ICMBA Exam Study Guide
The Time Value of Money (TVM) is a foundational concept in finance and is heavily tested in the ICMBA exam. The problems from this section of the textbook cover all the essential concepts, from single payments to annuities and mixed streams. To prepare effectively, focus on the following key areas:
- Understanding the Core Concepts: Ensure you are comfortable with the difference between Future Value (FV) and Present Value (PV), and when to apply each. Understand the distinction between an ordinary annuity and an annuity due.
- Formula Mastery: While financial calculators are useful, knowing the underlying formulas will help you solve more complex problems, especially those involving mixed streams or finding an unknown variable like the number of periods or interest rate.
- Cash Flow Timelines: A crucial first step for any TVM problem is drawing a clear timeline. This helps you visualize when each cash flow occurs and whether you should compound or discount.
Key Formulas:
Future Value (Single Amount): \( FV_n = PV \times (1+r)^n \)
Present Value (Single Amount): \( PV = FV_n \times \frac{1}{(1+r)^n} \)
Future Value (Ordinary Annuity): \( FV_n = PMT \times \left[ \frac{(1+r)^n - 1}{r} \right] \)
Present Value (Ordinary Annuity): \( PV = PMT \times \left[ \frac{1 - \frac{1}{(1+r)^n}}{r} \right] \)
Problems and Solutions (P5-1 to P5-20)
Problem P5-1: Using a time line
Problem Statement: The financial manager at Starbuck Industries is considering an investment that requires an initial outlay of $25,000 and is expected to result in cash inflows of $3,000 at the end of year 1, $6,000 at the end of years 2 and 3, $10,000 at the end of year 4, $8,000 at the end of year 5, and $7,000 at the end of year 6.
Solution:
Part a: Draw and label a time line.
A time line helps visualize the cash flows. Time 0 represents today, with cash outflows shown as negative values.
\[ \begin{array}{cccccccc} \text{Time (years)}: & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \text{Cash Flow}: & -\$25,000 & \$3,000 & \$6,000 & \$6,000 & \$10,000 & \$8,000 & \$7,000 \end{array} \]
Part b: Demonstrate compounding to find future value.
To find the future value at the end of year 6, each cash flow is compounded to time 6. The initial outlay of $25,000 would be compounded for 6 years, the $3,000 inflow for 5 years, and so on. The total future value would be the sum of all these compounded values.
Part c: Demonstrate discounting to find present value.
To find the present value at time 0, each future cash flow is discounted back to time 0. The $3,000 inflow is discounted for 1 year, the $6,000 for 2 years, and so on. The total present value would be the sum of these discounted values, which is then compared to the initial outlay.
Part d: Which approach do financial managers rely on most often?
Financial managers most often rely on the **present value** approach. This is because the present value of all future cash flows from an investment is directly comparable to its initial cost, which is a key part of decision-making tools like Net Present Value (NPV).
Tricky Area:
Understanding the distinction between cash inflows and outflows is critical. Cash outflows (investments) are typically shown as negative values on a timeline, while cash inflows (returns) are positive. This is essential for correctly calculating net present value.
Problem P5-2: Future value calculation
Problem Statement: Without referring to the preprogrammed function on your financial calculator, use the basic formula for future value along with the given interest rate, r, and the number of periods, n, to calculate the future value of $1 in each of the cases shown in the following table.
| Case | Interest rate, r | Number of periods, n |
|---|---|---|
| A | 12% | 2 |
| B | 6% | 3 |
| C | 9% | 2 |
| D | 3% | 4 |
Solution:
Theoretical Foundation: The future value of a single amount is calculated using the formula: \( FV_n = PV \times (1+r)^n \). Since the present value (PV) is $1 in all cases, the formula simplifies to \( FV_n = (1+r)^n \).
- **Case A:** \( FV_2 = 1 \times (1 + 0.12)^2 = 1.2544 \)
- **Case B:** \( FV_3 = 1 \times (1 + 0.06)^3 = 1.1910 \)
- **Case C:** \( FV_2 = 1 \times (1 + 0.09)^2 = 1.1881 \)
- **Case D:** \( FV_4 = 1 \times (1 + 0.03)^4 = 1.1255 \)
Tricky Area:
This problem is straightforward, but it's important to be careful with the order of operations and rounding, as small errors can compound. Ensure you convert the percentage interest rate to a decimal before using it in the formula.
Problem P5-3: Future value
Problem Statement: You have $100 to invest. If you can earn 12% interest, about how long does it take for your $100 investment to grow to $200? Suppose that the interest rate is just half that, at 6%. At half the interest rate, does it take twice as long to double your money? Why or why not? How long does it take?
Solution:
Theoretical Foundation: This problem demonstrates the concept of the Rule of 72, a quick approximation for how long it takes for an investment to double. The formula is \( \text{Years to double} \approx \frac{72}{\text{Interest Rate}} \). The problem also requires you to solve for 'n', the number of periods, using the future value formula and logarithms.
12% interest rate:
Using the Rule of 72: \( \frac{72}{12} = 6 \) years. The exact calculation is:
\( \$200 = \$100 \times (1+0.12)^n \)
\( 2 = (1.12)^n \)
\( \ln(2) = n \times \ln(1.12) \)
\( n = \frac{\ln(2)}{\ln(1.12)} \approx 6.12 \text{ years} \)
6% interest rate:
Using the Rule of 72: \( \frac{72}{6} = 12 \) years. The exact calculation is:
\( 2 = (1.06)^n \)
\( \ln(2) = n \times \ln(1.06) \)
\( n = \frac{\ln(2)}{\ln(1.06)} \approx 11.90 \text{ years} \)
At half the interest rate (6%), it does not take exactly twice as long to double your money (11.90 years is close to, but not exactly, twice 6.12 years). This is because of the effect of **compounding**. The interest earned in each period also earns interest in subsequent periods, which is a non-linear process.
Tricky Area:
The "why or why not" part of the question is the key. While the Rule of 72 provides a helpful approximation, the exact answer reveals that the relationship is not perfectly linear. The magic of compounding means the doubling time is not a direct inverse of the interest rate.
Problem P5-4: Future values
Problem Statement: For each of the cases shown in the following table, calculate the future value of the single cash flow deposited today at the end of the deposit period if the interest is compounded annually at the rate specified.
| Case | Single cash flow | Interest rate | Deposit period (years) |
|---|---|---|---|
| A | $200 | 5% | 20 |
| B | $4,500 | 8% | 7 |
| C | $10,000 | 9% | 10 |
| D | $25,000 | 10% | 12 |
| E | $37,000 | 11% | 5 |
| F | $40,000 | 12% | 9 |
Solution:
Theoretical Foundation: This problem is a direct application of the future value of a single amount formula: \( FV_n = PV \times (1+r)^n \).
- **Case A:** \( FV = \$200 \times (1 + 0.05)^{20} = \$530.66 \)
- **Case B:** \( FV = \$4,500 \times (1 + 0.08)^7 = \$7,712.21 \)
- **Case C:** \( FV = \$10,000 \times (1 + 0.09)^{10} = \$23,673.64 \)
- **Case D:** \( FV = \$25,000 \times (1 + 0.10)^{12} = \$78,460.71 \)
- **Case E:** \( FV = \$37,000 \times (1 + 0.11)^5 = \$62,284.14 \)
- **Case F:** \( FV = \$40,000 \times (1 + 0.12)^9 = \$110,920.19 \)
Tricky Area:
This is a fundamental problem. The main challenge is to use a financial calculator or spreadsheet correctly. If you're doing this manually, be careful with the exponents and decimal places to avoid calculation errors.
Problem P5-5: Time value
Problem Statement: You have $1,500 to invest today at 7% interest compounded annually.
- Find how much you will have accumulated in the account at the end of (1) 3 years, (2) 6 years, and (3) 9 years.
- Use your findings in part a to calculate the amount of interest earned in (1) the first 3 years (years 1 to 3), (2) the second 3 years (years 4 to 6), and (3) the third 3 years (years 7 to 9).
- Compare and contrast your findings in part b. Explain why the amount of interest earned increases in each succeeding 3-year period.
Solution:
Theoretical Foundation: This problem illustrates the power of compounding. The amount of interest earned accelerates over time because interest is earned not only on the principal but also on the previously accumulated interest. This is known as "interest on interest."
Part a: Accumulated amount
Using \( FV_n = PV \times (1+r)^n \):
- (1) 3 years: \( FV_3 = \$1,500 \times (1 + 0.07)^3 = \$1,837.56 \)
- (2) 6 years: \( FV_6 = \$1,500 \times (1 + 0.07)^6 = \$2,251.10 \)
- (3) 9 years: \( FV_9 = \$1,500 \times (1 + 0.07)^9 = \$2,757.90 \)
Part b: Interest earned
- (1) First 3 years: \( \text{Interest} = \$1,837.56 - \$1,500 = \$337.56 \)
- (2) Second 3 years: \( \text{Interest} = \$2,251.10 - \$1,837.56 = \$413.54 \)
- (3) Third 3 years: \( \text{Interest} = \$2,757.90 - \$2,251.10 = \$506.80 \)
Part c: Comparison and explanation
The amount of interest earned increases in each succeeding 3-year period ($337.56 < $413.54 < $506.80). This is a direct consequence of **compounding**. In the first period, interest is earned only on the initial principal. In the second period, interest is earned on both the principal and the interest accumulated from the first period. This "interest on interest" effect accelerates the growth of the investment over time, leading to larger absolute interest gains in later periods.
Tricky Area:
The explanation in part c is the most important part of this problem. Don't just state that the interest increases; explicitly state that this is due to compounding and the fact that interest is earned on both the original principal and accumulated interest.
Problem P5-6: Time value
Problem Statement: You wish to purchase a new car exactly 5 years from today. The car costs $14,000 today, and your research indicates that its price will increase by 2% to 4% per year over the next 5 years.
- Estimate the price of the car at the end of 5 years if inflation is (1) 2% per year and (2) 4% per year.
- How much more expensive will the car be if the rate of inflation is 4% rather than 2%?
- Estimate the price of the car if inflation is 2% for the next 2 years and 4% for 3 years after that.
Solution:
Theoretical Foundation: This problem applies the future value concept to inflation. The future cost of an item is found by compounding its current price by the inflation rate over the given period. The formula used is the same as the future value of a single amount.
Part a: Estimated price
Using \( FV_n = PV \times (1+r)^n \):
- (1) 2% inflation: \( FV_5 = \$14,000 \times (1 + 0.02)^5 = \$15,457.13 \)
- (2) 4% inflation: \( FV_5 = \$14,000 \times (1 + 0.04)^5 = \$17,033.26 \)
Part b: Price difference
The difference in price is the result from part a(2) minus the result from part a(1):
The car will be $1,576.13 more expensive.
Part c: Mixed inflation rates
We compound the initial price for each period's inflation rate sequentially:
\( \text{Price} = \$14,000 \times 1.0404 \times 1.124864 \approx \$16,400.99 \)
Tricky Area:
For part c, you cannot simply average the inflation rates. You must compound the value year by year using the appropriate rate for each period. The price at the end of year 2 becomes the starting price for the inflation calculation in year 3.
Problem P5-7: Time value
Problem Statement: You can deposit $10,000 into an account paying 9% annual interest either today or exactly 10 years from today. How much better off will you be at the end of 40 years if you decide to make the initial deposit today rather than 10 years from today?
Solution:
Theoretical Foundation: This problem highlights the immense power of a long investment horizon and compounding. By starting earlier, you give your money more time to grow, and the "interest on interest" effect becomes significant over a longer period.
Case 1: Deposit today
The initial deposit of $10,000 is compounded for 40 years:
Case 2: Deposit 10 years from today
The deposit of $10,000 is made at the end of year 10, so it only has 30 years to compound (40 total years - 10-year delay):
Difference:
You will be $181,417.42 better off by depositing the money today.
Tricky Area:
The key insight is to correctly identify the number of compounding periods for the second case. It's not 40 years, but rather 30 years (from year 10 to year 40). Miscalculating the number of periods is a very common error in these types of problems.
Problem P5-8: Time value
Problem Statement: Peter wants to buy a new sports car for $70,000. He has $3,000 to invest today. He is offered the following investment opportunities: (1) River Bank's savings account with an interest rate of 10.8% compounded monthly; (2) First State Bank's savings account with an interest rate of 11.5% compounded annually; and (3) Union Bank's saving account with an interest rate of 9.3% compounded weekly. How long will it take for Peter to accumulate enough money to buy the car in each of the above three cases?
Solution:
Theoretical Foundation: This problem requires you to solve for 'n', the number of periods, in the future value of a single amount formula. It also introduces the concept of different compounding frequencies. When compounding is not annual, you must adjust the interest rate and the number of periods accordingly.
Case 1: River Bank (Monthly)
Here, the monthly interest rate is \( \frac{10.8\%}{12} = 0.9\% = 0.009 \). Let \( n \) be the number of months.
\( \$70,000 = \$3,000 \times (1 + 0.009)^n \)
\( 23.3333 = (1.009)^n \)
\( \ln(23.3333) = n \times \ln(1.009) \)
\( n = \frac{\ln(23.3333)}{\ln(1.009)} \approx 357.75 \text{ months} \)
\( \text{Years} = \frac{357.75}{12} \approx 29.81 \text{ years} \)
Case 2: First State Bank (Annually)
The annual interest rate is 11.5%. Let \( n \) be the number of years.
\( 23.3333 = (1.115)^n \)
\( \ln(23.3333) = n \times \ln(1.115) \)
\( n = \frac{\ln(23.3333)}{\ln(1.115)} \approx 29.56 \text{ years} \)
Case 3: Union Bank (Weekly)
The weekly interest rate is \( \frac{9.3\%}{52} \approx 0.1788\% = 0.001788 \). Let \( n \) be the number of weeks.
\( 23.3333 = (1.001788)^n \)
\( \ln(23.3333) = n \times \ln(1.001788) \)
\( n = \frac{\ln(23.3333)}{\ln(1.001788)} \approx 1,848.43 \text{ weeks} \)
\( \text{Years} = \frac{1,848.43}{52} \approx 35.55 \text{ years} \)
Tricky Area:
The main challenge is correctly adjusting the interest rate and the number of periods for each compounding frequency. Remember to convert the annual interest rate to the periodic rate (e.g., divide by 12 for monthly) and that the resulting 'n' will be in that same periodic unit (e.g., months). You must convert 'n' back to years for a final comparison.
Problem P5-9: Single-payment loan repayment
Problem Statement: A person borrows $200 to be repaid in 8 years with 14% annually compounded interest. The loan may be repaid at the end of any earlier year with no prepayment penalty.
- What amount will be due if the loan is repaid at the end of year 1?
- What is the repayment at the end of year 4?
- What amount is due at the end of the eighth year?
Solution:
Theoretical Foundation: This is a simple future value of a single amount problem. The loan principal ($200) is the present value, and the final repayment amount is the future value. The number of compounding periods is simply the number of years the loan is outstanding.
Using the formula \( FV_n = PV \times (1+r)^n \):
- **Part a (Year 1):** \( FV_1 = \$200 \times (1 + 0.14)^1 = \$228.00 \)
- **Part b (Year 4):** \( FV_4 = \$200 \times (1 + 0.14)^4 = \$337.89 \)
- **Part c (Year 8):** \( FV_8 = \$200 \times (1 + 0.14)^8 = \$569.17 \)
Tricky Area:
This problem is a good test of whether you understand that the "future value" in a loan context is the repayment amount. The only variable that changes is the number of years, which serves as the exponent 'n' in the future value formula.
Problem P5-10: Present value calculation
Problem Statement: Without referring to the preprogrammed function on your financial calculator, use the basic formula for present value, along with the given opportunity cost, r, and the number of periods, n, to calculate the present value of $1 in each of the cases shown in the following table.
| Case | Opportunity cost, r | Number of periods, n |
|---|---|---|
| A | 2% | 4 |
| B | 10% | 2 |
| C | 5% | 3 |
| D | 13% | 2 |
Solution:
Theoretical Foundation: The present value of a single amount is calculated using the formula: \( PV = FV_n \times \frac{1}{(1+r)^n} \). Since the future value (FV) is $1 in all cases, the formula simplifies to \( PV = \frac{1}{(1+r)^n} \).
- **Case A:** \( PV = \frac{1}{(1 + 0.02)^4} = 0.9238 \)
- **Case B:** \( PV = \frac{1}{(1 + 0.10)^2} = 0.8264 \)
- **Case C:** \( PV = \frac{1}{(1 + 0.05)^3} = 0.8638 \)
- **Case D:** \( PV = \frac{1}{(1 + 0.13)^2} = 0.7831 \)
Tricky Area:
Similar to the future value problems, this is a test of your basic formula application. The key is to correctly use the reciprocal in the present value formula. It's often helpful to think of the discount factor as \( \frac{1}{(1+r)^n} \).
Problem P5-11: Present values
Problem Statement: For each of the cases shown in the following table, calculate the present value of the cash flow, discounting at the rate given and assuming that the cash flow is received at the end of the period noted.
| Case | Single cash flow | Discount rate | End of period (years) |
|---|---|---|---|
| A | $7,000 | 12% | 4 |
| B | $28,000 | 8% | 20 |
| C | $10,000 | 14% | 12 |
| D | $150,000 | 11% | 6 |
| E | $45,000 | 20% | 8 |
Solution:
Theoretical Foundation: This problem is a direct application of the present value of a single amount formula: \( PV = FV_n \times \frac{1}{(1+r)^n} \).
- **Case A:** \( PV = \$7,000 \times \frac{1}{(1 + 0.12)^4} = \$4,448.91 \)
- **Case B:** \( PV = \$28,000 \times \frac{1}{(1 + 0.08)^{20}} = \$6,006.12 \)
- **Case C:** \( PV = \$10,000 \times \frac{1}{(1 + 0.14)^{12}} = \$2,075.59 \)
- **Case D:** \( PV = \$150,000 \times \frac{1}{(1 + 0.11)^6} = \$80,317.06 \)
- **Case E:** \( PV = \$45,000 \times \frac{1}{(1 + 0.20)^8} = \$10,486.68 \)
Tricky Area:
Ensure you are using the correct discount factor for each case. In the real world, the discount rate (or opportunity cost) reflects the risk of the investment. A higher risk would be associated with a higher discount rate, leading to a lower present value.
Problem P5-12: Present value concept
Problem Statement: Answer each of the following questions.
- What single investment made today, earning 12% annual interest, will be worth $6,000 at the end of 6 years?
- What is the present value of $6,000 to be received at the end of 6 years if the discount rate is 12%?
- What is the most you would pay today for a promise to repay you $6,000 at the end of 6 years if your opportunity cost is 12%?
- Compare, contrast, and discuss your findings in parts a through c.
Solution:
Theoretical Foundation: This problem is designed to highlight the interconnectedness of TVM concepts. Parts a, b, and c are essentially the same question phrased differently, all requiring the calculation of a present value.
Part a, b, and c: Calculation
All three parts ask for the present value of a single amount of $6,000, discounted over 6 years at a 12% rate.
\( PV = \$6,000 \times \frac{1}{(1 + 0.12)^6} \)
\( PV = \$6,000 \times 0.50663 \approx \$3,039.78 \)
The answer to parts a, b, and c is **$3,039.78**.
Part d: Comparison and discussion
The findings from all three parts are identical because they are different ways of asking the same question. The **present value** of a future sum is the amount you would need to invest today to grow into that future sum (part a). It is also the current worth of a future payment, given a specific discount rate (part b). Finally, it represents the maximum price you would be willing to pay today for that future payment, because paying any more would mean your return is less than your opportunity cost (part c). This demonstrates that the present value is a unified concept in finance.
Tricky Area:
The trick is recognizing that these are not three different problems, but rather three different phrasings of the same core present value concept. The "opportunity cost" in part c is the same as the "interest rate" in parts a and b for the purpose of the calculation.
Problem P5-13: Time value
Problem Statement: Jim Nance has been offered an investment that will pay him $500 three years from today.
- If his opportunity cost is 7% compounded annually, what value should he place on this opportunity today?
- What is the most he should pay to purchase this payment today?
- If Jim can purchase this investment for less than the amount calculated in part a, what does that imply about the rate of return that he will earn on the investment?
Solution:
Theoretical Foundation: This problem reinforces the concept of present value as the intrinsic worth of a future cash flow. It also introduces the idea of a project's return relative to the opportunity cost, which is a key component of investment decisions.
Part a: Value today
This is a present value calculation with \( FV_3 = \$500 \), \( r = 7\% \), and \( n = 3 \).
\( PV = \$500 \times \frac{1}{(1 + 0.07)^3} \)
\( PV = \$500 \times 0.8163 \approx \$408.15 \)
Part b: Maximum price
The most Jim should pay is the present value calculated in part a, which is **$408.15**. Paying more than this amount would mean his return is less than his opportunity cost of 7%.
Part c: Implied rate of return
If Jim can purchase the investment for less than $408.15, it means his rate of return will be **greater than his 7% opportunity cost**. This is a financially sound decision, as he is earning a return higher than what he could get on an alternative investment of similar risk.
Tricky Area:
Part c is the critical thinking component. It connects the mathematical calculation of present value to the real-world decision of accepting or rejecting an investment. The present value is the maximum price you should ever pay, as it represents the point of indifference where your return equals your opportunity cost.
Problem P5-14: Time value
Problem Statement: You want to save money to pay for a down payment on an apartment in 5 years' time. One year from now, you will invest your $30,000 year-end bonus for the down payment. If you can invest at 15% per year, how much interest will you receive on your cash in 5 years? If you need $210,000 for the down payment, and you would like to top-up the remaining amount by investing a lump sum today, what is the amount you should invest?
Solution:
Theoretical Foundation: This problem combines future value and present value calculations. It requires you to first find the future value of a delayed cash flow, then determine the funding gap, and finally calculate the present value of that gap.
Part 1: Future value of the bonus
The $30,000 bonus is received at the end of year 1 and is compounded for 4 years to reach the end of year 5.
\( \text{Interest} = \$52,470 - \$30,000 = \$22,470 \)
You will receive approximately $22,470 in interest on your bonus.
Part 2: Lump sum needed today
First, find the remaining amount needed for the down payment:
Now, calculate the present value of this amount by discounting it back 5 years to today:
\( PV = \$157,530 \times \frac{1}{(1 + 0.15)^5} \)
\( PV = \$157,530 \times 0.49718 \approx \$78,359.88 \)
Tricky Area:
The main trick is in correctly identifying the time period for each cash flow. The bonus is received at the end of year 1, so it only has 4 years to grow to year 5. For the lump sum needed today, you must discount the remaining amount for the full 5 years. It's a common mistake to use the same 'n' for both calculations without considering the timing of the cash flows.
Problem P5-15: Time value and discount rates
Problem Statement: You just won a lottery that promises to pay you $1,000,000 exactly 10 years from today. Because the $1,000,000 payment is guaranteed by the state in which you live, opportunities exist to sell the claim today for an immediate single cash payment.
- What is the least you will sell your claim for if you can earn the following rates of return on similar-risk investments during the 10-year period? (1) 6%, (2) 9%, (3) 12%.
- Rework part a under the assumption that the $1,000,000 payment will be received in 15 rather than 10 years.
- On the basis of your findings in parts a and b, discuss the effect of both the size of the rate of return and the time until receipt of payment on the present value of a future sum.
Solution:
Theoretical Foundation: This problem demonstrates how the present value of a future cash flow is inversely related to both the discount rate and the time to maturity. A higher discount rate or a longer time to maturity will result in a lower present value.
Part a: Present value in 10 years
Using \( PV = FV \times \frac{1}{(1+r)^n} \) with \( FV = \$1,000,000 \) and \( n = 10 \):
- (1) 6%: \( PV = \$1,000,000 \times \frac{1}{(1.06)^{10}} = \$558,394.78 \)
- (2) 9%: \( PV = \$1,000,000 \times \frac{1}{(1.09)^{10}} = \$422,410.74 \)
- (3) 12%: \( PV = \$1,000,000 \times \frac{1}{(1.12)^{10}} = \$321,973.24 \)
Part b: Present value in 15 years
Using \( PV = FV \times \frac{1}{(1+r)^n} \) with \( FV = \$1,000,000 \) and \( n = 15 \):
- (1) 6%: \( PV = \$1,000,000 \times \frac{1}{(1.06)^{15}} = \$417,265.06 \)
- (2) 9%: \( PV = \$1,000,000 \times \frac{1}{(1.09)^{15}} = \$274,538.04 \)
- (3) 12%: \( PV = \$1,000,000 \times \frac{1}{(1.12)^{15}} = \$182,696.48 \)
Part c: Discussion
The results show a clear relationship. As the rate of return (discount rate) increases, the present value of the future sum decreases. This is because a higher discount rate implies a higher opportunity cost, making a future payment less valuable today. Similarly, as the time until the payment is received increases, the present value also decreases. This is because the money has more time to compound, and the "time value" effect is more pronounced. Both of these effects demonstrate the core principle of TVM: a dollar today is worth more than a dollar tomorrow.
Tricky Area:
The discussion part of the question is just as important as the calculations. You must be able to articulate the inverse relationship between present value and both the discount rate and the time period. Simply listing the numbers is not a complete answer.
Problem P5-16: Time value comparisons of single amounts
Problem Statement: In exchange for a $20,000 payment today, a well-known company will allow you to choose one of the alternatives shown in the following table. Your opportunity cost is 11%.
| Alternative | Single amount |
|---|---|
| A | $28,500 at end of 3 years |
| B | $54,000 at end of 9 years |
| C | $160,000 at end of 20 years |
- Find the value today of each alternative.
- Are all the alternatives acceptable? That is, are they worth $20,000 today?
- Which alternative, if any, will you take?
Solution:
Theoretical Foundation: This problem is an application of the **Net Present Value (NPV)** concept. An investment is acceptable if its present value is greater than or equal to its cost. You should choose the alternative with the highest positive NPV.
Part a: Present value of each alternative
Using \( PV = FV \times \frac{1}{(1+r)^n} \) with \( r = 11\% \):
- **Alternative A:** \( PV = \$28,500 \times \frac{1}{(1 + 0.11)^3} = \$20,830.40 \)
- **Alternative B:** \( PV = \$54,000 \times \frac{1}{(1 + 0.11)^9} = \$21,159.98 \)
- **Alternative C:** \( PV = \$160,000 \times \frac{1}{(1 + 0.11)^{20}} = \$20,627.51 \)
Part b: Acceptability
All three alternatives are acceptable because their present values ($20,830.40, $21,159.98, and $20,627.51) are all greater than the initial cost of $20,000. In other words, all alternatives have a positive NPV.
Part c: Decision
You should choose **Alternative B**. This alternative has the highest present value, which means it offers the highest return in today's dollars. This is the optimal choice as it maximizes the value of your investment.
Tricky Area:
The primary challenge is correctly identifying the best alternative. You should not compare the future values. Instead, you should bring all future cash flows back to a common point in time (the present) to make a direct comparison. The alternative with the highest present value is the best choice.
Problem P5-17: Cash flow investment decision
Problem Statement: Tom Alexander has an opportunity to purchase any of the investments shown in the following table. The purchase price, the amount of the single cash inflow, and its year of receipt are given for each investment. Which purchase recommendations would you make, assuming that Tom can earn 10% on his investments?
| Investment | Price | Single cash inflow | Year of receipt |
|---|---|---|---|
| A | $18,000 | $30,000 | 5 |
| B | $600 | $3,000 | 20 |
| C | $3,500 | $10,000 | 10 |
| D | $1,000 | $15,000 | 40 |
Solution:
Theoretical Foundation: This problem is another application of NPV. The decision rule is to accept any investment with a positive NPV (i.e., where the present value of the future cash flow is greater than the initial cost). You can accept multiple independent investments if they all meet this criterion.
Using \( PV = FV \times \frac{1}{(1+r)^n} \) with a discount rate of \( r = 10\% \):
- **Investment A:** \( PV = \$30,000 \times \frac{1}{(1.10)^5} = \$18,627.64 \). Since \( \$18,627.64 > \$18,000 \), **accept**.
- **Investment B:** \( PV = \$3,000 \times \frac{1}{(1.10)^{20}} = \$445.92 \). Since \( \$445.92 < \$600 \), **reject**.
- **Investment C:** \( PV = \$10,000 \times \frac{1}{(1.10)^{10}} = \$3,855.43 \). Since \( \$3,855.43 > \$3,500 \), **accept**.
- **Investment D:** \( PV = \$15,000 \times \frac{1}{(1.10)^{40}} = \$331.42 \). Since \( \$331.42 < \$1,000 \), **reject**.
Tom should purchase **Investments A and C**.
Tricky Area:
The key here is understanding the decision rule for independent projects. You don't choose just one best option. Instead, you evaluate each option on its own merits against the initial cost. Any investment with a positive NPV (or a present value greater than the cost) should be accepted.
Problem P5-18: Calculating deposit needed
Problem Statement: You put $10,000 in an account earning 5%. After 3 years, you make another deposit into the same account. Four years later (that is, 7 years after your original $10,000 deposit), the account balance is $20,000. What was the amount of the deposit at the end of year 3?
Solution:
Theoretical Foundation: This is a multi-step future value problem. You must first find the future value of the initial deposit at the time of the second deposit. Then, you can work backward from the final future value to find the value of the second deposit, and thus the amount of the deposit itself.
Let the unknown deposit at year 3 be \( X \). Let's first find the value of the initial $10,000 deposit at the end of year 7.
Now, let's find the value of the second deposit, \( X \), at the end of year 7. Since it was made at the end of year 3, it compounds for \( 7-3=4 \) years.
The total balance at the end of year 7 is the sum of these two future values. We know this total is $20,000.
\( X \times 1.2155 = \$20,000 - \$14,071.00 = \$5,929 \)
\( X = \frac{\$5,929}{1.2155} \approx \$4,877.83 \)
The amount of the deposit at the end of year 3 was **$4,877.83**.
Tricky Area:
The most common mistake is miscalculating the number of periods for the second deposit. A timeline is extremely helpful here. Remember that the second deposit is made at the end of year 3, and the total value is measured at the end of year 7. This means the second deposit only compounds for 4 years, not 7.
Problem P5-19: Future value of an annuity
Problem Statement: For each case in the accompanying table, answer the questions that follow.
| Case | Amount of annuity | Interest rate | Deposit period (years) |
|---|---|---|---|
| A | $2,500 | 8% | 10 |
| B | $500 | 12% | 6 |
| C | $30,000 | 20% | 5 |
| D | $11,500 | 9% | 8 |
| E | $6,000 | 14% | 30 |
- Calculate the future value of the annuity, assuming that it is (1) An ordinary annuity. (2) An annuity due.
- Compare your findings in parts a(1) and a(2). All else being identical, which type of annuity-ordinary or annuity due-is preferable? Explain why.
Solution:
Theoretical Foundation: This problem tests your understanding of the difference between an **ordinary annuity** and an **annuity due**. An annuity due receives one extra period of compounding compared to an ordinary annuity, as payments are made at the beginning of each period rather than at the end. The formula for an annuity due is \( FV_{n(\text{due})} = FV_{n(\text{ordinary})} \times (1+r) \).
Part 1: Future Value
Using the ordinary annuity formula \( FV_n = PMT \times \left[ \frac{(1+r)^n - 1}{r} \right] \) and the annuity due formula:
- **Case A:**
- Ordinary: \( FV_{10} = \$2,500 \times \left[ \frac{(1.08)^{10} - 1}{0.08} \right] = \$36,216.41 \)
- Annuity Due: \( FV_{10(\text{due})} = \$36,216.41 \times (1.08) = \$39,113.72 \)
- **Case B:**
- Ordinary: \( FV_{6} = \$500 \times \left[ \frac{(1.12)^6 - 1}{0.12} \right] = \$4,046.04 \)
- Annuity Due: \( FV_{6(\text{due})} = \$4,046.04 \times (1.12) = \$4,531.57 \)
- **Case C:**
- Ordinary: \( FV_{5} = \$30,000 \times \left[ \frac{(1.20)^5 - 1}{0.20} \right] = \$223,200.00 \)
- Annuity Due: \( FV_{5(\text{due})} = \$223,200.00 \times (1.20) = \$267,840.00 \)
- **Case D:**
- Ordinary: \( FV_{8} = \$11,500 \times \left[ \frac{(1.09)^8 - 1}{0.09} \right] = \$114,896.25 \)
- Annuity Due: \( FV_{8(\text{due})} = \$114,896.25 \times (1.09) = \$125,236.91 \)
- **Case E:**
- Ordinary: \( FV_{30} = \$6,000 \times \left[ \frac{(1.14)^{30} - 1}{0.14} \right] = \$2,140,721.13 \)
- Annuity Due: \( FV_{30(\text{due})} = \$2,140,721.13 \times (1.14) = \$2,440,422.09 \)
Part 2: Comparison
An **annuity due** is always preferable to an ordinary annuity, all else being equal. This is because every cash flow in an annuity due is received one period earlier, giving it an additional period of compounding interest. This results in a larger future value compared to an ordinary annuity with the same payment amount, interest rate, and number of periods.
Tricky Area:
Remembering to multiply the result of the ordinary annuity calculation by \( (1+r) \) to find the annuity due value is the key step. Forgetting this simple multiplier is a very common mistake.
Problem P5-20: Present value of an annuity
Problem Statement: Consider the following cases.
| Case | Amount of annuity | Interest rate | Period (years) |
|---|---|---|---|
| A | $12,000 | 7% | 3 |
| B | $55,000 | 12% | 15 |
| C | $700 | 20% | 9 |
| D | $140,000 | 8% | 7 |
| E | $22,500 | 10% | 5 |
- Calculate the present value of the annuity, assuming that it is (1) An ordinary annuity. (2) An annuity due.
- Compare your findings in parts a(1) and a(2). All else being identical, which type of annuity-ordinary or annuity due-is preferable? Explain why.
Solution:
Theoretical Foundation: This problem is the inverse of the previous one, focusing on the present value of annuities. The present value of an annuity due is always higher than an ordinary annuity because the cash flows are received earlier, and are therefore discounted for fewer periods. The relationship is \( PV_{n(\text{due})} = PV_{n(\text{ordinary})} \times (1+r) \).
Part 1: Present Value
Using the ordinary annuity formula \( PV = PMT \times \left[ \frac{1 - \frac{1}{(1+r)^n}}{r} \right] \) and the annuity due formula:
- **Case A:**
- Ordinary: \( PV_{3} = \$12,000 \times \left[ \frac{1 - \frac{1}{(1.07)^3}}{0.07} \right] = \$31,491.56 \)
- Annuity Due: \( PV_{3(\text{due})} = \$31,491.56 \times (1.07) = \$33,695.97 \)
- **Case B:**
- Ordinary: \( PV_{15} = \$55,000 \times \left[ \frac{1 - \frac{1}{(1.12)^{15}}}{0.12} \right] = \$374,597.40 \)
- Annuity Due: \( PV_{15(\text{due})} = \$374,597.40 \times (1.12) = \$419,549.09 \)
- **Case C:**
- Ordinary: \( PV_{9} = \$700 \times \left[ \frac{1 - \frac{1}{(1.20)^9}}{0.20} \right] = \$2,821.84 \)
- Annuity Due: \( PV_{9(\text{due})} = \$2,821.84 \times (1.20) = \$3,386.21 \)
- **Case D:**
- Ordinary: \( PV_{7} = \$140,000 \times \left[ \frac{1 - \frac{1}{(1.08)^7}}{0.08} \right] = \$728,922.75 \)
- Annuity Due: \( PV_{7(\text{due})} = \$728,922.75 \times (1.08) = \$787,236.57 \)
- **Case E:**
- Ordinary: \( PV_{5} = \$22,500 \times \left[ \frac{1 - \frac{1}{(1.10)^5}}{0.10} \right] = \$85,292.73 \)
- Annuity Due: \( PV_{5(\text{due})} = \$85,292.73 \times (1.10) = \$93,822.00 \)
Part 2: Comparison
An **annuity due** is always preferable to an ordinary annuity, all else being equal. This is because the payments for an annuity due occur at the beginning of each period, meaning they are received sooner. When calculating present value, this means each payment is discounted for one less period, resulting in a higher present value. From an investment standpoint, this is more valuable today.
Tricky Area:
Just like the future value problem, the core concept here is that a cash flow received earlier is more valuable. For present value, this translates to a higher present value because the cash flow is discounted over a shorter period. A common mistake is to think that the same rule applies to both future and present value without understanding the underlying time difference.