ICMBA Exam Study Guide
The problems in this section of the textbook build upon the foundational concepts of Time Value of Money (TVM) and introduce more complex scenarios, including perpetuities, mixed cash flow streams, and various compounding frequencies. As you prepare for your ICMBA exam, it's crucial to understand these advanced applications. Here are some key areas to focus on for effective study:
- Perpetuities and Annuities: Understand how to calculate the value of an infinite stream of payments (perpetuity) and how this concept can be applied to creating endowments.
- Mixed Streams of Cash Flows: These problems are a key test of your ability to apply TVM techniques to non-uniform cash flows. Practice breaking down these streams into individual present or future values and summing them up.
- Compounding Frequency: Pay close attention to how compounding frequency (annual, semiannual, quarterly, or continuous) impacts both the future value and the effective annual rate (EAR). Remember to adjust your interest rate and the number of periods accordingly.
- Solving for Unknowns: Many problems will require you to solve for a variable other than present or future value, such as an unknown payment, interest rate, or number of periods. Master the algebraic manipulation of the core TVM formulas.
Key Formulas:
Present Value (Perpetuity): \( PV = \frac{PMT}{r} \)
Future Value (Annuity Due): \( FV_{n(\text{due})} = FV_{n(\text{ordinary})} \times (1+r) \)
Present Value (Annuity Due): \( PV_{n(\text{due})} = PV_{n(\text{ordinary})} \times (1+r) \)
Future Value (Continuous Compounding): \( FV = PV \times e^{rt} \)
Problems and Solutions (P5-21 to P5-40)
Problem P5-21: Time value: Annuities
Problem Statement: Marian Kirk wishes to select the better of two 10-year annuities, C and D. Annuity C is an ordinary annuity of $2,500 per year for 10 years. Annuity D is an annuity due of $2,200 per year for 10 years.
- Find the future value of both annuities at the end of year 10 assuming that Marian can earn (1) 10% annual interest and (2) 20% annual interest.
- Use your findings in part a to indicate which annuity has the greater future value at the end of year 10 for both the (1) 10% and (2) 20% interest rates.
- Find the present value of both annuities, assuming that Marian can earn (1) 10% annual interest and (2) 20% annual interest.
- Use your findings in part c to indicate which annuity has the greater present value for both (1) 10% and (2) 20% interest rates.
- Briefly compare, contrast, and explain any differences between your findings using the 10% and 20% interest rates in parts b and d.
Solution:
Theoretical Foundation: This problem directly compares an ordinary annuity with an annuity due. The key takeaway is that an annuity due, with its payments at the beginning of each period, benefits from an extra period of compounding, making it more valuable than an equivalent ordinary annuity.
Part a: Future Value at Year 10
Using the ordinary annuity formula \( FV_n = PMT \times \left[ \frac{(1+r)^n - 1}{r} \right] \) and the annuity due multiplier \( (1+r) \):
- (1) 10% Interest:
- Annuity C (Ordinary): \( FV_{10} = \$2,500 \times \left[ \frac{(1.10)^{10} - 1}{0.10} \right] = \$39,843.56 \)
- Annuity D (Annuity Due): \( FV_{10} = \$2,200 \times \left[ \frac{(1.10)^{10} - 1}{0.10} \right] \times (1.10) = \$38,367.75 \)
- (2) 20% Interest:
- Annuity C (Ordinary): \( FV_{10} = \$2,500 \times \left[ \frac{(1.20)^{10} - 1}{0.20} \right] = \$64,896.86 \)
- Annuity D (Annuity Due): \( FV_{10} = \$2,200 \times \left[ \frac{(1.20)^{10} - 1}{0.20} \right] \times (1.20) = \$68,529.74 \)
Part b: Which annuity has the greater future value?
- (1) 10% Interest: Annuity C ($39,843.56) is greater than Annuity D ($38,367.75).
- (2) 20% Interest: Annuity D ($68,529.74) is greater than Annuity C ($64,896.86).
Part c: Present Value
Using the ordinary annuity formula \( PV = PMT \times \left[ \frac{1 - \frac{1}{(1+r)^n}}{r} \right] \) and the annuity due multiplier \( (1+r) \):
- (1) 10% Interest:
- Annuity C (Ordinary): \( PV = \$2,500 \times \left[ \frac{1 - \frac{1}{(1.10)^{10}}}{0.10} \right] = \$15,361.34 \)
- Annuity D (Annuity Due): \( PV = \$2,200 \times \left[ \frac{1 - \frac{1}{(1.10)^{10}}}{0.10} \right] \times (1.10) = \$14,942.50 \)
- (2) 20% Interest:
- Annuity C (Ordinary): \( PV = \$2,500 \times \left[ \frac{1 - \frac{1}{(1.20)^{10}}}{0.20} \right] = \$10,482.02 \)
- Annuity D (Annuity Due): \( PV = \$2,200 \times \left[ \frac{1 - \frac{1}{(1.20)^{10}}}{0.20} \right] \times (1.20) = \$11,093.07 \)
Part d: Which annuity has the greater present value?
- (1) 10% Interest: Annuity C ($15,361.34) is greater than Annuity D ($14,942.50).
- (2) 20% Interest: Annuity D ($11,093.07) is greater than Annuity C ($10,482.02).
Part e: Comparison and explanation
The results show that the choice of the better annuity depends on the interest rate. At a lower interest rate (10%), the higher payment of Annuity C ($2,500) outweighs the benefit of earlier payments in Annuity D. However, at a higher interest rate (20%), the compounding effect becomes more pronounced, and the extra compounding period for Annuity D's payments makes it more valuable, despite its lower payment amount. The crossover point is the interest rate at which the two annuities have the same value.
Tricky Area:
The primary challenge is to not assume that one type of annuity is always better. The larger payment amount of Annuity C must be weighed against the earlier payments of Annuity D. The interest rate is the key factor that determines which of these two benefits is more valuable.
Problem P5-22: Retirement planning
Problem Statement: Hal Thomas, a 25-year-old college graduate, wishes to retire at age 65. To supplement other sources of retirement income, he can deposit $2,000 each year into a tax-deferred individual retirement arrangement (IRA). The IRA will earn a 10% return over the next 40 years.
- If Hal makes annual end-of-year $2,000 deposits into the IRA, how much will he have accumulated by the end of his sixty-fifth year?
- If Hal decides to wait until age 35 to begin making annual end-of-year $2,000 deposits into the IRA, how much will he have accumulated by the end of his sixty-fifth year?
- Using your findings in parts a and b, discuss the impact of delaying making deposits into the IRA for 10 years (age 25 to age 35) on the amount accumulated by the end of Hal's sixty-fifth year.
- Rework parts a, b, and c, assuming that Hal makes all deposits at the beginning, rather than the end, of each year. Discuss the effect of beginning-of-year deposits on the future value accumulated by the end of Hal's sixty-fifth year.
Solution:
Theoretical Foundation: This problem demonstrates the critical importance of starting to invest early. The power of compounding over a long time horizon means that a small delay in starting can lead to a massive difference in the final accumulated amount.
Part a: Deposits from age 25 to 65 (40 years)
This is an ordinary annuity calculation. \( PMT = \$2,000 \), \( r = 10\% \), \( n = 40 \).
Part b: Deposits from age 35 to 65 (30 years)
This is an ordinary annuity calculation with a shorter time horizon. \( PMT = \$2,000 \), \( r = 10\% \), \( n = 30 \).
Part c: Impact of delay
Delaying the deposits for just 10 years (from age 25 to 35) results in a massive loss of accumulated funds. The difference is \( \$885,185.11 - \$328,988.05 = \$556,197.06 \). This loss of over half a million dollars for only 10 fewer contributions highlights the extraordinary power of compounding over time. The interest on the initial deposits makes up a huge portion of the final value.
Part d: Beginning-of-year deposits (Annuity Due)
We use the annuity due formula by multiplying the ordinary annuity result by \( (1+r) \).
- **Age 25-65 (40 years):** \( FV_{40(\text{due})} = \$885,185.11 \times (1.10) = \$973,703.62 \)
- **Age 35-65 (30 years):** \( FV_{30(\text{due})} = \$328,988.05 \times (1.10) = \$361,886.86 \)
- **Discussion:** Making deposits at the beginning of the year is always preferable as it gives the money one extra period to compound. The effect is particularly significant over long time horizons, as shown by the increase of over $88,000 in the first case and over $32,000 in the second case.
Tricky Area:
The most important part of this problem is not just the numbers, but the financial wisdom they represent. The immense difference between starting at age 25 versus 35 is not intuitive until you see the numbers and understand how compounding works. Also, remember to correctly adjust the number of periods and the annuity type for each part of the problem.
Problem P5-23: Calculating the number of periods
Problem Statement: You want to borrow $600,000 to buy an apartment, and you can only afford $4,000 a month to repay the loan. Suppose the bank charges you a fixed interest rate of 4% with monthly compounding. How long will it take you to pay off the loan?
Solution:
Theoretical Foundation: This problem requires solving for the number of periods (\( n \)) in the present value of an annuity formula. It also involves a non-annual compounding frequency, which requires you to adjust the interest rate to a monthly rate.
Given: \( PV = \$600,000 \), \( PMT = \$4,000 \), nominal annual rate = 4%, compounding monthly.
The monthly interest rate is \( r = \frac{0.04}{12} \approx 0.003333 \).
Using the present value of an annuity formula and solving for \( n \):
\( \$600,000 = \$4,000 \times \left[ \frac{1 - \frac{1}{(1.003333)^n}}{0.003333} \right] \)
\( 150 = \frac{1 - (1.003333)^{-n}}{0.003333} \)
\( 0.5 = 1 - (1.003333)^{-n} \)
\( (1.003333)^{-n} = 0.5 \)
\( -n \times \ln(1.003333) = \ln(0.5) \)
\( n = \frac{-\ln(0.5)}{\ln(1.003333)} \approx 208.28 \text{ months} \)
It will take approximately \( \frac{208.28}{12} \approx 17.36 \) years to pay off the loan.
Tricky Area:
The main challenge is the algebraic manipulation to solve for \( n \). Remember to isolate the term with \( n \), take the natural logarithm of both sides, and then solve. Also, it's easy to forget to convert the interest rate to a monthly rate and the final answer from months to years.
Problem P5-24: Funding your retirement
Problem Statement: You plan to retire in exactly 20 years. Your goal is to create a fund that will allow you to receive $20,000 at the end of each year for the 30 years between retirement and death. You know that you will be able to earn 11% per year during the 30-year retirement period.
- How large a fund will you need when you retire in 20 years to provide the 30-year, $20,000 retirement annuity?
- How much will you need today as a single amount to provide the fund calculated in part a if you earn only 9% per year during the 20 years preceding retirement?
- What effect would an increase in the rate you can earn both during and prior to retirement have on the values found in parts a and b? Explain.
- Now assume that you will earn 10% from now through the end of your retirement. You want to make 20 end-of-year deposits into your retirement account that will fund the 30-year stream of $20,000 annual annuity payments. How large do your annual deposits have to be?
Solution:
Theoretical Foundation: This is a classic multi-stage TVM problem. It involves calculating the present value of a retirement annuity (at the time of retirement) and then finding the present or future value of a savings plan to reach that goal. The solution requires a clear understanding of cash flow timelines.
Part a: Fund needed at retirement (Year 20)
This is the present value of a 30-year ordinary annuity of $20,000 at a discount rate of 11%.
You will need a fund of $170,183.05 at the time of retirement.
Part b: Lump sum needed today (Year 0)
This is the present value of the amount found in part a, discounted over the 20-year pre-retirement period at a 9% rate.
You need to invest $30,344.59 today.
Part c: Effect of interest rate increase
An increase in the interest rate would **decrease** the values in both parts a and b. In part a, a higher rate means you need a smaller fund at retirement to generate the same $20,000 annual payments, as the fund itself will earn a higher return. In part b, a higher interest rate means your initial lump sum will grow faster, so you need to invest a smaller amount today to reach the required retirement fund.
Part d: Annual deposits needed (10% rate)
First, find the present value of the retirement annuity at the time of retirement (Year 20) using the new rate of 10%:
Now, this amount ($188,538.45) is the future value of your 20-year savings annuity. We can solve for the annual payment (PMT).
\( \$188,538.45 = PMT \times \left[ \frac{(1.10)^{20} - 1}{0.10} \right] \)
\( \$188,538.45 = PMT \times 57.275 \)
\( PMT = \frac{\$188,538.45}{57.275} = \$3,291.60 \)
Tricky Area:
The most important step is to treat the retirement period and the savings period as two separate, linked TVM problems. The final value of your savings plan must equal the present value of your retirement spending plan, with that present value being calculated at the exact moment of retirement.
Problem P5-25: Value of an annuity versus a single amount
Problem Statement: Assume that you just won the state lottery. Your prize can be taken either in the form of $40,000 at the end of each of the next 25 years (that is, $1,000,000 over 25 years) or as a single amount of $500,000 paid immediately.
- If you expect to be able to earn 5% annually on your investments over the next 25 years, ignoring taxes and other considerations, which alternative should you take? Why?
- Would your decision in part a change if you could earn 7% rather than 5% on your investments over the next 25 years? Why?
- On a strictly economic basis, at approximately what earnings rate would you be indifferent between the two plans?
Solution:
Theoretical Foundation: This problem is a classic example of comparing two different cash flow streams by bringing them to a common point in time. The present value calculation is the best method to make this comparison, as it determines the value of each option today.
Part a: Decision at 5% rate
Calculate the present value of the annuity using \( PMT = \$40,000 \), \( r = 5\% \), \( n = 25 \).
Since the present value of the annuity ($563,757.78) is greater than the lump sum offer ($500,000), you should take the **annuity**. It is the economically superior choice.
Part b: Decision at 7% rate
Calculate the new present value of the annuity using \( r = 7\% \).
Yes, the decision would change. At a 7% rate, the present value of the annuity ($466,143.20) is less than the lump sum offer ($500,000). This is because the higher interest rate makes the future payments less valuable in today's dollars. In this case, you should take the **lump sum**.
Part c: Indifference point
To find the indifference rate, set the present value of the annuity equal to the lump sum and solve for \( r \).
Using a financial calculator or spreadsheet, you can find that the rate is approximately **6.13%**. At this rate, the two options have the same present value, and you would be indifferent between them.
Tricky Area:
The key insight is to bring both cash flow streams to the same point in time (the present). You cannot simply compare the total nominal value of the annuity ($1M) to the lump sum ($500k). The final part of the problem also requires you to understand that as the discount rate increases, the value of the future annuity stream decreases, and at a certain point, the lump sum becomes more attractive.
Problem P5-26: Perpetuities
Problem Statement: Consider the data in the following table.
| Perpetuity | Annual amount | Discount rate |
|---|---|---|
| A | $20,000 | 8% |
| B | $100,000 | 10% |
| C | $3,000 | 6% |
| D | $60,000 | 5% |
Determine the present value of each perpetuity.
Solution:
Theoretical Foundation: This problem is a direct application of the present value of a perpetuity formula: \( PV = \frac{PMT}{r} \). A perpetuity is an annuity that is expected to continue forever.
- **Perpetuity A:** \( PV = \frac{\$20,000}{0.08} = \$250,000 \)
- **Perpetuity B:** \( PV = \frac{\$100,000}{0.10} = \$1,000,000 \)
- **Perpetuity C:** \( PV = \frac{\$3,000}{0.06} = \$50,000 \)
- **Perpetuity D:** \( PV = \frac{\$60,000}{0.05} = \$1,200,000 \)
Tricky Area:
This is a straightforward problem as long as you remember the correct formula. The primary trick is recognizing the term "perpetuity" and applying the correct formula, which is much simpler than the annuity formula. Also, ensure you convert the percentage discount rate to a decimal before calculating.
Problem P5-27: Creating an endowment
Problem Statement: Marla Lee's parents want to create an endowment to fund three needy students' tuition and books each year in perpetuity. The annual cost is $600 per student. The endowment will be funded by a single payment today, and the university expects to earn 6% per year on the funds.
- How large an initial single payment must Marla's parents make to the university to fund the endowment?
- What amount would be needed to fund the endowment if the university could earn 9% rather than 6% per year on the funds?
Solution:
Theoretical Foundation: This problem is a practical application of the perpetuity concept. An endowment fund's required size is the present value of all future payments it will make, which, for a perpetual stream, is simply the annual payment divided by the interest rate.
Part a: Initial payment at 6%
First, find the total annual payment: \( PMT = 3 \text{ students} \times \$600/\text{student} = \$1,800 \).
Using the perpetuity formula: \( PV = \frac{PMT}{r} \)
Part b: Initial payment at 9%
Using the perpetuity formula with the new interest rate:
Tricky Area:
The initial amount needed for a perpetuity is inversely related to the interest rate. A higher interest rate means the fund's principal will generate more income each year, so a smaller initial principal is needed to fund the same annual payment. It's also important to correctly calculate the total annual payment first before applying the perpetuity formula.
Problem P5-28: Value of a mixed stream
Problem Statement: For each of the mixed streams of cash flows shown in the following table, determine the future value at the end of the final year if deposits are made into an account paying annual interest of 12%, assuming that no withdrawals are made during the period and that the deposits are made
| Year | A | B | C |
|---|---|---|---|
| 1 | $900 | $30,000 | $1,200 |
| 2 | $1,000 | $25,000 | $1,200 |
| 3 | $1,200 | $20,000 | $1,000 |
| 4 | $10,000 | $1,900 | |
| 5 | $5,000 |
- At the end of each year.
- At the beginning of each year.
Solution:
Theoretical Foundation: This problem involves calculating the future value of a mixed stream of cash flows. The principle is to calculate the future value of each individual cash flow and then sum them up. For an ordinary annuity (end-of-year payments), you compound each payment from its receipt date to the final year. For an annuity due (beginning-of-year payments), each payment compounds for one additional year.
Part a: End-of-Year Deposits (Ordinary Annuity)
The total future value is the sum of the future values of each cash flow compounded to the final year (Year 3 for A, Year 5 for B, Year 4 for C) at a 12% rate.
- Stream A:
- \( FV_{3} = \$900(1.12)^2 + \$1,000(1.12)^1 + \$1,200(1.12)^0 = \$1,128.96 + \$1,120 + \$1,200 = \$3,448.96 \)
- Stream B:
- \( FV_{5} = \$30,000(1.12)^4 + \$25,000(1.12)^3 + \$20,000(1.12)^2 + \$10,000(1.12)^1 + \$5,000(1.12)^0 = \$47,205.58 + \$35,123.20 + \$25,088 + \$11,200 + \$5,000 = \$123,616.78 \)
- Stream C:
- \( FV_{4} = \$1,200(1.12)^3 + \$1,200(1.12)^2 + \$1,000(1.12)^1 + \$1,900(1.12)^0 = \$1,685.99 + \$1,505.28 + \$1,120 + \$1,900 = \$6,211.27 \)
Part b: Beginning-of-Year Deposits (Annuity Due)
The cash flows are received one period earlier, so each compounds for one extra period. We can simply multiply the result from Part a by \( (1+r) \).
- **Stream A:** \( FV_{3(\text{due})} = \$3,448.96 \times (1.12) = \$3,862.84 \)
- **Stream B:** \( FV_{5(\text{due})} = \$123,616.78 \times (1.12) = \$138,450.80 \)
- **Stream C:** \( FV_{4(\text{due})} = \$6,211.27 \times (1.12) = \$6,956.62 \)
Tricky Area:
For mixed streams, it's essential to compound each individual cash flow for the correct number of periods. Drawing a timeline can prevent mistakes. For the annuity due version, remember that the last payment still compounds for one period, unlike a present value calculation for an annuity due where the first payment is not discounted at all.
Problem P5-29: Value of a single amount versus a mixed stream
Problem Statement: Gina Vitale has just contracted to sell a small parcel of land. The buyer offers either a single payment of $24,000 at the closing or a mixed stream of payments at the beginning of each of the next 5 years. Gina wants to choose the option that provides the higher future value at the end of 5 years, assuming a 7% annual interest rate.
| Beginning of year | Cash flow |
|---|---|
| 1 | $2,000 |
| 2 | $4,000 |
| 3 | $6,000 |
| 4 | $8,000 |
| 5 | $10,000 |
Solution:
Theoretical Foundation: This problem compares a single lump sum to a mixed stream of payments. To make a fair comparison, both must be evaluated at the same point in time. Since the goal is to buy a house at the end of year 5, calculating the future value of each option is the appropriate approach.
Option 1: Single Amount Future Value
The single payment of $24,000 is received at time 0 and compounded for 5 years at 7%.
Option 2: Mixed Stream Future Value
The payments are at the beginning of each year, making this a mixed stream annuity due. Each cash flow must be compounded from its receipt date to the end of year 5.
- Year 1 payment ($2,000) compounds for 5 years: \( \$2,000(1.07)^5 = \$2,805.10 \)
- Year 2 payment ($4,000) compounds for 4 years: \( \$4,000(1.07)^4 = \$5,243.19 \)
- Year 3 payment ($6,000) compounds for 3 years: \( \$6,000(1.07)^3 = \$7,350.25 \)
- Year 4 payment ($8,000) compounds for 2 years: \( \$8,000(1.07)^2 = \$9,159.20 \)
- Year 5 payment ($10,000) compounds for 1 year: \( \$10,000(1.07)^1 = \$10,700.00 \)
Total Future Value = \( \$2,805.10 + \$5,243.19 + \$7,350.25 + \$9,159.20 + \$10,700.00 = \$35,257.74 \)
Conclusion
Since the future value of the mixed stream ($35,257.74) is greater than the future value of the single amount ($33,656.70), Gina should choose the **mixed stream of payments**.
Tricky Area:
The trickiest part is correctly compounding the cash flows from the mixed stream. The payments are at the *beginning* of the year, so the first payment compounds for all 5 years, the second for 4 years, and so on. A common mistake is to compound the first payment for only 4 years, which would be incorrect for an annuity due structure.
Problem P5-30: Value of mixed streams
Problem Statement: Find the present value of the streams of cash flows shown in the following table. Assume that the firm's opportunity cost is 12%.
| Year | A | Year | B | Year | C |
|---|---|---|---|---|---|
| 1 | -$2,000 | 1 | $10,000 | 1-5 | $10,000/yr |
| 2 | $3,000 | 2-5 | $5,000/yr | 6-10 | $8,000/yr |
| 3 | $4,000 | 6 | $7,000 | ||
| 4 | $6,000 | ||||
| 5 | $8,000 |
Solution:
Theoretical Foundation: This problem requires finding the present value of three different types of cash flow streams: a standard mixed stream, a mixed stream with an embedded annuity, and a deferred annuity. Each requires a different approach to discounting to get the correct present value at time 0.
Stream A: Standard Mixed Stream
Discount each cash flow individually to the present at a 12% rate and sum them up. The initial outflow of $2,000 at year 1 is also discounted.
- \( PV_A = -\$2,000(1.12)^{-1} + \$3,000(1.12)^{-2} + \$4,000(1.12)^{-3} + \$6,000(1.12)^{-4} + \$8,000(1.12)^{-5} \)
- \( PV_A = -\$1,785.71 + \$2,391.58 + \$2,847.16 + \$3,813.11 + \$4,539.63 = \$11,805.77 \)
Stream B: Mixed Stream with an embedded annuity
This stream can be broken into two parts: a single payment of $10,000 at year 1, and a 4-year ordinary annuity of $5,000 starting in year 2, followed by a single payment of $7,000 in year 6.
- \( PV_B = \$10,000(1.12)^{-1} + \left( \$5,000 \times \left[ \frac{1 - \frac{1}{(1.12)^4}}{0.12} \right] \right)(1.12)^{-1} + \$7,000(1.12)^{-6} \)
- \( PV_B = \$8,928.57 + (\$15,188.57)(0.892857) + \$3,542.43 = \$26,981.82 \)
Stream C: Two annuities
This can be broken down into a 5-year ordinary annuity and a deferred 5-year ordinary annuity. The deferred annuity must be discounted an additional 5 years.
- \( PV_{C} = \left( \$10,000 \times \left[ \frac{1 - \frac{1}{(1.12)^5}}{0.12} \right] \right) + \left( \$8,000 \times \left[ \frac{1 - \frac{1}{(1.12)^5}}{0.12} \right] \right)(1.12)^{-5} \)
- \( PV_C = (\$36,047.76) + (\$28,838.21)(0.567427) = \$52,389.02 \)
Tricky Area:
Stream A is straightforward but requires careful calculation. Stream B is tricky because the 4-year annuity starts at year 2, not time 0, so its present value at time 1 must be discounted back to time 0. Stream C's trick is to realize you can treat the second stream as a deferred annuity. A common mistake is to calculate the present value of the second annuity at time 0 and not at the beginning of the deferred period, or simply to add them up without considering the timing.
Problem P5-31: Present value: Mixed streams
Problem Statement: Consider the mixed streams of cash flows shown in the following table.
| Year | A | B |
|---|---|---|
| 1 | $50,000 | $10,000 |
| 2 | $40,000 | $20,000 |
| 3 | $30,000 | $30,000 |
| 4 | $20,000 | $40,000 |
| 5 | $10,000 | $50,000 |
| Totals | $150,000 | $150,000 |
- Find the present value of each stream using a 15% discount rate.
- Compare the calculated present values and discuss them in light of the undiscounted cash flows totaling $150,000 in each case.
Solution:
Theoretical Foundation: This problem emphasizes that the timing of cash flows is a key component of their value. Even if two streams have the same total nominal value, the one with cash flows received earlier will have a higher present value.
Part a: Present Value at 15%
- **Stream A:**
- \( PV_A = \$50,000(1.15)^{-1} + \$40,000(1.15)^{-2} + \$30,000(1.15)^{-3} + \$20,000(1.15)^{-4} + \$10,000(1.15)^{-5} \)
- \( PV_A = \$43,478.26 + \$30,245.42 + \$19,726.06 + \$11,435.09 + \$4,971.77 = \$109,856.60 \)
- **Stream B:**
- \( PV_B = \$10,000(1.15)^{-1} + \$20,000(1.15)^{-2} + \$30,000(1.15)^{-3} + \$40,000(1.15)^{-4} + \$50,000(1.15)^{-5} \)
- \( PV_B = \$8,695.65 + \$15,122.71 + \$19,726.06 + \$22,870.18 + \$24,858.84 = \$91,273.44 \)
Part b: Comparison
The total undiscounted cash flows for both streams are the same ($150,000). However, the present value of Stream A ($109,856.60) is significantly higher than the present value of Stream B ($91,273.44). This is because Stream A has larger cash flows in the earlier years, while Stream B has larger cash flows in the later years. Because of the time value of money, cash flows received sooner are more valuable today. This result reinforces the core principle that the timing of cash flows is crucial in financial analysis.
Tricky Area:
The trick is not in the calculation but in the interpretation. The problem explicitly gives you the same total nominal value for both streams to see if you fall into the trap of thinking they are equally valuable. The correct approach is to realize that discounting makes earlier cash flows more valuable.
Problem P5-32: Value of a mixed stream
Problem Statement: CCTech is bidding on a flood recovery project. The government will pay $5 million this year and $2 million for the following four years. The discount rate is 8% per year. The project is expected to cost $10 million today.
- Draw the time line for the stream of cash flows.
- If the discount rate is 8% per year, what is the present value of the project?
- Suppose the project is expected to cost $10 million today. Should CCTech take the project if it is offered? Why or why not?
Solution:
Theoretical Foundation: This problem requires you to calculate the Net Present Value (NPV) of a project. The cash flow stream is a mixed stream, with a different payment in the first year. The decision rule is simple: if the NPV is positive, accept the project; if it is negative, reject it.
Part a: Time line
The time line shows the cash flows from the government. Since the payments start "this year" (Time 1), this is a mixed stream, not an annuity due.
\[ \begin{array}{cccccc} \text{Time (years)}: & 1 & 2 & 3 & 4 & 5 \\ \text{Cash Flow}: & \$5M & \$2M & \$2M & \$2M & \$2M \end{array} \]
Part b: Present value of the project
The project's cash flows can be broken into a single payment of $5M at year 1 and a 4-year ordinary annuity of $2M starting at year 2. We can discount each part individually.
\( PV = \$4.6296M + (\$6.6242M)(0.9259) \approx \$4.6296M + \$6.132M = \$10.7616M \)
Part c: Project decision
The Net Present Value (NPV) of the project is the present value of the cash inflows minus the initial cost:
Since the NPV is positive ($0.7616M), CCTech should **take the project**. It adds value to the firm.
Tricky Area:
The main trap here is misidentifying the cash flow stream. The payment of $5M is a single payment at year 1, and the $2M payments are a 4-year annuity that starts at year 2. A common mistake is to treat the entire stream as a 5-year annuity, which would be incorrect.
Problem P5-33: Funding budget shortfalls
Problem Statement: You need to fund a series of budget shortfalls over the next 5 years, with amounts of $5,000, $4,000, $6,000, $10,000, and $3,000 at the end of each year, respectively. You can earn 8% on your investments and want to make a single deposit today to cover all future shortfalls. How large must this deposit be?
Solution:
Theoretical Foundation: This problem is a direct application of the present value of a mixed stream. The single deposit you need to make today is simply the sum of the present values of each future cash outflow.
We need to discount each budget shortfall back to time 0 at an 8% rate.
\( PV = \$4,629.63 + \$3,429.35 + \$4,763.15 + \$7,350.30 + \$2,041.75 = \$22,214.18 \)
The single deposit today must be **$22,214.18**.
Tricky Area:
This problem is a straightforward application of present value for a mixed stream. The trick is to not overthink it. You don't need to find a single lump sum that grows into the total of the shortfalls; you need to find the single lump sum today that, when discounted back, is equal to the present value of all the future shortfalls. The total of the shortfalls ($28,000) is irrelevant in the present value calculation.
Problem P5-34: Relationship between future value and present value: Mixed stream
Problem Statement: Using the information in the accompanying table, answer the questions that follow.
| Year (1) | Cash flow |
|---|---|
| 1 | $800 |
| 2 | $900 |
| 3 | $1,000 |
| 4 | $1,500 |
| 5 | $2,000 |
- Determine the present value of the mixed stream of cash flows using a 5% discount rate.
- How much would you be willing to pay for an opportunity to buy this stream, assuming that you can at best earn 5% on your investments?
- What effect, if any, would a 7% rather than a 5% opportunity cost have on your analysis? (Explain verbally.)
Solution:
Theoretical Foundation: This problem reinforces the concept of present value as the intrinsic worth of a future cash flow stream. It also highlights the inverse relationship between the discount rate and the present value.
Part a: Present value at 5%
Discount each cash flow individually to the present at a 5% rate and sum them up.
\( PV = \$761.90 + \$816.33 + \$863.84 + \$1,234.03 + \$1,567.05 = \$5,243.15 \)
Part b: Willing to pay
You would be willing to pay up to the present value of the stream, which is **$5,243.15**. Paying this amount would mean your return on the investment is exactly equal to your opportunity cost of 5%.
Part c: Effect of a 7% opportunity cost
A higher opportunity cost of 7% would **decrease** the present value of the stream. This is because a higher discount rate makes future cash flows less valuable today. If the present value is lower, you would be willing to pay less for the opportunity. This demonstrates the inverse relationship between the discount rate and present value.
Tricky Area:
Part b and c are about financial decision-making. The present value is a decision-making metric; it's the maximum you should be willing to pay. Part c tests your understanding of the core TVM principle that a higher discount rate (opportunity cost) reduces the present value of future cash flows.
Problem P5-35: Relationship between future value and present value: Mixed stream
Problem Statement: The table below shows a mixed cash flow stream except that the cash flow for year 3 is missing. The present value of the entire stream is $32,911.03 and the discount rate is 4%. What is the amount of the missing cash flow in year 3?
| Year | Cash Flow |
|---|---|
| 1 | $10,000 |
| 2 | $5,000 |
| 3 | ? |
| 4 | $20,000 |
| 5 | $3,000 |
Solution:
Theoretical Foundation: This problem requires you to use the present value formula in reverse to find a missing cash flow. The total present value of a mixed stream is the sum of the present values of its individual cash flows. You can find the unknown cash flow by isolating its present value and then solving for the cash flow itself.
First, find the present value of all the known cash flows.
\( PV_{\text{known}} = \$9,615.38 + \$4,622.78 + \$17,094.88 + \$2,465.75 = \$33,798.79 \)
The total present value of the stream is given as $32,911.03. We can find the present value of the missing cash flow in year 3 by subtracting the present value of the known cash flows from the total present value.
\( PV_{\text{missing}} = \$32,911.03 - \$33,798.79 = -\$887.76 \)
Now, we solve for the missing cash flow, \( X \), by using the present value formula for a single amount. The present value of the missing cash flow is $887.76, and it is discounted for 3 years at 4%.
\( -\$887.76 = \frac{X}{1.124864} \)
\( X = -\$887.76 \times 1.124864 = -\$998.60 \)
The missing cash flow in year 3 is approximately **-$998.60**. The negative sign indicates a cash outflow.
Tricky Area:
The main challenge is the backward logic. Instead of finding the total present value, you use the total present value to find a missing component. A common error is to forget that the present value of the missing cash flow is its final value discounted back to the present. The present value of the missing cash flow is negative because the sum of the known cash flows is greater than the given total present value.
Problem P5-36: Changing compounding frequency
Problem Statement: Using annual, semiannual, and quarterly compounding periods for each of the following, (1) calculate the future value if $5,000 is deposited initially, and (2) determine the effective annual rate (EAR).
- At 12% annual interest for 5 years.
- At 16% annual interest for 6 years.
- At 20% annual interest for 10 years.
Solution:
Theoretical Foundation: This problem demonstrates how compounding frequency impacts both the future value and the effective annual rate (EAR). A higher compounding frequency leads to a higher future value and a higher EAR, all else being equal. The formulas are \( FV = PV \times \left(1 + \frac{r}{m}\right)^{m \times n} \) for future value and \( EAR = \left(1 + \frac{r}{m}\right)^m - 1 \) for the effective annual rate.
Part 1: 12% for 5 years
- **Annual (m=1):**
- \( FV = \$5,000 \times (1 + 0.12)^5 = \$8,811.71 \)
- \( EAR = (1 + 0.12)^1 - 1 = 12.00\% \)
- **Semiannual (m=2):**
- \( FV = \$5,000 \times \left(1 + \frac{0.12}{2}\right)^{2 \times 5} = \$8,954.24 \)
- \( EAR = \left(1 + \frac{0.12}{2}\right)^2 - 1 = 12.36\% \)
- **Quarterly (m=4):**
- \( FV = \$5,000 \times \left(1 + \frac{0.12}{4}\right)^{4 \times 5} = \$9,030.56 \)
- \( EAR = \left(1 + \frac{0.12}{4}\right)^4 - 1 = 12.55\% \)
Part 2: 16% for 6 years
- **Annual (m=1):**
- \( FV = \$5,000 \times (1 + 0.16)^6 = \$12,187.05 \)
- \( EAR = 16.00\% \)
- **Semiannual (m=2):**
- \( FV = \$5,000 \times \left(1 + \frac{0.16}{2}\right)^{2 \times 6} = \$12,590.85 \)
- \( EAR = \left(1 + \frac{0.16}{2}\right)^2 - 1 = 16.64\% \)
- **Quarterly (m=4):**
- \( FV = \$5,000 \times \left(1 + \frac{0.16}{4}\right)^{4 \times 6} = \$12,816.59 \)
- \( EAR = \left(1 + \frac{0.16}{4}\right)^4 - 1 = 16.99\% \)
Part 3: 20% for 10 years
- **Annual (m=1):**
- \( FV = \$5,000 \times (1 + 0.20)^{10} = \$30,958.68 \)
- \( EAR = 20.00\% \)
- **Semiannual (m=2):**
- \( FV = \$5,000 \times \left(1 + \frac{0.20}{2}\right)^{2 \times 10} = \$33,637.50 \)
- \( EAR = \left(1 + \frac{0.20}{2}\right)^2 - 1 = 21.00\% \)
- **Quarterly (m=4):**
- \( FV = \$5,000 \times \left(1 + \frac{0.20}{4}\right)^{4 \times 10} = \$35,178.68 \)
- \( EAR = \left(1 + \frac{0.20}{4}\right)^4 - 1 = 21.55\% \)
Tricky Area:
The main trap is forgetting to adjust both the interest rate and the number of periods for the new compounding frequency. For example, for semiannual compounding over 5 years, you must use \( r/2 \) and \( n \times 2 \). Failing to do both will lead to an incorrect answer.
Problem P5-37: Compounding frequency, time value, and effective annual rates
Problem Statement: For each of the cases in the following table: a. Calculate the future value at the end of the specified deposit period. b. Determine the effective annual rate, EAR. c. Compare the nominal annual rate, r, to the effective annual rate, EAR. What relationship exists between compounding frequency and the nominal and effective annual rates?
| Case | Amount of initial deposit | Nominal annual rate, r | Compounding frequency, m (times/year) | Deposit period (years) |
|---|---|---|---|---|
| A | $2,500 | 6% | 2 | 5 |
| B | $50,000 | 12% | 6 | 3 |
| C | $1,000 | 5% | 1 | 10 |
| D | $20,000 | 16% | 4 | 6 |
Solution:
Theoretical Foundation: This problem is a direct application of the future value and EAR formulas for different compounding frequencies. The core principle is that as the compounding frequency increases, the EAR also increases, which in turn leads to a higher future value.
Part a: Future Value
Using \( FV = PV \times \left(1 + \frac{r}{m}\right)^{m \times n} \):
- **Case A:** \( FV = \$2,500 \times \left(1 + \frac{0.06}{2}\right)^{2 \times 5} = \$3,359.79 \)
- **Case B:** \( FV = \$50,000 \times \left(1 + \frac{0.12}{6}\right)^{6 \times 3} = \$71,280.97 \)
- **Case C:** \( FV = \$1,000 \times \left(1 + \frac{0.05}{1}\right)^{1 \times 10} = \$1,628.89 \)
- **Case D:** \( FV = \$20,000 \times \left(1 + \frac{0.16}{4}\right)^{4 \times 6} = \$51,215.11 \)
Part b: Effective Annual Rate (EAR)
Using \( EAR = \left(1 + \frac{r}{m}\right)^m - 1 \):
- **Case A:** \( EAR = \left(1 + \frac{0.06}{2}\right)^2 - 1 = 6.09\% \)
- **Case B:** \( EAR = \left(1 + \frac{0.12}{6}\right)^6 - 1 = 12.62\% \)
- **Case C:** \( EAR = \left(1 + \frac{0.05}{1}\right)^1 - 1 = 5.00\% \)
- **Case D:** \( EAR = \left(1 + \frac{0.16}{4}\right)^4 - 1 = 16.99\% \)
Part c: Comparison and Relationship
For Case C, where compounding is annual, the nominal rate and the EAR are identical. For all other cases where compounding is more frequent than annually, the EAR is higher than the nominal rate. This illustrates the relationship that as the compounding frequency (\( m \)) increases, the EAR also increases. This higher EAR results in a higher future value, as seen in part a. The difference between the nominal and effective rate is the extra interest earned on the interest itself.
Tricky Area:
The most important part of this problem is the final comparison. While the calculations are straightforward, the key is to articulate the relationship between compounding frequency, the nominal rate, the EAR, and the final future value. A common mistake is to calculate the EAR correctly but fail to explain its significance.
Problem P5-38: Continuous compounding
Problem Statement: For each of the cases in the following table, find the future value at the end of the deposit period, assuming that interest is compounded continuously at the given nominal annual rate.
| Case | Amount of initial deposit | Nominal annual rate, r | Deposit period (years), n |
|---|---|---|---|
| A | $1,000 | 9% | 2 |
| B | $600 | 10% | 10 |
| C | $4,000 | 8% | 7 |
| D | $2,500 | 12% | 4 |
Solution:
Theoretical Foundation: Continuous compounding is the theoretical limit of compounding, where interest is calculated and added infinitely often. The formula for future value with continuous compounding is \( FV = PV \times e^{rt} \), where \( e \) is the base of the natural logarithm (approximately 2.71828).
- **Case A:** \( FV = \$1,000 \times e^{(0.09 \times 2)} = \$1,197.22 \)
- **Case B:** \( FV = \$600 \times e^{(0.10 \times 10)} = \$1,630.97 \)
- **Case C:** \( FV = \$4,000 \times e^{(0.08 \times 7)} = \$6,962.19 \)
- **Case D:** \( FV = \$2,500 \times e^{(0.12 \times 4)} = \$4,043.20 \)
Tricky Area:
The main trick here is simply remembering the correct formula for continuous compounding and knowing how to use the 'e' key on your calculator. Continuous compounding is always the highest possible future value for a given nominal rate and time period.
Problem P5-39: Compounding frequency and time value
Problem Statement: You plan to invest $2,000 in an individual retirement arrangement (IRA) today at a nominal annual rate of 8% for 10 years.
- How much will you have in the account at the end of 10 years if interest is compounded (1) annually, (2) semiannually, (3) daily (assume a 365-day year), and (4) continuously?
- What is the effective annual rate (EAR) for each compounding period in part a?
- How much greater will your IRA balance be at the end of 10 years if interest is compounded continuously rather than annually?
- How does the compounding frequency affect the future value and effective annual rate for a given deposit? Explain in terms of your findings in parts a through c.
Solution:
Theoretical Foundation: This problem provides a comprehensive look at how different compounding frequencies impact an investment. It's a great exercise to solidify your understanding of the relationship between nominal rate, EAR, and future value.
Part a: Future Value (FV)
Using \( FV = PV \times \left(1 + \frac{r}{m}\right)^{m \times n} \) and \( FV = PV \times e^{rt} \):
- (1) Annually (m=1): \( FV = \$2,000(1 + 0.08)^{10} = \$4,317.85 \)
- (2) Semiannually (m=2): \( FV = \$2,000\left(1 + \frac{0.08}{2}\right)^{20} = \$4,382.25 \)
- (3) Daily (m=365): \( FV = \$2,000\left(1 + \frac{0.08}{365}\right)^{3650} = \$4,451.07 \)
- (4) Continuously: \( FV = \$2,000 \times e^{(0.08 \times 10)} = \$4,451.08 \)
Part b: Effective Annual Rate (EAR)
Using \( EAR = \left(1 + \frac{r}{m}\right)^m - 1 \) and \( EAR = e^r - 1 \):
- (1) Annually: \( EAR = (1 + 0.08)^1 - 1 = 8.00\% \)
- (2) Semiannually: \( EAR = (1 + \frac{0.08}{2})^2 - 1 = 8.16\% \)
- (3) Daily: \( EAR = (1 + \frac{0.08}{365})^{365} - 1 = 8.3277\% \)
- (4) Continuously: \( EAR = e^{0.08} - 1 = 8.3287\% \)
Part c: Difference between Continuous and Annual
Part d: Effect of compounding frequency
As the compounding frequency increases, both the future value and the effective annual rate increase. This is because interest is earned on the interest more often. The difference is minimal between daily and continuous compounding but is significant when compared to annual compounding. This demonstrates the "interest on interest" effect and shows that all else being equal, a higher compounding frequency is better for an investor.
Tricky Area:
This problem is a solid test of your mastery of all compounding methods. A common mistake is to miscalculate the number of periods or the periodic interest rate. Also, be sure to use the correct formula for continuous compounding. The final part requires you to synthesize the results and explain the underlying principle.
Problem P5-40: Accumulating a growing future sum
Problem Statement: You have $30,000 and are considering buying a new car or investing the amount. You have two investment options: a savings account at 5% compounded monthly or stocks with an expected return of 6% compounded continuously. The car's price inflates by 2% per year.
- If you choose to invest, how much will you have 6 years later if you invested in (1) the savings account, or (2) stocks?
- If you invested in the savings account, how long will it take for you to double your money?
- Suppose the price of the car inflates by 2% per year. If you choose to invest your money in stocks, how long will it take for you to be able to afford 2 cars?
Solution:
Theoretical Foundation: This is a multi-part problem that combines different compounding frequencies, future value calculations, and solving for an unknown number of periods. It's a comprehensive test of your TVM knowledge.
Part a: Future value in 6 years
Using the appropriate future value formulas for each compounding method:
- (1) Savings Account (monthly): \( FV = \$30,000 \times \left(1 + \frac{0.05}{12}\right)^{12 \times 6} = \$40,470.92 \)
- (2) Stocks (continuously): \( FV = \$30,000 \times e^{(0.06 \times 6)} = \$43,331.06 \)
Part b: Time to double in savings account
We need to solve for \( n \) in months, with \( PV = \$30,000 \), \( FV = \$60,000 \), and a monthly rate of \( r = \frac{0.05}{12} \).
\( 2 = \left(1 + \frac{0.05}{12}\right)^n \)
\( n = \frac{\ln(2)}{\ln(1 + \frac{0.05}{12})} \approx 166.72 \text{ months} \)
\( \text{Years} = \frac{166.72}{12} \approx 13.89 \text{ years} \)
Part c: Time to afford two cars (stocks)
First, find the future cost of two cars today: \( \$30,000 \times 2 = \$60,000 \). Then, find the future value of the car price after \( n \) years of 2% inflation: \( FV_{car} = \$30,000 \times (1 + 0.02)^n \). The future value of two cars is \( 2 \times FV_{car} = \$60,000 \times (1.02)^n \).
Now, we need to find the number of years (\( n \)) where the future value of the stock investment equals the future cost of two cars.
\( e^{0.06n} = 2 \times (1.02)^n \)
\( 0.06n = \ln(2) + n \times \ln(1.02) \)
\( 0.06n - n \times 0.0198 = 0.6931 \)
\( 0.0402n = 0.6931 \)
\( n = \frac{0.6931}{0.0402} \approx 17.24 \text{ years} \)
Tricky Area:
Part c is the most challenging. You need to set up an equation where the future value of the investment is equal to the future value of the two cars. This requires you to handle both continuous compounding for the investment and discrete compounding for the inflation. The final algebraic step involves isolating 'n' after applying logarithms to both sides.