Please review the following case and use it as the basis for your analysis for this unit. Pay careful attention to the situation and note the specific events as they might apply to the information from your chapter. You will use this data to form the basis of your analysis. The specific architecture of your submission may be found in the course Syllabus on page x.
George Thriftless is 45 years old, earns $50,000 per year, and expects that his future earnings will keep pace with inflation, but will not exceed inflation. He has not yet saved anything toward his retirement. His company does not offer any pension plan. George pays Social Security taxes equal to 7.5% of his salary, and he assumes that when he retires at age 65, he will receive $ 12,000 per year in inflation-adjusted Social Security benefits for the rest of his life. His life expectancy is age 85.
George buys a book on retirement planning that recommends saving enough so that when private savings and Social Security are combined, he can replace 80% of his preretirement salary. George buys a financial calculator and goes through the following calculations:
First, he computes the amount he will need to receive in each year of retirement to replace 80% of his salary: 0.8 X $50,000 = $40,000.
Since he expects to receive $12,000 per year in Social Security benefits, he calculates that he will have to provide the other $28,000 per year from his own retirement fund.
Using the 8% interest rate on long-term default-free bonds, George computes the amount he will need to have at age 65 as $274,908 (the present value of $28,000 for 20 years at 8% per year). Then he computes the amount he will have to save in each of the next 20 years to reach that future accumulation as $6,007 (the annual payment that will produce a future value of $274,908 at an interest rate of 8% per year). George feels pretty confident that he can save 12% of his salary (i.e., $6,007/$50,000) in order to ensure a comfortable retirement.
Questions:
* If the expected long-term real interest rate is 3% per year, approximately what is the long-term expected rate of inflation?
* Has George correctly taken account of inflation in his calculations? If not, how
would you correct him?
* How much should George save in each of the next 20 years (until age 65) if he wants to maintain a constant level of consumption over the remaining 40 years of his life (from age 45 to age 85)? Ignore income taxes.
To answer the questions, let's analyze the given information step-by-step:
1. If the expected long-term real interest rate is 3% per year, approximately what is the long-term expected rate of inflation?
The real interest rate is the nominal interest rate minus the expected inflation rate. In this case, the real interest rate is given as 3%. If we assume that the long-term real interest rate equals the nominal interest rate, then the expected inflation rate can be estimated as 3%. Therefore, the long-term expected rate of inflation is approximately 3% per year.
2. Has George correctly taken account of inflation in his calculations? If not, how would you correct him?
No, George has not correctly accounted for inflation in his calculations. He assumed that his future earnings will keep pace with inflation but will not exceed inflation. However, he did not consider adjusting the future value of his retirement fund or savings for inflation.
To correct this, George should account for inflation by using real interest rates instead of nominal interest rates in his calculations. Real interest rates are obtained by adjusting the nominal interest rates for inflation. By using real interest rates, George can accurately calculate the future value of his retirement fund, taking into account the impact of inflation.
3. How much should George save in each of the next 20 years (until age 65) if he wants to maintain a constant level of consumption over the remaining 40 years of his life (from age 45 to age 85)? Ignore income taxes.
To determine the amount George should save, we need to consider his desired constant level of consumption over the remaining 40 years of his life. Since George wants to replace 80% of his preretirement salary, he will need $40,000 (80% of $50,000) per year in retirement.
To find out how much George should save each year, we can use the concept of present value. Assuming an annual real interest rate of 3%, the present value of $40,000 per year for 40 years would be:
PV = PMT / r = $40,000 / 0.03,
PV = $1,333,333.33.
This means that George should aim to have a retirement fund value of $1,333,333.33 by the time he reaches age 65 in order to maintain a constant level of consumption over the remaining 40 years.
To calculate how much he should save each year, we can use the future value of an annuity formula. Plugging in the values, we find:
FV = PV * (1 + r) ^ n,
$1,333,333.33 = PMT * (1 + 0.03) ^ 20.
Solving for PMT, we get:
PMT = $1,333,333.33 / ((1 + 0.03) ^ 20),
PMT ≈ $35,080.11.
Therefore, George should save approximately $35,080.11 each year for the next 20 years (until age 65) in order to maintain a constant level of consumption over the remaining 40 years of his life.
To answer the questions posed in the case, we'll go through the steps as described in the text.
1. If the expected long-term real interest rate is 3% per year, approximately what is the long-term expected rate of inflation?
To find the long-term expected rate of inflation, we need to subtract the real interest rate from the nominal interest rate. In this case, the nominal interest rate is given as 8% and the real interest rate is 3%.
Long-term expected rate of inflation = Nominal interest rate - Real interest rate
= 8% - 3%
= 5%
Therefore, the long-term expected rate of inflation is approximately 5% per year.
2. Has George correctly taken account of inflation in his calculations? If not, how would you correct him?
No, George has not taken account of inflation in his calculations. In his calculations, he assumes that his future earnings will keep pace with inflation and he uses the nominal rate of return (8%) to determine the amount he needs to have at age 65.
To account for inflation, George should have used the real rate of return (8% - 5% = 3%) instead of the nominal rate of return. This is because the real rate of return adjusts for inflation and reflects the purchasing power of his savings. By using the nominal rate, George has overestimated the amount he needs to have at age 65.
To correct him, George should use the real rate of return to determine the future accumulation required.
3. How much should George save in each of the next 20 years (until age 65) if he wants to maintain a constant level of consumption over the remaining 40 years of his life (from age 45 to age 85)? Ignore income taxes.
To calculate the amount George should save in each of the next 20 years, we need to determine the future value of his retirement fund at age 65, taking into account the real rate of return.
George wants to maintain a constant level of consumption over the remaining 40 years of his life, so he needs to accumulate enough savings to replace 80% of his current salary ($40,000). Since he will receive $12,000 per year from Social Security, he needs to provide the other $28,000 per year from his own retirement fund.
Using the formula for the future value of an ordinary annuity, we can calculate the annual savings required:
Future value of annuity = Annual savings × [((1 + Real rate of return)^(Number of years) - 1) / Real rate of return]
Plugging in the values:
Annual savings × [((1 + 0.03)^20 - 1) / 0.03] = $28,000
Solving for the annual savings:
Annual savings = $28,000 / [((1 + 0.03)^20 - 1) / 0.03]
Using a financial calculator or spreadsheet, we can calculate this value:
Annual savings ≈ $28,000 / 12.462
≈ $2,247.74
Therefore, George should save approximately $2,247.74 each year for the next 20 years in order to maintain a constant level of consumption over the remaining 40 years of his life.