Key Takeaways
- A dollar today is worth more than a dollar in the future due to its earning potential (opportunity cost)
- Present value discounts future cash flows using a discount rate; future value compounds current money forward
- A positive NPV indicates an investment adds value and should be accepted; negative NPV means reject
- IRR is the discount rate where NPV equals zero — a bond's IRR equals its yield to maturity (YTM)
- The Rule of 72 estimates doubling time: Years to Double = 72 / Interest Rate
Time Value of Money
The time value of money (TVM) is one of the most fundamental concepts in finance: a dollar received today is worth more than a dollar received in the future. Why? Because money received today can be invested to earn returns.
Core Concept: Opportunity Cost
If you receive $1,000 today and invest it at 5% annually, you'll have $1,050 in one year. If you wait a year to receive that $1,000, you've lost the opportunity to earn that $50. This lost opportunity is the essence of TVM.
Present Value (PV)
Present value is the current worth of a future sum of money, discounted at a specific rate.
Formula: PV = FV / (1 + r)^n
Where:
- FV = Future Value
- r = Discount Rate (per period)
- n = Number of periods
Example
What is the present value of $10,000 received in 5 years at a 6% discount rate?
PV = $10,000 / (1.06)^5 = $10,000 / 1.338 = $7,473
This means $7,473 today is equivalent to $10,000 in 5 years at 6%.
Future Value (FV)
Future value is what a current sum will be worth at a future date, compounded at a specific rate.
Formula: FV = PV × (1 + r)^n
Example
If you invest $5,000 today at 8% annual return, what will it be worth in 10 years?
FV = $5,000 × (1.08)^10 = $5,000 × 2.159 = $10,795
Net Present Value (NPV)
NPV is the difference between the present value of an investment's future cash inflows and its initial cost. It determines whether an investment creates or destroys value.
Formula: NPV = Present Value of Cash Inflows - Initial Investment Cost
| NPV Result | Decision | Meaning |
|---|---|---|
| NPV > 0 | Accept | Investment adds value |
| NPV < 0 | Reject | Investment destroys value |
| NPV = 0 | Indifferent | Investment breaks even |
When to Use NPV
NPV is most appropriate for investments with predictable future cash flows:
- Bonds — Fixed coupon payments and principal repayment
- Preferred stock — Fixed, regular dividend payments
- Real estate — Expected rental income
NPV is less useful for common stock because future dividends are unpredictable.
Internal Rate of Return (IRR)
IRR is the discount rate that makes the NPV of an investment equal to zero. It represents the investment's expected rate of return.
| Comparison | Decision |
|---|---|
| IRR > Required Return | Accept the investment |
| IRR < Required Return | Reject the investment |
IRR and Bonds
For bonds, the IRR equals the yield to maturity (YTM). Exam questions may use these terms interchangeably.
| Bond Trading At | IRR vs. Coupon Rate |
|---|---|
| Discount (below par) | IRR > Coupon Rate |
| Par | IRR = Coupon Rate |
| Premium (above par) | IRR < Coupon Rate |
The Rule of 72
The Rule of 72 provides a quick estimate of how long it takes money to double at a given interest rate.
Formula: Years to Double = 72 / Interest Rate
| Interest Rate | Years to Double |
|---|---|
| 4% | 18 years |
| 6% | 12 years |
| 8% | 9 years |
| 10% | 7.2 years |
| 12% | 6 years |
Example
At 9% annual return, an investment doubles in approximately 72 / 9 = 8 years.
Exam Tip: You won't need to perform complex NPV or IRR calculations on the exam, but you must understand the concepts. Positive NPV = accept; IRR > required return = accept. Remember that a bond's IRR equals its YTM.
Using the Rule of 72, approximately how long will it take to double an investment earning 6% annually?
An investment has a Net Present Value (NPV) of -$5,000. The investment adviser should recommend:
For a bond, the Internal Rate of Return (IRR) is equal to:
1.3 Descriptive Statistics and Risk Measures
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