Time value of money - Calculations Formulas and Derivations

Time Value of Money: Standard Calculations and Definitions Introduction The concept of time value of money is fundamental to finance. It recognizes a simple truth: a dollar today is worth more than a dollar in the future. Why? Because you can invest that dollar today and earn returns. This principle underlies nearly every financial valuation problem you'll encounter. To compare amounts of money occurring at different times, we use two core operations: discounting (converting future amounts to present value) and compounding (converting present amounts to future value). This section teaches you the standard formulas and when to apply them. Single Sum Calculations Present Value of a Single Future Sum Present value (PV) is the current worth of money you will receive in the future. To find it, we discount the future amount backward using a discount rate—essentially reversing the growth that would occur if you invested that money today. The formula is: $$PV = FV \cdot (1+i)^{-n}$$ Where: $FV$ = future value (amount you'll receive later) $i$ = discount rate per period (as a decimal) $n$ = number of periods into the future Key insight: The higher the discount rate, the lower the present value. A 5% discount rate makes future money worth less (relative to a 3% rate) because you'd need less money today to grow to that same future amount. Example: You expect to receive $1,000 in 3 years. If the appropriate discount rate is 5% per year, what's the present value? $$PV = 1000 \cdot (1.05)^{-3} = 1000 \cdot 0.8638 = \$863.80$$ This means $863.80 invested today at 5% would grow to $1,000 in three years. The image above shows how present value decreases as time extends into the future, with steeper drops at higher discount rates. Notice how a 9% discount rate creates a much more dramatic decrease than a 3% rate. Future Value of a Single Present Sum Future value (FV) is the opposite operation: it tells you what a sum of money today will grow to in the future. The formula is: $$FV = PV \cdot (1+i)^{n}$$ Where: $PV$ = present value (amount you have today) $i$ = interest rate per period $n$ = number of periods Example: If you invest $1,000 today at 5% annual interest for 3 years: $$FV = 1000 \cdot (1.05)^{3} = 1000 \cdot 1.1576 = \$1,157.60$$ Notice that the future value formula is just the inverse of the present value formula. If you discount $1,157.60 back three periods at 5%, you'll get $1,000. Annuity Calculations An annuity is a series of equal payments occurring at evenly spaced intervals. Think of rent payments, loan payments, or retirement withdrawals. Annuities are everywhere in finance, making these formulas essential. Two Types of Annuities There's an important distinction based on when payments occur: Ordinary annuity (annuity in arrears): Payments occur at the end of each period Annuity due (annuity in advance): Payments occur at the beginning of each period This matters because payments at the beginning of a period have an extra period to grow, so an annuity due is always worth more than an ordinary annuity. Present Value of an Ordinary Annuity Present value of an annuity (PVA) is the current worth of a stream of equal future payments. The formula is: $$PV = A \cdot \frac{1-(1+i)^{-n}}{i}$$ Where: $A$ = annuity payment (same each period) $i$ = discount rate per period $n$ = number of periods Example: You'll receive $500 at the end of each year for 4 years. If the discount rate is 6%, what's the present value? $$PV = 500 \cdot \frac{1-(1.06)^{-4}}{0.06} = 500 \cdot \frac{1-0.7921}{0.06} = 500 \cdot 3.4651 = \$1,732.55$$ This means $1,732.55 today is equivalent to receiving $500 per year for four years, assuming a 6% discount rate. Why this formula works: You're essentially discounting each $500 payment individually: Year 1: $500 × (1.06)^{-1}$ Year 2: $500 × (1.06)^{-2}$ Year 3: $500 × (1.06)^{-3}$ Year 4: $500 × (1.06)^{-4}$ The formula condenses this sum into a single calculation using geometric series algebra. Present Value of an Annuity Due An annuity due simply requires adjusting the ordinary annuity result by one period: $$PV{\text{due}} = PV{\text{ordinary}} \cdot (1+i)$$ Why? Each payment is received one period earlier, so it compounds for one additional period. Multiply by $(1+i)$ to shift forward by one period. Example: Using the previous example but with payments at the beginning of each year: $$PV{\text{due}} = 1,732.55 \cdot 1.06 = \$1,836.50$$ The annuity due is worth $103.95 more because each payment arrives earlier. Future Value of an Ordinary Annuity Future value of an annuity (FVA) is the accumulated value of regular payments invested at a given interest rate. The formula is: $$FV = A \cdot \frac{(1+i)^{n}-1}{i}$$ Where the variables have the same meanings as before. Example: You invest $500 at the end of each year for 4 years at 6% interest. What's the future value? $$FV = 500 \cdot \frac{(1.06)^{4}-1}{0.06} = 500 \cdot \frac{1.2625-1}{0.06} = 500 \cdot 4.3746 = \$2,187.30$$ Notice the comparison: the ordinary annuity's present value was $1,732.55, but its future value is $2,187.30. The extra $454.75 is interest earned on the payments as they accumulate. Future Value of an Annuity Due Similarly, for an annuity due: $$FV{\text{due}} = FV{\text{ordinary}} \cdot (1+i)$$ Each payment compounds for one additional period, so: $$FV{\text{due}} = 2,187.30 \cdot 1.06 = \$2,318.54$$ Growing Annuities A growing annuity is a series of payments that increase by a constant percentage each period. For example, a salary that increases 3% annually, or rent that escalates each year. Present Value of a Growing Annuity When payments grow at a constant rate $g$ each period (where $g \neq i$): $$PV = A \cdot \frac{1-\left(\frac{1+g}{1+i}\right)^{n}}{i-g}$$ Where: $A$ = first payment $g$ = growth rate of payments $i$ = discount rate $n$ = number of periods Example: Your salary starts at $50,000 and grows 3% annually. You'll work for 20 years. What's the present value of your salary stream at a 6% discount rate? $$PV = 50,000 \cdot \frac{1-\left(\frac{1.03}{1.06}\right)^{20}}{0.06-0.03}$$ $$PV = 50,000 \cdot \frac{1-0.5438}{0.03} = 50,000 \cdot 15.207 = \$760,350$$ This tells you the lump sum that would be equivalent to receiving those growing salary payments. Critical assumption: The discount rate must exceed the growth rate ($i > g$), or the formula breaks down. This makes intuitive sense: if your salary grew faster than the discount rate, the present value would be infinite (or negative). Perpetuities A perpetuity is a stream of equal payments that continues forever, never ending. While perpetual payments sound theoretical, real examples include preferred stocks that pay perpetual dividends or government bonds that never mature. Present Value of a Perpetuity The present value formula is remarkably simple: $$PV = \frac{A}{i}$$ Where: $A$ = the perpetual payment $i$ = discount rate Example: A preferred stock pays $10 annually forever. What's its value if the required return is 5%? $$PV = \frac{10}{0.05} = \$200$$ This is elegant: pay $200 today, and you'll receive $10 per year forever, earning exactly 5% annually. How this formula arises: It's the limit of the ordinary annuity formula as $n$ approaches infinity. The term $(1+i)^{-n}$ approaches zero, leaving only $\frac{A}{i}$. Present Value of a Growing Perpetuity A growing perpetuity has payments that increase at a constant rate $g$ forever, provided $i > g$: $$PV = \frac{A}{i-g}$$ This is also called the Gordon growth model in dividend valuation contexts. Example: A stock pays a $2 dividend today, expected to grow 4% annually forever. With a 10% required return: $$PV = \frac{2 \cdot 1.04}{0.10 - 0.04} = \frac{2.08}{0.06} = \$34.67$$ Note: We multiply the dividend by $(1+g)$ because $A$ represents the next payment, not today's. Why $i > g$ is necessary: If growth exceeded the discount rate, the value would be infinite. This violates economic intuition about valuation. Continuous Compounding <extrainfo> When to Use Continuous Compounding In most of this course, you'll use the standard formulas with discrete compounding (annual, monthly, etc.). However, some advanced applications assume continuous compounding, where interest compounds instantaneously. Present Value with Continuous Compounding With a continuously compounded rate $r$, the present value formula becomes: $$PV = FV \cdot e^{-rt}$$ Where: $e$ = Euler's number (≈ 2.71828) $r$ = continuously compounded rate $t$ = time in years This replaces the $(1+i)^{-n}$ term from discrete compounding. Example: What's the present value of $1,000 due in 3 years at a continuous rate of 5%? $$PV = 1000 \cdot e^{-0.05 \times 3} = 1000 \cdot e^{-0.15} = 1000 \cdot 0.8607 = \$860.71$$ Compare this to discrete annual compounding ($863.80 from our earlier example)—the difference is small. Variable Discount Rates Over Time Continuous compounding truly shines when discount rates change over time. If $r(t)$ varies with time: $$PV = FV \cdot \exp\left(-\int{0}^{T} r(t)\,dt\right)$$ This formula elegantly handles any time-varying discount rate using calculus, which would be cumbersome with discrete compounding. Why Use Continuous Compounding? Mathematical convenience: Continuous rates simplify calculus-based derivations Handling variable rates: Time-varying rates integrate smoothly into formulas Approximation quality: Continuous compounding closely approximates daily compounding (the practical limit for most financial markets) For exam purposes, focus on discrete formulas unless your course specifically covers continuous compounding. Most applications use annual, quarterly, or monthly compounding. </extrainfo> Summary of Key Formulas | Calculation | Formula | |---|---| | Present Value of Single Sum | $PV = FV \cdot (1+i)^{-n}$ | | Future Value of Single Sum | $FV = PV \cdot (1+i)^{n}$ | | PV of Ordinary Annuity | $PV = A \cdot \frac{1-(1+i)^{-n}}{i}$ | | PV of Annuity Due | $PV{\text{due}} = PV{\text{ordinary}} \cdot (1+i)$ | | FV of Ordinary Annuity | $FV = A \cdot \frac{(1+i)^{n}-1}{i}$ | | FV of Annuity Due | $FV{\text{due}} = FV{\text{ordinary}} \cdot (1+i)$ | | PV of Growing Annuity | $PV = A \cdot \frac{1-\left(\frac{1+g}{1+i}\right)^{n}}{i-g}$ | | PV of Perpetuity | $PV = \frac{A}{i}$ | | PV of Growing Perpetuity | $PV = \frac{A}{i-g}$ | Test-Taking Tips Identify the annuity type: Always determine whether payments occur at period's end (ordinary) or period's beginning (due) before selecting your formula. Check for growth: If payments change each period, you need a growing annuity formula, not an ordinary one. Verify rate vs. growth assumptions: For perpetuities and growing annuities, confirm that the discount rate exceeds the growth rate. Watch your discount rate period: If payments are monthly but you're given an annual rate, convert the annual rate to a monthly rate ($i{\text{monthly}} = i{\text{annual}}/12$) and adjust $n$ accordingly. Annuity due adjustment: Remember that annuity due is always worth more than ordinary annuity. If your due calculation yields a lower value, you've made an error.