Question

Peter Goodman invested $40,000 in the Carters balanced mutual fund 5 years ago and he is looking for advice on whether to sell this investment. He is really concerned about his return being influenced by inflation. His tax rate is 30% and the inflation rate is 1.5% p.a. Assume that taxes will be paid when he sells the investment.  His investment accounts earned the following annual rates of return: +2%, +8%, +5%, +1%, -4%.

1. Calculate the arithmetic and geometric mean rate of return on his investments before tax.
2. As his financial advisor, what mean (geometric vs. arithmetic) would you present to him and why?
3. Calculate Peter’s real after-tax rate of return.

Answer 1

Part (a)

Arithmetic mean = (2% + 8% + 5% + 1% - 4%) / 5 = 2.4%

Geometric mean = [(1 + 2%)(1 + 8%)(1 + 5%)(1 + 1%)(1 - 4%)]^1/5 - 1 = 2.32%

Part (b)

As a financial advisor, I will present the geometric mean, because returns are compounded at the end of year, returns are not discreet. Hence, geometric mean is a better measure than arithmetic mean.

Part (c)

Peter's maturity value, FV = PV x (1 + geometric mean return)⁵ = 4,000 x (1 + 2.32%)⁵ = 4,486.07

Hence, gain = FV - PV = 4,486.07 - 4,000 = 486.07

Tax on gain = Gain x tax rate = 486.07 x 30% = 145.82

hence, post tax value, FV* = FV - tax on gain = 4,486.07 - 145.82 = 4,340.25

Hence, after tax nominal annual return = (FV* / PV)^1/n - 1 = (4,340.25 / 4,000)^1/5 - 1 = 1.65%

Hence, after tax annual real rate of return = (1 + nominal annual return) / (1 + inflation) - 1 = (1 + 1.65%) / (1 + 1.5%) - 1 = 0.1440%