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Question 723 mean and median returns, return distribution, arithmetic and geometric averages, continuously compounding rate

Here is a table of stock prices and returns. Which of the statements below the table is NOT correct?

Price and Return Population Statistics
Time	Prices	LGDR	GDR	NDR
0	100
1	99	-0.010050	0.990000	-0.010000
2	180.40	0.600057	1.822222	0.822222
3	112.73	0.470181	0.624889	0.375111

Arithmetic average		0.0399	1.1457	0.1457
Arithmetic standard deviation		0.4384	0.5011	0.5011

(a) The geometric average of the gross discrete returns (GAGDR) equals 1.04075 which is 104.075%.

(b) ##\exp \left( \text{GAGDR} \right) = \text{AALGDR}##. The natural exponent of the GAGDR equals the arithmetic average of the logarithms of the gross discrete returns (AALGDR).

(c) ##\text{GAGDR}^T = \left( P_T/P_0 \right)##. The GAGDR raised to the power of the number of time period that it's measured over equals the ratio of the first and last prices.

(d) ##\text{GAGDR} = \exp \left( \text{AALGDR} \right)##. This is always true, regardless of the distribution of the prices or returns. The logarithm of the geometric average of the gross discrete returns (LGAGDR) is always equal to the arithmetic average of the logarithm of the gross discrete returns (AALGDR).

(e) ##\text{AAGDR} \approx \exp \left( \text{AALGDR} + \text{SDLGDR}^2/2 \right)##. This is only approximately true and depends on the distribution of the prices and returns. It's asymptotically true as the time period that the returns are measured over reaches infinity.

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