Question

Money is invested in a savings account with a nominal interest rate of 2.4% convertible monthly for three years. The rate of inflation is 1.5% for the first year, 2.8% for the second year, and 3.4% for the third year. Find the percentage of purchasing power lost during the time the money is invested; that is, find p so that if you could purchase exactly u units at the time the money was invested, three years later you could purchase u(1 ? 0.01p).

Answer 1

To calculate the percentage of purchasing power lost during an investment over three years with a nominal interest rate of 2.4% and variable inflation rates, compound the monthly interest rate for three years and compound the inflation rates annually. Then subtract the compounded inflation rate from the effective interest rate of the investment to find the loss in purchasing power in percentage terms. Lastly, use the formula u(1 - 0.01p) to determine how many units could be purchased after three years.

Step-by-step explanation:

To find the percentage of purchasing power lost during three years of investment considering variable inflation rates, we need to compare the nominal interest rate with the effective inflation rates over the period. Even though the nominal interest rate of the savings account is 2.4% convertible monthly, we must adjust this rate for inflation to understand its real impact on purchasing power. To do this, the monthly interest rate needs to be compounded over the three years and the inflation rates need to be compounded respectively for each year.

Firstly, since the nominal annual interest rate is 2.4% compounded monthly, the monthly interest rate is 2.4%/12 = 0.2%. Compounding this monthly over three years gives us an effective interest rate EIR which can be calculated using the formula:

EIR = [(1 + 0.2/100)12*3 - 1] * 100

To calculate the overall effect of inflation, we must adjust the individual annual inflation rates to find the compounded rate over the three years. Assuming the inflation rates are compounded annually, the compounded inflation rate over the three years can be found using:

Compounded Inflation = [(1 + 1.5/100) * (1 + 2.8/100) * (1 + 3.4/100) - 1] * 100

To determine the percentage of purchasing power lost, subtract the compounded inflation rate from the effective interest rate. The result will give us the loss in purchasing power as a percentage.

Percentage of Purchasing Power Lost (p) = EIR - Compounded Inflation

Finally, to determine how many units could be purchased after three years relative to the initial number, use the formula:

u(1 - 0.01p), where p is the percentage of purchasing power lost.

Answer 2

real rate of return -0.48%

if before I could purchase X units now I purchase X*(1 - 0.0048)

Step-by-step explanation:

We solve using the fishcer model:

`(1+r_n)/(1+ \theta ) = 1+r_e`

As we have more than one inflation period we multiply each other as it was a succession of interest

`(1.024^3)/(1.015 * 1.028 * 1.034) - 1 = r_e`

real rate -0.004777527 = -0.48%