Time Value of Money: Core Concepts, Formulas, and Annuity Applications

Time Value of Money: Core Concepts

• Money has time value: a dollar today can be invested to earn more in the future; therefore future value (FV) and present value (PV) are linked through interest and compounding.
• Key terms:
  • Principal (P): initial amount invested.
  • Interest rate (i): per-period rate; when rate is annual, we may have multiple compounding periods per year.
  • Nominal annual rate (I): stated annual rate; frequency of compounding is m times per year.
  • Per-period rate: $i_{per} = \frac{I}{m}.$
  • Number of periods: $n = t \times m,$ where t is time in years and m is the number of compounding periods per year.
  • Present value (PV): current value of a future sum or cash flows.
  • Future value (FV): value of an investment after compounding over n periods.
  • Effective annual yield (EAY): the equivalent annual yield when compounding frequency is more than once per year.
• Core idea: FV and PV depend on compounding frequency and the time horizon.

Compounding and Future Value

• Basic formula for future value with periodic compounding (ordinary annuity with one lump sum):
  • $FV = P(1+i)^{n}.$ where $i = i_{per} = \dfrac{I}{m}$ and $n = t m.$
• Example 1: Start with $P = 100$, annual rate 10%, for 5 years (annual compounding, m = 1):
  • $FV = 100(1+0.10)^5 = 100(1.1)^5 \approx 161.05.$
• Example 2: Same start, but compounding semiannually (m = 2, I = 10%):
  • Per-period rate: $i_{per} = \frac{0.10}{2} = 0.05.$
  • Number of periods: $n = 5 \times 2 = 10.$
  • $FV = 100(1+0.05)^{10} \approx 162.89.$
• Insight: Increasing compounding frequency within a year generally increases FV for the same nominal annual rate.
• Important distinction:
  • The per-period rate is not the same as the annual rate; compounding more frequently creates more opportunities for interest-on-interest.

Effective Yield and Frequency of Compounding

• Effective annual yield (EAY) captures the true annual return when compounding occurs more than once per year:
  • $\text{EAY} = \left(1+\frac{I}{m}\right)^{m} - 1.$
• Example: I = 10%, m = 2 (semiannual):
  • $\text{EAY} = (1+0.10/2)^2 - 1 = 1.1025 - 1 = 0.1025 = 10.25\%.$
• Observations:
  • As frequency m increases, EAY increases (up to the limit as m grows large, for fixed I).
  • More frequent compounding increases both FV (for a given PV) and the nominal yield seen by lenders/investors.
• Real-world analogies:
  • Annual compounding: like a once-a-year interest payout (e.g., some bonds or loans).
  • Semiannual, quarterly, monthly: more frequent interest accrual (used by bonds, mortgages, credit cards).
  • Very high frequency (daily, 365 times/year or continuous) approaches a continuous compounding limit.

Present Value and Discounting

• Present value relates future cash flows to today using the interest factor:
  • For a single future sum FV, with per-period rate i and n periods:
  • $PV = \frac{FV}{(1+i)^{n}}.$
  • For a stream of equal payments R per period (ordinary annuity, payments at end of period):
  • Present value factor: $PV = R \cdot \frac{1 - (1+i)^{-n}}{i}.$
• Example: $20{,}000 in 3 years; annual compounding at 15%.
  • Per-period rate: since compounding annually, i = 0.15, n = 3.
  • $PV = \frac{20000}{(1+0.15)^3} = \frac{20000}{1.15^3} \approx 13{,}150.$
• Another key point from the discussion:
  • Increasing compounding frequency at a fixed nominal rate lowers the present value of a future cash flow (discounting becomes more aggressive with more periods).
• Practical takeaway:
  • PV soundly reflects risk, inflation, and opportunity costs; asset pricing values are the present value of future cash flows.

Annuities: Future Value and Present Value

• Annuity concept: a sequence of equal payments (or receipts) at regular intervals.
• Annuity types:
  • Ordinary annuity (payments at end of period, arrears).
  • Annuity due (payments at beginning of period, advance).
• Future value of an ordinary annuity (payments R at end of each period, for n periods):
  • $FV_{ann} = R \frac{(1+i)^n - 1}{i}.$
• Present value of an ordinary annuity:
  • $PV_{ann} = R \frac{1 - (1+i)^{-n}}{i}.$
• Annuity due (payments at beginning of each period) adjusts by a factor of (1+i):
  • $FV_{ann, due} = R \frac{(1+i)^n - 1}{i} \cdot (1+i).$
  • $PV_{ann, due} = R \frac{1 - (1+i)^{-n}}{i} \cdot (1+i).$
• Example 1 (ordinary annuity): annual payments of $1{,}000 for 5 years at i = 5%
  • FV: $FV_{ann} = 1000 \frac{(1+0.05)^5 - 1}{0.05} \approx 5{,}525..$
  • PV: $PV_{ann} = 1000 \frac{1 - (1+0.05)^{-5}}{0.05} \approx 4{,}329..$
• Example 2 (PV of rents): monthly rent payments of $2,000 for 12 months at annual discount rate 10%
  • Monthly rate: $i_m = \frac{0.10}{12} \approx 0.0083333.$
  • PV factor for 12 months: $\text{PV factor} = \frac{1 - (1+i<em>m)^{-12}}{i</em>m} \approx 11.47.$
  • Present value: $PV = 2000 \times 11.47 \approx 22{,}940.$
• Example 3 (property with rent and sale): End-of-year rents of $30{,}000 for 6 years + sale price $750{,}000 at year 6; discount rate 10%
  • PV of rents (ordinary annuity): $PV_{rent} = 30000 \frac{1 - (1+0.10)^{-6}}{0.10} \approx 130{,}658.$
  • PV of sale: $PV_{sale} = \frac{750000}{(1+0.10)^6} \approx 423{,}420.$
  • Total PV: $PV<em>{total} = PV</em>{rent} + PV_{sale} \approx 554{,}078.$

Worked Calculator-oriented Procedures (Financial Calculators)

• General workflow (ordinary annuity, payments at end):
  • Set N = number of total periods (e.g., 5 for five years with annual payments).
  • Set I/Y = per-period interest rate (e.g., 5 for 5%).
  • Set PMT = payment per period (negative if cash inflow to you, positive if paying out; convention varies by model).
  • Set FV = 0 for present value problems focused on PV; or compute FV for accumulation problems.
  • Compute PV or FV as required.
• For annuity due (payments at beginning):
  • Either enable BGN mode (Begin) on the calculator or adjust the final result by multiplying by (1+i).
  • On many calculators, to remove Begin mode, use: 2nd PMT, 2nd ENTER (varies by model).
• Key takeaway about BGN vs END: BGN mode assumes payments occur at the beginning of each period (annuity due); END mode assumes payments occur at end of each period (ordinary annuity).
• Present value factor (the denominator in PV) is the “interest factor”: the discounting term that reduces future cash flows to today.

Practical Insight: Why This Matters in Real World

• Asset pricing perspective: the value of an asset equals the present value of all cash flows it can generate; paying more than that PV implies you’re overpaying relative to risk-adjusted return.
• The choice of compounding frequency affects yield and portfolio outcomes (loans, mortgages, bonds, credit cards, savings):
  • More frequent compounding generally increases the effective yield, increasing the amount owed or the return earned.
  • Higher compounding frequency lowers the PV of future cash inflows for a given nominal rate, increasing the cost of waiting (discounting effect).

Quick Recap: Takeaways and Formulas to Remember

• Future value with periodic compounding: $FV = P(1+i)^n, \, i = \frac{I}{m}, \ n = t m.$
• Effective annual yield: $\text{EAY} = \left(1+ \frac{I}{m}\right)^m - 1.$
• Present value of a single future sum: $PV = \frac{FV}{(1+i)^n}.$
• Present value of an ordinary annuity: $PV_{ann} = R \frac{1 - (1+i)^{-n}}{i}.$
• Future value of an ordinary annuity: $FV_{ann} = R \frac{(1+i)^n - 1}{i}.$
• Annuity due adjustments: multiply by (1+i) to convert from ordinary to due, for both PV and FV.
• Examples illustrate the same formulas with concrete numbers (e.g., 5% vs 10% rates, annual vs monthly compounding, 6-year horizons, etc.).
• When solving with a calculator, be mindful of: N, I/Y, PMT, PV, FV, and whether payments are in arrears (END) or advance (BGN/BEGIN).