Resource: Ch. 5 of Financial Management in Health Care Organizations.
„h Calculate the future value of the following:
o $5,000 compounded annually at 6% for 5 years
o $5,000 compounded semiannually at 6% for 5 years
o $5,000 compounded quarterly at 6% for 5 years
o $5,000 compounded annually at 6% for 6 years
„h Answer the following: What conclusions can be drawn about the frequency of compounding interest? What conclusions can be drawn about the length of time an amount is compounding?
„h Calculate the present value of the following:
o $7,000 in 5 years at an annual discount rate of 6%
o $7,000 in 5 years at a semiannual discount rate of 6%
o $7,000 in 5 years at a quarterly discount rate of 6%
o $7,000 in 6 years at an annual discount rate of 6%
„h Answer the following: What conclusions can be drawn about the frequency of the discounting interval? What conclusions can be drawn about the length of time until the receipt of that value?
„h Answer the following: Assume you have a choice between two annuity contracts. Contract A pays $5,000 per year for 5 years starting one year from today. Contract B pays $5,000 per year for 5 years starting today. The discount rate for each is 6%. Which annuity contract would you choose for your retirement? Why?
the formula is used was $5000(1+r)^n
To calculate the future value of an amount compounded at a given rate over a certain period of time, you can use the formula:
FV = PV * (1 + r/n)^(n*t)
Where:
- FV is the future value
- PV is the present value (initial investment)
- r is the annual interest rate
- n is the number of compounding periods per year
- t is the number of years
1. Calculate the future value of $5,000 compounded annually at 6% for 5 years:
FV = 5000 * (1 + 0.06/1)^(1*5)
2. Calculate the future value of $5,000 compounded semiannually at 6% for 5 years:
FV = 5000 * (1 + 0.06/2)^(2*5)
3. Calculate the future value of $5,000 compounded quarterly at 6% for 5 years:
FV = 5000 * (1 + 0.06/4)^(4*5)
4. Calculate the future value of $5,000 compounded annually at 6% for 6 years:
FV = 5000 * (1 + 0.06/1)^(1*6)
To answer the question about the conclusions drawn from the frequency of compounding interest and the length of time an amount is compounding, you can observe the different future values for each scenario. Generally, increasing the frequency of compounding (from annually to semiannually to quarterly) will result in a higher future value. Increasing the length of time the amount is compounded also leads to a higher future value. This demonstrates the power of compounding and highlights the advantage of more frequent compounding and longer-term investments.
To calculate the present value of an amount discounted at a given rate over a certain period of time, you can use the formula:
PV = FV / (1 + r/n)^(n*t)
Where:
- PV is the present value
- FV is the future value
- r is the annual discount rate (interest rate)
- n is the number of discounting periods per year
- t is the number of years
1. Calculate the present value of $7,000 in 5 years at an annual discount rate of 6%:
PV = 7000 / (1 + 0.06/1)^(1*5)
2. Calculate the present value of $7,000 in 5 years at a semiannual discount rate of 6%:
PV = 7000 / (1 + 0.06/2)^(2*5)
3. Calculate the present value of $7,000 in 5 years at a quarterly discount rate of 6%:
PV = 7000 / (1 + 0.06/4)^(4*5)
4. Calculate the present value of $7,000 in 6 years at an annual discount rate of 6%:
PV = 7000 / (1 + 0.06/1)^(1*6)
To answer the question about the conclusions drawn from the frequency of the discounting interval and the length of time until the receipt of that value, you can observe the different present values for each scenario. Generally, decreasing the frequency of discounting (from annually to semiannually to quarterly) will result in a lower present value. Increasing the length of time until the receipt of the value also leads to a lower present value. This reflects the time value of money concept, where the longer it takes to receive a future amount, the less valuable it is in present terms.
To compare annuity contracts and choose the better one for retirement, we can compare the present values of the cash flows from each contract.
1. Contract A: $5,000 per year for 5 years starting one year from today.
2. Contract B: $5,000 per year for 5 years starting today.
Calculate the present value of both contracts using the appropriate discount rate of 6% for retirement planning:
PV_A = 5000 / (1+0.06)^1 + 5000 / (1+0.06)^2 + 5000 / (1+0.06)^3 + 5000 / (1+0.06)^4 + 5000 / (1+0.06)^5
PV_B = 5000 / (1+0.06)^0 + 5000 / (1+0.06)^1 + 5000 / (1+0.06)^2 + 5000 / (1+0.06)^3 + 5000 / (1+0.06)^4
Compare the present values of both contracts. The contract with a higher present value would be the better choice for retirement as it would provide a higher amount of money in present terms.
In conclusion, the frequency of compounding interest and the length of time an amount compounds have significant impacts on future value and present value calculations. The more frequent the compounding or discounting intervals, the higher the resulting value. Similarly, the longer the time the amount is compounded or discounted, the greater the resulting value. These concepts are essential in financial management to analyze investment options and make informed decisions.