Initial rate question?

Growth rate question

• Something grows at an annualized rate of 5% for 4 months and then declines at an annualized rate of 4% for the next four months. What would be the annualized growth rate over the entire 8 month period? I am conflicted if I should use a geometric mean of 5% an -4% or if I should do (1+0.05)(1-0.04)-1 = 0.8%. Thanks for the help

• Answer:

  If it grows at an annualized $5\%$ for $4$ months it grows by $\frac{5}{3}\%$ (or a bit more if you compound), then falling at $4\%$ for $4$ months it falls by $\frac{4}{3}\%$. Your last is correct if you get the correct growth rates: $(1+\frac{5}{3}\%)(1-\frac{4}{3}\%)-1$, which I make growing by $\frac{7}{2250}$ or about $0.31111\%$ in $8$ months. To annualize this, we should mutiply by $\frac{3}{2},$ getting $\frac{21}{4500}\approx 0.46667\%$