Why GBM Sharpe Ratio Uses Instantaneous Expected Returns

Summary

The question concerns deriving the Sharpe ratio for a stock modeled by geometric Brownian motion alongside a riskless zero-coupon bond. A portfolio investing fraction w in the stock and the remainder in the bond has an instantaneous expected return formed by weighting the two assets’ drift rates, while its instantaneous volatility is w times the stock’s volatility, assuming the stated nonnegative weight range. This gives the same excess-return-to-volatility ratio as the stock for such scaled combinations.

The apparent conflict is with comparing these drift rates to finite-horizon expected price growth, which is exponential in time. The portfolio argument uses instantaneous return rates in the stochastic differential equation, or equivalently a short-period approximation; it is not claiming that finite-horizon expected price levels grow linearly. The source poses the conceptual issue but supplies no answer or derivation beyond the lecture-note argument, so further assumptions and horizon-specific definitions are not examined.

Key ideas

• The GBM drift parameter is an instantaneous expected return rate, not a finite-horizon price-level expectation.
• A portfolio’s instantaneous expected return combines stock and bond drift rates by their weights.
• With a nonnegative stock weight, portfolio volatility scales in proportion to that weight.
• The Sharpe-ratio cancellation applies to instantaneous excess return and volatility under the stated setup.
• Finite-horizon expected price growth is exponential and should not be confused with the instantaneous-rate calculation.

Tags

# Relationship between risk and return for GBM and riskless bond


# Relationship between risk and return for GBM and riskless bond












Suppose we have $S$, a stock following geometric Brownian motion ($dS_t = S_t (\mu dt + \sigma dZ_t)$ for $Z =$ Brownian motion) and $B$, a zero coupon bond with rate $r$, i.e. $dB_t = rB_t dt$.

In trying to explain/derive the Sharpe ratio using these two assets ($= (\mu - r)/\sigma$), a set of lecture notes that I'm reading states that if we invest some proportion $w \in [0,1]$ in $S$, then the expected return is $w\mu + (1-w) r$ and the volatility is $w \sigma$ and hence any security with this volatility should give the same expected return. i.e. Any asset with volatility $w \sigma$ must give return excess of $r$ of $w(\mu - r)$ and thus

$$\frac{\text{Excess return}}{\text{Volatility}} = \frac{w(\mu-r)}{w \sigma} = \frac{\mu -r}{\sigma}$$

This confuses me, because the expected return of the stock is actually $\exp(\mu t)$ and of the bond is $\exp(rt)$. What is the rationale here? I've attached the slide I'm referring to in particular. Is the argument supposed to be purely heuristic over a short period? I've attached the slide I'm interested in below.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.