Introduction & Learning Objectives
This guide is your interactive companion to understanding **Discounted Cash Flow (DCF) Valuation**, based on **Chapter 6 of Ross's textbook**. It builds upon the basics of time value of money by introducing how to value investments with multiple cash flows, a much more realistic scenario for financial analysis.
Learning Objectives (Ross, page 156)
- Determine the future and present value of investments with multiple cash flows.
- Explain how loan payments are calculated and how to find the interest rate on a loan.
- Describe how loans are amortized or paid off.
- Show how interest rates are quoted (and misquoted).
1. Future & Present Values with Multiple Cash Flows
Annuities & Perpetuities (Ross, page 164)
An **annuity** is a series of constant cash flows for a fixed number of periods. A **perpetuity** is a constant stream of cash flows that lasts forever. These cash flow patterns are very common in finance, and there are useful shortcuts for their valuation.
Exam Hint:
Remember to distinguish between an **ordinary annuity** (payments at the end of the period) and an **annuity due** (payments at the beginning). This was tested in **Question 53 from the basic problems**.
2. Comparing Rates and Loan Types
EAR vs. APR (Ross, page 175)
The **Annual Percentage Rate (APR)** is the quoted, or stated, interest rate. The **Effective Annual Rate (EAR)** is the actual interest rate you pay or receive after accounting for compounding. For financial decisions, the **EAR is the only relevant rate** for comparison.
Where $m$ is the number of times interest is compounded per year.
Exam Hint:
You must be able to convert between APR and EAR to compare different loans or investments accurately. This was tested in **Question 14 from the basic problems**.
Loan Amortization (Ross, page 181)
An **amortized loan** is paid off by making regular principal reductions over time. The most common type involves equal periodic payments. You can create an amortization schedule to see how each payment is split between interest and principal.
Exam Hint:
Be able to calculate the loan payment amount and create an amortization schedule. This was tested in **Question 6.5** of the chapter review problems.
3. Problems & Solutions
Problem Statement: A first-round draft choice quarterback has been signed to a three-year, $25 million contract. The details provide for an immediate cash bonus of $2 million. The player is to receive $5 million in salary at the end of the first year, $8 million the next, and $10 million at the end of the last year. Assuming a 15 percent discount rate, is this package worth $25 million? If not, how much is it worth?
Solution:
To find the true worth, we calculate the present value of all payments using a 15% discount rate. The bonus is an immediate payment, so its value is its face value.
PV = $2M + \$4.35M + \$6.05M + \$6.58M = \$18.98 \text{ million}$
The package is **not** worth $25 million. It is worth only $18.98 million.
Problem Statement: You want to buy a new sports coupe for $84,500, and the finance office at the dealership has quoted you an APR of 4.7 percent for a 60-month loan to buy the car. What will your monthly payments be? What is the effective annual rate on this loan?
Solution:
1. Monthly Payment:
This is an annuity problem where we need to find the payment (C).
Loan Amount (PV) = $84,500
Monthly Rate ($r$) = $4.7\% / 12 = 0.3917\%$
Number of Payments ($t$) = 60 months
PV = $C \times \frac{1 - [1/(1+r)^t]}{r}$
$84,500 = C \times \frac{1 - [1/(1.003917)^{60}]}{0.003917} = C \times 54.76$
$C = 84,500 / 54.76 = \$1,543.10
2. Effective Annual Rate (EAR):
We use the APR and compounding frequency to find the EAR.
$EAR = (1 + \frac{APR}{m})^m - 1 = (1 + \frac{0.047}{12})^{12} - 1 = (1.003917)^{12} - 1 = 0.0480$ or 4.80%
Problem Statement: You receive a credit card application from Shady Banks Savings and Loan offering an introductory rate of 1.25 percent per year, compounded monthly for the first six months, increasing thereafter to 17.8 percent compounded monthly. Assuming you transfer the $9,000 balance from your existing credit card and make no subsequent payments, how much interest will you owe at the end of the first year?
Solution:
We need to find the future value of the $9,000 balance after one year, with two different interest rates applied.
1. First 6 Months:
Monthly Rate ($r_1$) = $1.25\% / 12 = 0.1042\%$
$FV_6 = \$9,000 \times (1 + 0.001042)^6 = \$9,056.40$
2. Next 6 Months:
Monthly Rate ($r_2$) = $17.8\% / 12 = 1.4833\%$
$FV_{12} = \$9,056.40 \times (1 + 0.014833)^6 = \$9,890.31$
3. Total Interest Owed:
Total Interest = $FV_{12} - \text{Principal} = \$9,890.31 - \$9,000 = \$890.31
Problem Statement: Ben Bates is considering two MBA programs (Wilton and Mount Perry) versus staying in his current job. You need to perform a financial analysis to determine the best option. Key variables include current salary, tuition, living expenses, and expected post-MBA salary and bonuses.
Solution:
To evaluate the options, we need to find the NPV of each. This involves calculating all relevant cash flows (costs and salaries) for each option and discounting them back to today's value.
1. Status Quo (Stay at Current Job):
This is the baseline. Ben's after-tax salary today is $57,000 \times (1 - 0.26) = \$42,180$. We need to find the present value of this growing annuity of salaries for the next 38 years at a 5.5% discount rate.
NPV = $702,968
2. Wilton University (2-Year Program):
Costs: Tuition, books, and lost salary for 2 years. Gains: Signing bonus and higher future salary for 36 years.
NPV = $734,519
3. Mount Perry College (1-Year Program):
Costs: Tuition, books, and lost salary for 1 year. Gains: Signing bonus and higher future salary for 37 years.
NPV = $741,438
Conclusion:
From a strictly financial standpoint, the **Mount Perry College** option has the highest NPV and is the best choice.
Question 4: Future Value Analysis
Future value analysis is flawed because it does not account for the **time value of money** properly. The appropriate method is to discount all cash flows back to a single point in time (the present) to make a direct and accurate comparison.