Returns Additivity

Let

• \(p_t\) = price at time \(t\)

• \(\hat r_t = \frac{p_t}{p_{t-1}} - 1\) be the actual (arithmetic) return at time \(t\)

• \(r_t = \ln\frac{p_t}{p_{t-1}}\) be the log return at time \(t\).

Consider the following table of TQQQ prices for ten trading days and their corresponding returns:

Period	Date	\(p_t\)	Actual return \(\hat r_t\)	Log return \(r_t\)
0	2025‑04‑25	53.86	–	–
1	2025‑04‑28	53.82	−0.000743	−0.000743
2	2025‑04‑29	54.87	0.019509	0.019322
3	2025‑04‑30	54.88	0.000182	0.000182
4	2025‑05‑01	56.77	0.034439	0.033859
5	2025‑05‑02	59.43	0.046856	0.045791
6	2025‑05‑05	58.40	−0.017331	−0.017483
7	2025‑05‑06	56.74	−0.028425	−0.028836
8	2025‑05‑07	57.40	0.011632	0.011565
9	2025‑05‑08	59.11	0.029791	0.029356

Analysis of Returns

• Sum of actual returns:

  \[ \sum_{t=1}^{9} \hat r_t = -0.000743 + 0.019509 + \cdots + 0.029791 = 0.09591\;(=9.59\%). \]

• Compounded actual return:

  \[ \prod_{t=1}^{9}(1+\hat r_t) - 1 = 1.02\ldots - 1 = 0.09747\;(=9.75\%). \]

• Sum of log returns:

  \[ \sum_{t=1}^{9} r_t = -0.000743 + 0.019322 + \cdots + 0.029356 = 0.09301\;(=9.30\%). \]

• Converted to actual return:

  \[ e^{\sum_{t=1}^{9} r_t} - 1 = e^{0.09301} - 1 = 0.09747\;(=9.75\%). \]

Key Insights

• The sum of log returns equals the log of the total price ratio: \(\displaystyle \sum_{t=1}^T r_t = \ln\frac{p_T}{p_0}.\)

• Exponentiating the sum of log returns reproduces the compounded return.

• Simply summing arithmetic returns (9.59%) underestimates the true compounded return (9.75%).

• The return obtained via exponentiating the sum of log returns (9.75%) and the compounded actual return (9.75%) are effectively identical—showing how closely these methods agree.

• The gap between summed and compounded returns widens with more periods or greater volatility.