä tiedoilla voidaan valita tuoton maksimoiva sijoitusaste.

#sijoittaminen 2/
@hkeskiva ketju käteisen optioarvosta innosti tutkimaan mikä voisi olla sijoitusaseen optimi pitkällä aikavälillä.

Yritänkin myös tässä ketjussani tutkia CAPEn kykyä ennustaa optimaalinen sijoitusaste.

https://twitter.com/hkeskiva/status/1484806544063614977?s=21