The first one is "Siegel's Paradox", which seem to be basically a prolonged discussion around the fat that losing 50% of your investment means that you need to make 100% to get back to where you started, expressed in the terms of foreign exchange rate moves (or I might have just missed something on this one). But of course, the actual purchasing parity moves the opposite direction, so there is no free lunch to have based on exchange rate movements.

The next chapter discusses, how the likelihood of loss can be about anything, depending on how exactly the criteria is spelled out.

Chapter 3 on time diversification discusses if/how the investment time horizon effects risk (this is dependent on how we defined risk) and the surprising answer might be, that longer term investing carries the same risk as shorter term (in particular if risk is measured in using log wealth utility function). The log wealth utility function (first expressed in Daniel Bernoulli's St. Petersburg Paradox, the Bernoullis also had a major role in defining

Chapter 4 demonstrates, using bootstrapping (which is resampling of the results of a past period repeatedly) of real returns on securities, that geometric average/mean of a compounding distribution is higher than its median, so your expected return on your investment is not to be expected. Chapter 4 also demonstrates the asymetricity in compounding effects of geometric returns, where the high end of returns are further away from the middle value, than the low end of returns (both in percentage and actual values). At the same time, the percentage of continuous returns stays symmetrical.

The next chapter offers the view, that an investor is better off using a balanced strategy (investing in a riskless and risky asset at the same time), as contrasted to switching between those same assets for equal periods. While the cumulative wealth are the same, the switching strategy has higher variance (it is riskier). To arrive to the same Sharpe ratio (as expressed by dividing the excess return of an investment by the standard deviation of that same excess return), the switching 100% each year strategy would need to bring an extra 5% of return (this is the market timing skill needed to make switching worthwhile). These results are even more pronounced if log wealth utility is applied to these returns. Another demonstration of increasing risk for longer timeframe is shown by option pricing, prices for options increases with volatility, but it also increases with longer expiration (even if volatility stays the same).

Chapter 6 explains how the pricing of options is independent from the expected return of the underlier. The Black-Scholes formula indeed has no reference to the expected return (although one probably should argue that volatility, which is assumed to be known and constant by the formula is greatly influenced by the changing perceptions and expectations of the expected return), instead the riskless return determines the option price (based on equivalence of hedged positions). Volatility of an asset can be expressed using beta, which represents the asset's non-diversifiable risk (which relates the return of the asset to the market's expected return). The chapter offers a good history of the formula from botany, through heat distribution to corporate finance. The law of one price is also related, stating that price of assets is independent of the capital structure and dividend payment policies.

Good brainteasers.

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