EY

Chapter 04 Notes on Time Value of Money

Chapter 04 — Introduction to Valuation: The Time Value of Money (Notes)

  • Overview and objectives

    • Learn to determine future value (FV) and present value (PV) of cash flows.

    • Learn to compute return on investment and the time required to reach a target value.

    • Solve time value of money (TVM) problems using formulas, financial calculators, and spreadsheets.

    • Understand implications of discounting, compounding, opportunity cost, risk, and practical applications.

4-1 Future Value and Compounding

  • Basic concepts

    • FV and compounding describe how money grows over time when it earns interest.

    • FV is the amount an investment is worth after t periods; it is the value “later” money.

    • FV grows according to interest earned on both principal and accumulated interest (compound interest).

  • Key formulas

    • General future value formula: FV = PV \, (1 + r)^t where

    • PV = present value (initial investment at t = 0),

    • r = period interest rate (as a decimal),

    • t = number of periods.

    • Future value factor: $(1 + r)^t$.

    • Simple vs. compound interest (conceptual):

    • Simple interest earns interest only on the original principal.

    • Compound interest earns interest on the principal and on previously earned interest ("interest on interest").

  • Illustrative example (Example 1)

    • Invest PV = 100 for 1 year at r = 10 extrm{%} = 0.10.

    • Interest earned: 100 imes 0.10 = 10.

    • FV after 1 year: FV = 100 imes (1 + 0.10) = 110.

    • FV = 110.

  • Example 1 continued (two-year horizon)

    • If money remains invested for 5 years at 10%, use the compound formula: FV = 100 imes (1.10)^5 = 161.051 (from Table 4.1, FV after 5 years is 161.05, with total interest 61.05).

  • Effects of compounding

    • Simple interest example: FV under simple interest after 2 years on a $100 principal at 10% is FV_{ ext{simple}} = 100 + 2 imes 10 = 120.

    • Compound interest example: FV with compounding after 2 years is FV_{ ext{compound}} = 100 imes (1.10)^2 = 121.

    • The extra 1.00 in the 2-year example comes from the interest on the first year’s interest: 0.10 imes 10 = 1.00.

  • Tabulated example

    • Table 4.1: Future value of $100 at 10% for years 1–5:

    • Year 1: Beginning $100, Interest $10, Ending $110

    • Year 2: Beginning $110, Interest $11, Ending $121

    • Year 3: Beginning $121, Interest $12.10, Ending $133.10

    • Year 4: Beginning $133.10, Interest $13.31, Ending $146.41

    • Year 5: Beginning $146.41, Interest $14.64, Ending $161.05

    • Total interest: $61.05

  • Worked example with a 5-year horizon (Example 2)

    • If you invest $100 for 5 years at 10%, FV = $161.05 (as above).

  • Practical tool: TI BAII Plus and Excel

    • BAII Plus basics for TVM (Example setup):

    • I/Y = period rate r (as a percent, not a decimal).

    • N = number of periods.

    • PV and FV: PMT typically 0 for single lump-sum problems in this chapter.

    • Clear registers before each problem: 2nd → CLR TVM.

    • Set P/Y to 1 (payments per year).

    • End vs Begin cash flow timing: End (default) vs Begin mode (2nd → BGN).

    • Calculator input example to get FV for 10% over 5 years with PV = 100: N = 5, I/Y = 10, PV = -100, PMT = 0, FV → FV = 161.05.

  • Excel TVM functions (for TVM problems)

    • FV(rate, nper, pmt, pv)

    • PV(rate, nper, pmt, fv)

    • RATE(nper, pmt, pv, fv) computes the implied rate.

    • NPER(rate, pmt, pv, fv) computes the number of periods.

    • Example: FV with rate = 0.10, nper = 5, pv = -100, pmt = 0 gives 161.05.

  • Quick note on a general growth problem (growth of widgets)

    • If a company expects sales to increase by 15% per year for 5 years starting from 3 million widgets, the future quantity is: FV = PV imes (1 + 0.15)^5 = 3 imes (1.15)^5 ext{ widgets}

    • With sign conventions used in their example, Excel/Calc outputs may show negative FV when the PV is treated as a present outflow.

4-2 Present Value and Discounting

  • Basic idea

    • PV = present value: the current value of future cash flows discounted at the appropriate discount rate.

    • PV answers questions like: What do I have to invest today to achieve a future cash flow? What is the current value of a future amount?

  • Reasons why PV is less than future value

    • Opportunity cost, risk and uncertainty, and time preference.

  • Time line of cash flows (PV context)

    • Time 0 is today; Time 1 is end of Period 1; tick marks at period ends.

    • +CF denotes cash inflow; −CF denotes cash outflow; PMT = constant cash flow per period.

  • PV formula and discounting

    • For a single future amount: PV = rac{FV}{(1 + r)^t}

    • For multiple cash flows: PV = rac{CF1}{(1 + r)^1} + rac{CF2}{(1 + r)^2} + \