Fundamentals of Finance M1T2: Present Value

I. Time Value of Money: Foundations and Formulas

A. Simple vs. Compound Interest

  1. Simple Interest:
    Gains are linear over time:

FV=P(1+rt)

  1. Compound Interest:
     Returns are exponential due to reinvestment:

FV= P(1 + r)^t

Insight: The reinvestment effect becomes dramatic over long horizons (e.g., 100 years: $100 grows to $800 under simple interest, but $86,771 under compounding at 7%).

Conclusion: Compound interest dominates in real-world finance, as it reflects reinvestment and compounding frequency.

B. Present Value (PV)

To reverse compound growth and determine today’s equivalent of a future cash flow:


PV=Ct(1+r)−tPV=\frac{C_{t}}{\left(1+r_{}\right)^{-t}}

Where:

  • Ct: cash flow at time t

  • r: discount rate

  • (1+r)−t\left(1+r\right)^{-t} : Discount factor

II. Annuities and Perpetuities

A. Ordinary Annuities (Finite Streams)

Definition: A series of fixed, periodic payments.

PV Formula:

                                PV=C⋅[1−(1+r)−tr]PV=C\cdot[\frac{1-(1+r)^{-t}}{r}]

Application: Mortgage payments, retirement savings, coupon bonds.

Annuity Factor Notation:

                                  AFtr=[1r−1r(1+r)t]AF_{t}^{r}=[\frac{1}{r}-\frac{1}{r\left(1+r\right)^{t}}]

Case Example:
$0.5M loan over 15 years at 4% ⇒ Annual payment ≈ $45,000.

Annual payment = Cashflow x Annuity Factor

B. Future Value of Annuity

                                    FV=C⋅[(1+r)t−1r]FV=C\cdot\left\lbrack\frac{\left(1+r\right)^{t}-1}{r}\right\rbrack

C. Perpetuities

Definition: A stream of fixed payments forever.

                                PV=CrPV=\frac{C}{r}

If r doubles, PV halves — explains high interest rate sensitivity in valuation (e.g., preferred stock, utilities).

III. Growing Annuities and Perpetuities

A. Growing Annuity (Finite Horizon with Growth)

                               PV=C⋅[1−(1+g1+r)tr−g],r≠gPV=C\cdot\left\lbrack\frac{1-\left(\frac{1+g}{1+r}\right)^{t}}{r-g}\right\rbrack,r\ne g

IF r = g:                                        PV=t⋅C1+gPV=\frac{t\cdot C}{1+g}

Use Case: Salaries, dividends, education costs with consistent growth.

B. Growing Perpetuity

                                                PV=Cr−gPV=\frac{C}{r-g} , only valid if r > g

Application: Dividend discount model for equities (Gordon Growth Model).

C. Delayed Annuities and Perpetuities

PV of a perpetuity starting in s years:

                                    PV=Cr⋅1(1+r)s−1PV=\frac{C}{r}\cdot\frac{1}{\left(1+r\right)^{s-1}}

PV of t-year annuity starting in s years:

                                    PV=(C⋅AFtr)⋅1(1+r)s−1PV=\left(C\cdot AF_{t}^{r}\right)\cdot\frac{1}{\left(1+r\right)^{s-1}}

Insight: Break complex cash flows into present-valued components to simplify valuation.

IV. Compounding Frequency and Interest Rate Comparisons

A. Compounding within the Year

Let:

  • ra : Stated Annual Interest Rate (SAIR)

  • m: Number of compounding periods per year

  • Period Rate: ra/m

Key Variables in Present Value and Annuity Formulas

Symbol

Meaning

C

Cash Flow – The amount of money received or paid at each time period (usually annually).

r

Discount Rate / Interest Rate – The rate used to discount future cash flows to the present. Also called the required rate of return or opportunity cost of capital.

t

Time Periods – The total number of periods (usually years) over which the cash flows occur.

s

Start Delay – The number of periods before the first cash flow begins (used in delayed annuities/perpetuities).

g

Growth Rate – The rate at which cash flows grow over time in a growing annuity or growing perpetuity.