Question

The incredible shrinking $50 bill in 1957 was worth $50, but in 2007 it is worth only $6.34. a. What was the compounded average annual inflation rate (loss of purchasing power) during this period of time? b. Fifty dollars invested in the stock market in 1957 was worth $1,997 in 2007. In view of your answer to Part (a), what was the annual real interest rate earned on this investment? a. The loss of purchasing power is %. (Round to two decimal places.)

Answer 1

(a)

Value of incredible shrinking bill in 1957 = $50

Value of increadible shrinking bill in 2007 = $6.34

Time period = 50 years

Calculate the compounded average annual inflation rate -

Compounded average annual inflation rate = [Value of incredible shrinking bill in 2007/Value of incredible shrinking bill in 1957]^{1/time period} - 1

Compounded average annual inflation rate = [6.34/50]^1/50 - 1

Compouned average annual inflation rate = [6.34/50]^0.02 - 1

Compounded average annual inflation rate = 0.9595 - 1 = -0.0405

The compounded average annual inflation rate is -4.05%

Negative value indicates the loss of purchasing power.

Thus,

The loss of purchasing power is 4.05%.

(b)

Value of investment in 1957 = $50

Value of investment in 2007 = $1,997

Time period = 50 years

Calculate the average annual nominal interest rate -

Average annual nominal interest rate = [Value of investment in 2007/Value of investment in 1957]^{1/time period} - 1

Average annual nominal interest rate = [1997/50]^1/50 - 1

Average annual nominal interest rate = [1997/50]^0.02 - 1

Average annual nominal interest rate = 1.0765 - 1 = 0.0765 or 7.65%

The average annual nominal interest rate is 7.65%

Calculate the annual real interest rate -

Annual real interest rate = Annual nominal interest rate - Annual inflation rate

Annual real interest rate = 7.65% - 4.05% = 3.60%

Thus,

The annual real interest rate is 3.60%