Comparing Investment Growth with Compound Interest and Annuities
Mia is 25 and wants to compare two savings strategies over 30 years. Strategy A: invest a lump sum of $20 000 now at 5.4% per annum compounded monthly. Strategy B: deposit $150 at the end of each month into an account earning 4.8% per annum compounded monthly, starting one month from now. Mia also considers Strategy C: deposit $150 at the beginning of each month into the same account as B. She wants to know which strategy gives the largest balance at age 55, and by how much. She is also concerned about inflation, which she assumes averages 2.5% per annum over the period, and wants to know the real (inflation-adjusted) value of the best strategy's final balance in today's dollars.
1. Calculate the final balance of each strategy after 30 years and identify which is largest.
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Strategy A: monthly rate i = 0.054/12 = 0.0045, n = 360. FV = 20000(1.0045)^360. Using logarithms: ln(1.0045) ≈ 0.0044899, times 360 = 1.61637, so (1.0045)^360 ≈ e^1.61637 ≈ 5.034. FV ≈ $100 680. Strategy B: ordinary annuity, i = 0.048/12 = 0.004, n = 360. FV = 150[((1.004)^360 - 1)/0.004]. (1.004)^360: ln(1.004) ≈ 0.0039920, times 360 = 1.43713, e^1.43713 ≈ 4.209. So FV = 150[(4.209 - 1)/0.004] = 150 × 802.25 ≈ $120 338. Strategy C: annuity due, FV = 120338 × 1.004 ≈ $120 819. Strategy C is largest, exceeding A by about $20 139 and B by about $481.
2. Calculate the real value of Strategy C's final balance in today's dollars, and explain what assumption this calculation depends on.
Read the worked answer
Real value = FV / (1 + inflation)^30 = 120819 / (1.025)^30. (1.025)^30: ln(1.025) ≈ 0.0246926, times 30 = 0.740778, e^0.740778 ≈ 2.0976. So real value ≈ 120819 / 2.0976 ≈ $57 600. The calculation assumes a constant inflation rate of 2.5% per annum compounding annually over the entire 30 years, and that the nominal balance is measured at the same point in time as the price level. It also assumes inflation affects all goods equally and that Mia's consumption basket matches the general index; changing any of these assumptions (for example, variable inflation or a different compounding frequency) would change the real value. The real value is much lower than the nominal figure, showing that nominal growth overstates purchasing power.