Timing

5. Timing#

5.1. Static Single Asset Example#

We model preferences using mean-variance utility

Consider a mean-variance investor allocating between a factor (the market) and a risk-free asset.

The investor solves:

\[\max_{w} E[w r_{t+1} + (1-w)r^f] - \frac{\gamma}{2} Var(w r_{t+1} + (1-w)r^f)\]

where:

  • \(w\) = weight on the risky portfolio

  • \(r_{t+1}\) = risky asset return

    • Not known at date t. it is a random variable: \(E[r_{t+1}]\neq r_{t+1}\)

  • \(r^f\) = risk-free rate from t to t+1

  • known at date t: \(E[r^f]= r^f\)

  • \(\gamma\) = risk aversion coefficient

  • \(\text{E}(x)\) = Expectation of random variable x

  • \(\text{Var}(x)\) = variance of random variable x

The solution is elegantly simple:

💡 Key Insight: At the optimal, the slope of your objective cannot change with the weight

\[(E[r_{t+1}]-r^f) - \gamma w Var(r_{t+1})=0\]

💡 Key Insight: The optimal allocation formula

\[w^* = \frac{E[r_{t+1} - r^f]}{\gamma \cdot Var(r_{t+1})}\]

You invest proportionally to the risk-return trade-off:

  • 📈 Lever up when expected returns are higher

  • 📉 Lever down when variance is higher

5.2. 📈 Sharpe Ratio Targeting#

A beautiful result emerges: you target portfolio volatility to move with the asset Sharpe ratio:

\[w^* \cdot \sqrt{Var(r_{t+1})} = \frac{1}{\gamma} \cdot \frac{E[r_{t+1}] - r^f_{t+1}}{\sqrt{Var(r_{t+1})}} = \frac{1}{\gamma} \cdot SR(r_{t+1})\]

💡 Key Insight: When the Sharpe ratio is high, you choose a more volatile portfolio.

Note that \(|w^*| \cdot \sqrt{Var(r_{t+1})}\) is the portfolio volatility

If SR<0 you still have a postive volatility portfolio–you just short the asset instead of going long!

This makes sense—it’s more attractive to take risk when you’re better compensated for it!

The practical implications:

Scenario

Action

Risk premium ↑ more than variance

Lever up

Variance ↑ more than risk premium

Lever down

The theory tells us it’s all about the ratio of premium to variance.

🤔 Stop and Think: What if I don’t care about variance?

  • what you would do if there are no leverage limits?

  • What if you can’t borrow?

  • What if you can borrow but your leverage depends on your portfolio vol?

  • What if it does not depend on it?

5.3. Single asset with time-varying moments#

Same problem but now expected returns, risk-free, and variance move around over time

\[\max_{w_t} E_t[w_t r_{t+1} + (1-w)r^f_{t+1}] - \frac{\gamma}{2} Var_t(w_t r_{t+1} + (1-w)r^f_{t+1})\]

where:

  • \(w_t\) = weight on the risky portfolio at date t

  • \(r_{t+1}\) = risky asset return from t to t+1

    • Not known at date t. it is a random variable: \(E_t[r_{t+1}]\neq r_{t+1}\)

  • \(r^f_{t+1}\) = risk-free rate from t to t+1 (known at date t. )

  • known at date t: \(E[r^f_{t+1}]= r^f_{t+1}\)

  • \(E_t(x)\) = Expectation of random variable x at date t given all I know

  • \(Var_t(x)\) = E_t[(x-E_t[x])^2] variance of random variable x around what I expect the vriable to be given all I know

This can be simplified to:

\[\max_{w} rf_{t+1}+w_tE_t[r^e_{t+1}] - \frac{\gamma}{2} w_t^2Var_t(r^e_{t+1})\]

Do you understand why this follows?

  • We get the same result!

\[w^*_t = \frac{E_t[r^e_{t+1}]}{\gamma \cdot Var_t(r^e_{t+1})}\]

  • The volatility of you portfolio should track the Sharpe ratio of the portfolio over time

  • What is the intuition?

\[w^* \cdot \sqrt{Var_t(r^e_{t+1})} = \frac{1}{\gamma} \cdot SR_t(r^e_{t+1})\]

  • Note that given the behavior of excess returns, variation in the risk-free rate does not matter at all

5.4. ⚠️ Caution: The risk-free rate#

  • Is \(Var(r_{t+1})=Var(r^e_{t+1})\)?

  • If I compute both in the data, will they be the same?

  • Should it matter if the risk-free is constant or time-varying?

  • If it moves around does it mean it is risky?

  • What is the correct one to use?

import pandas as pd
import pandas_datareader as pdr
import matplotlib.pyplot as plt

# Import 3-month Treasury Bill rate from FRED (entire time series)
tbill_3m = pdr.data.DataReader('TB3MS', 'fred', start='1934-01-01')

# Plot the 3-month T-Bill rate over time
plt.figure(figsize=(12, 6))
plt.plot(tbill_3m.index, tbill_3m['TB3MS'])
plt.title('3-Month Treasury Bill Rate Over Time')
plt.xlabel('Date')
plt.ylabel('Rate (%)')
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

print(f'The variance of the 3-month Treasury Bill rate is: {tbill_3m['TB3MS'].var()}')
../../_images/cea73c89a75984eab3306592606cdbdeda9b29677471e5b56cb07c16793ec1a6.png 
The variance of the 3-month Treasury Bill rate is: 9.583294505022533

5.5. 📌 Key Takeaways#

Before diving into implementation, remember these core principles:

  1. Time-varying moments create opportunity: If expected returns and volatility change over time, you can exploit this through dynamic allocation

  2. The optimal allocation is intuitive: Invest proportionally to expected excess return, inversely to variance and risk aversion

  3. Target the Sharpe ratio: Adjust portfolio volatility based on how attractive the risk-return trade-off is

  4. Think carefully about what you do: What you do in the data needs to be guided by an economic understanding

In the following notebooks, we’ll explore two practical approaches:

  • Expected Returns Timing: Predicting when market returns will be high or low

  • Volatility Timing: Adjusting exposure based on time-varying risk

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