13 — Sharpe Ratio, Drawdown, and Benchmark Basics¶

Free learning pack¶

1. MIT OCW Portfolio Theory I–III
   https://ocw.mit.edu/courses/15-401-finance-theory-i-fall-2008/pages/video-lectures-and-slides/portfolio-theory/
2. CFA Portfolio Risk and Return Part I
   https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/portfolio-risk-return-part-1
3. resources/portfolio_foundations.md
4. resources/active_management.md

Do not search for more material until these are insufficient.

PREDICT (Sharpe)¶

A strategy returns +2%, +2%, +2% every month — no variation at all. Its mean monthly return equals a second strategy that returns +6%, -2%, +2% over the same three months. Which strategy has the higher Sharpe ratio, and why?

Formula¶

Sharpe = (mean(r) - rf) / std(r)

Annualized: multiply by sqrt(periods_per_year).

In [ ]:

import numpy as np

monthly_returns = np.array([0.02, 0.04, 0.03, -0.01, 0.05, 0.00, 0.02, 0.03, -0.02, 0.04, 0.01, 0.02])
risk_free_rate = 0.0
periods_per_year = 12

# MANUAL FIRST:
# compute excess returns, their mean and sample std (ddof=1),
# then the annualized Sharpe ratio.
sharpe = None
print(sharpe)

import numpy as np monthly_returns = np.array([0.02, 0.04, 0.03, -0.01, 0.05, 0.00, 0.02, 0.03, -0.02, 0.04, 0.01, 0.02]) risk_free_rate = 0.0 periods_per_year = 12 # MANUAL FIRST: # compute excess returns, their mean and sample std (ddof=1), # then the annualized Sharpe ratio. sharpe = None print(sharpe)

PREDICT (Drawdown)¶

A portfolio returns +10%, -20%, +5% over three periods. Is the max drawdown the same magnitude as the single worst period return? Why or why not?

Formula¶

Wealth path: W_t = prod(1+r_s) for s<=t

DD_t = W_t / max(W_0..W_t) - 1

Max drawdown = min(DD_t)

In [ ]:

import numpy as np

period_returns = np.array([0.10, -0.20, 0.05])

# MANUAL FIRST:
# build the cumulative wealth path, the running peak, the drawdown series,
# and the max drawdown.
max_dd = None
print(max_dd)

import numpy as np period_returns = np.array([0.10, -0.20, 0.05]) # MANUAL FIRST: # build the cumulative wealth path, the running peak, the drawdown series, # and the max drawdown. max_dd = None print(max_dd)

Benchmark basics¶

No new code here — this is vocabulary you need before Phase 3 (active management).

• Benchmark weight w_b vs portfolio weight w_p
• Active weight: a = w_p - w_b
• Active return: r_p - r_b

Read reference/concepts/benchmark_basics.md in full, then answer:

PREDICT (Benchmark)¶

A portfolio is 60% US equity / 40% international equity. Its benchmark is 50% / 50%. Without calculating, is the portfolio's active return positive or negative in a period where US equity outperforms international equity?

Experiment¶

After your hand implementations work:

• Re-order the three periods in the drawdown example (e.g. -0.20, 0.10, 0.05). Does the max drawdown change even though total return doesn't?
• Hold the mean of monthly_returns fixed but increase its volatility. Watch the Sharpe ratio fall.

Reference¶

reference/concepts/sharpe_ratio.md
reference/concepts/drawdown.md
reference/concepts/benchmark_basics.md

Promote¶

Use src/pm/returns.py (sharpe_ratio, max_drawdown) and src/pm/active.py (active_return) only after your own implementation.

Test¶

pytest tests/test_returns.py tests/test_active.py

ORAL CHECK¶

Explain to a PM why two strategies with identical Sharpe ratios can have very different max drawdowns, and why that matters for a client who cannot tolerate a 20% loss.