Finance I - Lesson 4: Time Value of Money – Concepts and Calculations
The time value of money (TVM) is a foundational concept in finance, stating that a dollar today is worth more than a dollar received in the future. This arises from inflation, the opportunity cost of capital, and the inherent risk in deferring consumption or investment. Mastering present value (PV), future value (FV), and related calculations provides critical skills for valuing projects, investments, and any financial contract involving cash flows over time.
Why Money Has a Time Value
A dollar in hand today can be invested to earn interest or returns. Waiting for that dollar in the future means missing out on potential earnings in the meantime. Additionally:
- Inflation typically reduces purchasing power over time.
- Risk compounds as the waiting period increases, necessitating higher returns to compensate.
- Opportunity cost reflects the foregone earnings from alternative investments.
Key Concepts: Present Value and Future Value
Present Value (PV) translates future sums back to today’s purchasing power by discounting at a given interest or discount rate. Future Value (FV) projects current sums forward, compounding the principal to account for earned interest. The relationships can be summarized by:
Compounding and Discounting
Compounding applies an interest rate to principal over multiple periods, effectively earning interest on accumulated interest. Discounting is the opposite: pulling future cash flows backward in time by applying a discount rate that compensates for time, risk, and opportunity cost.
The flowchart represents a simple timeline where an initial amount grows (compounds) through multiple time periods until it reaches a future sum.
Net Present Value (NPV) and Internal Rate of Return (IRR)
Both NPV and IRR are critical tools for project and investment evaluation:
- NPV discounts all future cash flows back to the present. A positive NPV indicates that the project’s return exceeds the discount rate (often the firm’s required return or cost of capital), suggesting value creation for the firm.
- IRR is the rate at which the present value of a project’s inflows equals the present value of its outflows (NPV = 0). If IRR exceeds the required rate of return, the project is generally attractive.
Sample Calculations and Examples
Below are a few illustrative examples demonstrating the use of time value of money formulas. These examples clarify how to handle single cash flows, annuities, and project valuations using NPV.
| Example | Description | Solution Outline |
|---|---|---|
| Single Sum FV | Invest $2,000 at 8% annual interest for 4 years. How much will it grow to? | FV = 2,000 × (1 + 0.08)^4 = 2,000 × 1.36049 = $2,720.98 |
| Single Sum PV | You want $5,000 in 5 years, and can earn 6% annually. How much must you invest now? | PV = 5,000 ÷ (1 + 0.06)^5 = 5,000 ÷ 1.33823 = $3,736.30 |
| Present Value of an Annuity | Receive $1,000 at the end of each year for 5 years, at 10% discount rate. What is it worth today? | PV(annuity) = 1,000 × [1 − (1 + 0.10)^(-5)] ÷ 0.10 = $3,790.79 |
Visualizing Compounding Growth
The chart below illustrates how an initial investment can grow over a set number of years under a fixed interest rate. Even a small increase in the rate can significantly accelerate growth over longer periods, highlighting the power of compounding.
Summary
The time value of money anchors a vast array of financial decisions, from personal savings plans to corporate capital budgeting. By understanding how to compute present and future values, managers and investors can compare cash flows occurring at different points in time, assess the attractiveness of projects, and properly price assets. Comprehending NPV and IRR further refines these decisions, revealing which opportunities create or destroy value. These skills form the basis for more advanced topics, including bond and equity valuation, capital budgeting, and portfolio management.
Suggested Reading:
Principles of Corporate Finance by Brealey, Myers, and Allen (chapters covering time value of money,
NPV, and IRR).
