Time Value of Money (TVM) Comprehensive Study Notes

TVM overview

• Time Value of Money (TVM): money has earning potential; the timing of a payment matters because money available today can be invested to earn interest, making it worth more than the same amount received in the future.
• Key intuition: receiving money today allows investment and compounding; delaying receipt creates opportunity cost and reduces purchasing power over time due to inflation.
• Practical context: how firms manage accounts payable; given a rate, we can compute present value (PV) of a future sum or future value (FV) of a present sum.

Core concepts and definitions

• Future Value (FV) of a lump sum: $FV = PV\times (1+i)^n$
• Present Value (PV) of a lump sum: $PV = \frac{FV}{(1+i)^n}$
• Annuities: a sequence of equal payments over time; TVM applies differently depending on whether payments occur at the end or the beginning of periods.
• Compounding vs discounting: compounding converts present cash flows to future cash flows; discounting converts future cash flows to present values.
• Rates and periods: i is the per-period rate; n is the number of periods.

Why TVM matters (Key takeaways)

• A sum of money is worth more now than later because it can be invested to grow.
• Delaying investment represents a missed opportunity; growth comes from investing and compounding.
• The TVM formula considers amount, future value, earning potential (rate), and time frame.
• Number of compounding periods affects outcomes; more frequent compounding increases FV/PV.
• Inflation erodes purchasing power; higher inflation reduces the real value of money over time.

Risk factors in TVM

• Opportunity risk (time value of waiting): e.g., choose $10,000 today vs $10,000 in 2 years; today’s money has higher value due to missed opportunities if delayed.
• Inflation risk: higher inflation can erode purchasing power; if investment growth is below inflation, real value declines.
• Maturity risk: longer-term investments may require higher compensation for delay and higher risk.
• Liquidity risk: less liquid investments require higher returns to compensate for difficulty converting to cash quickly.

Inflation and purchasing power

• Inflation reduces the value of a dollar over time; if investment return is below the inflation rate, real purchasing power falls.
• Example: an asset growing at 4% annually loses purchasing power if inflation > 4%.

Maturity risk and liquidity risk (descriptions)

• Maturity premium: longer maturities imply higher sensitivity to interest rate changes; investors demand higher yields for longer commitments.
• Liquidity premium: illiquid investments must offer higher expected returns to compensate for liquidity risk.

TVM in practice: applications

• TVM underpins financial planning, stock/bond pricing, pension valuation, and project evaluation.
• TVM calculations are essential for comparing prospective projects, loans, mortgages, and savings goals.

The TVM toolkit: main calculation types

• Present value of known future values (discounting) and future value of present cash (compounding).
• Lump sums vs. annuities (payments over time).
• Net Present Value (NPV), Internal Rate of Return (IRR) are derived TVM concepts used in financial planning.

Discount rate and opportunity cost

• The discount rate measures the trade-off between present and future cash flows.
• It reflects:
• Preference for current consumption (higher preference increases the discount rate)
• Expected inflation (higher inflation raises the discount rate)
• Uncertainty of future cash flows (risk raises the discount rate)
• Opportunity cost (the returns from the next best alternative)
• A higher discount rate lowers the present value of future cash flows.
• Discounting transforms future cash flows into present value; compounding transforms present cash flows into future value.

Interest rates: components and concepts

• Real risk-free rate: baseline return with no risk, reflecting timing differences.
• Expected inflation: anticipated rise in prices; reduces purchasing power.
• Liquidity premium: compensation for less liquid investments.
• Maturity premium: compensation for interest-rate sensitivity with longer maturities.
• Nominal rate INOM: quoted rate that ignores compounding; used in contracts. Periodicity M must be specified.
• Periodic rate IPER: rate charged each period; IPER = INOM / M.
• Effective annual rate EAR: annual rate actually earned, accounting for compounding: $EAR = (1 + \frac{INOM}{M})^{M} - 1.$
• Continuous compounding gives EAR_cont = $e^{r} - 1,$ where r is the (continuous) growth rate.

Effective rates and comparisons

• Different compounding intervals yield different effective returns even if nominal rate is the same.
• To compare investments, use EAR (or EFFECT in Excel) to account for compounding.
• Table-style intuition: EAR increases with more frequent compounding for the same nominal rate.
• When M = 1 (annual compounding), EAR = INOM; otherwise EAR > INOM.

The frequency of compounding (example concepts)

• Annual: FV = PV(1+i)^n with i = INOM, n = years.
• Semiannual, monthly, daily: use IPER = INOM/M and number of periods nM = nM; EAR is computed as above.
• Continuous: use exponential growth with e as the base.

Why TVM matters for real-world decisions

• The choice between a lump sum today vs in the future depends on the opportunity cost and inflation.
• TVM informs savings goals, loan decisions, mortgage amortization, and retirement planning.

Time value of money primer: step-by-step examples

• Step 1: convert present values to future values or discount future values to present values depending on goal.
• Step 2: determine whether the asset is a lump sum or a series of payments (annuities).
• Step 3: use appropriate formulas for PV or FV of lump sums or annuities; derive NPV/IRR as needed.

Simple vs. compound interest (definitions and formulas)

• Simple interest: I = P r t; interest is earned on principal only.
• Compound interest: interest earned on principal plus accrued interest; effective growth accelerates over time.
• Compound interest formula (FV for a lump sum): $FV = P(1 + i)^n,$ where i is the periodic rate and n is the number of periods.

Practical example: a money growth intuition

• Money deposited in a savings account earns interest; over time, interest compounds and adds to principal, increasing future value.
• If money is hidden (e.g., under a mattress), inflation erodes purchasing power and the money loses potential growth.

Annuities: basics and terminology

• Annuity: a series of equal payments over time.
• Ordinary annuity (end-of-period payments): payments occur at the end of each period (type = 0 in Excel).
• Annuity due (beginning-of-period payments): payments occur at the start of each period (type = 1 in Excel).
• Perpetuity: a never-ending annuity; PV = \frac{PMT}{i}.

Key annuity formulas (per period rate i)

• FV of ordinary annuity: $FV_{annuity} = PMT \times \frac{(1+i)^n - 1}{i}.$
• PV of ordinary annuity: $PV_{annuity} = PMT \times \frac{1 - (1+i)^{-n}}{i}.$
• FV of annuity due: $FV_{due} = PMT \times \frac{(1+i)^n - 1}{i} \times (1+i).$
• PV of annuity due: $PV_{due} = PMT \times \frac{1 - (1+i)^{-n}}{i} \times (1+i).$
• PV of perpetuity: $PV_{perp} = \frac{PMT}{i}.$
• Present value of a growing annuity: $PV = \frac{C}{i - g}\left[1 - \left(\frac{1+g}{1+i}\right)^n\right],$ where C is the first payment and g is the growth rate per period.

Example problems and results (from the transcript)

• Ordinary annuity PV for $5,000 per year for 10 years at 9%: $PV = 5000 \times \frac{1 - (1+0.09)^{-10}}{0.09} \approx 32{,}088.$ FV of the same annuity: $FV = 5000 \times \frac{(1+0.09)^{10} - 1}{0.09} \approx 75{,}965.$
• 5-year ordinary annuity with $100 payments at 4%: PV ≈ $445.18; FV ≈ $312.28.
• Annuity due with $100 payments at 4% for 3 years: FVdue ≈ $324.65; PVdue ≈ $288.61.
• With 5-year ordinary annuity and 8% withdrawal or growth, use EAR adjustments for nonstandard timing (semiannual, quarterly) to compute FV/PV.

Uneven cash flows and NPV

• Uneven cash flow streams require bringing each cash flow to a common time via discounting.
• Example approach: use CF0, CF1, CF2, … in a calculator’s CFLO register and compute NPV at a given rate.
• Example result: PV of irregular cash flow sequence 0, 100, 300, 300, -50 at 4% equals $597.48 (PV).

Solving for timing and rates in FV problems

• Solving for FV with a given rate and time: FV_N = PV(1+i)^N.
• Solving for PV given FV: PV = \frac{FV}{(1+i)^N}.
• Solving for interest rate i given PV, FV, N: use a financial calculator or numerical methods (Excel RATE).
• Solving for N given PV, FV, and i: use NPER function or Rule of 72 for quick doubling estimates.
• Rule of 72: approximate years to double as $\text{Years to double} \approx \frac{72}{r}\,$ where r is the annual growth rate in percent.

Excel and calculator methods (summary)

• General FV: FV(rate, nper, pmt, pv, type)
• General PV: PV(rate, nper, pmt, fv, type)
• RATE to solve for i: RATE(nper, pmt, pv, fv, type, guess)
• NPER to solve for N: NPER(rate, pmt, pv, fv, type)
• PMT to solve for periodic payment: PMT(rate, nper, pv, fv, type)
• EAR and EFFECT functions in Excel to compare effective rates across compounding schemes.

Timelines and cash-flow visualization

• Timelines show the timing of cash flows; CF0, CF1, CF2, … mark cash flows at times 0, 1, 2, …
• End-of-period (ordinary) vs beginning-of-period (annuity due) cash flows are depicted on the timeline.

Amortization (loan repayment schedule)

• Amortization tables show how each payment is split into interest and principal.
• Example: a $1{,}000 loan at 4% with 3 equal payments of $360.35 (end of each year)
• Step 1: PMT = $360.35 (FV = 0) using PMT(0.04, 3, -1000, 0, 0).
• Step 2: Interest in Year 1: INT1 = BegBal × 0.04 = $1,000 × 0.04 = $40.
• Step 3: Principal repaid in Year 1: PRIN1 = PMT − INT1 = $360.35 − $40 = $320.35.
• Step 4: Ending balance Year 1: END BAL1 = BegBal − PRIN1 = $1,000 − $320.35 = $679.65.
• Repeat steps for subsequent years; total interest paid over loan is sum of yearly interest components; tax implications may apply.
• Illustrative amortization table shows decreasing interest and increasing principal portions over time.

Practical applications: problem examples with numbers

• Present value example: PV of $10{,}000 to receive in 10 years at 9%: PV ≈ $\frac{10000}{(1+0.09)^{10}} \approx 4{,}224.$
• Future value example: FV of $10{,}000 invested for 10 years at 9%: FV ≈ $10000\times (1+0.09)^{10} \approx 23{,}674.$
• Ordinary annuity PV example (10 years, 9%): PV ≈ $32{,}088;$ FV ≈ $75{,}965.$
• Stacey withdrawal problem: withdrawing $3,000 at the beginning of each year for 5 years at 8% requires a starting lump sum of about $11{,}978.$ End-of-year balances step through to ending balance of $3{,}000 withdrawals and growth to $0 at the end of Year 5.
• Mortgage example: $300{,}000 home price with 20% down and 30-year loan at 8% results in total interest over the life of the loan of about 399{,}570$ (roughly $21{,}319 per year times 30 years, depending on exact payment calculation).
• Cumulative interest function (loan accumulator): shows total interest paid over a 30-year period for a given principal and rate; useful for understanding total cost of debt.

Quick problem results (from the transcript)

• PV of $10,000 in 10 years at 9%: present value ≈ $4,224.
• FV of $10,000 in 10 years at 9%: future value ≈ $23,674.
• PV of a 10-year ordinary annuity of $5,000 per year at 9%: ≈ $32,088.
• FV of a 10-year ordinary annuity of $5,000 per year at 9%: ≈ $75,965.
• Present value of a 5-year ordinary annuity with $3,000 payments at 8% (beginning of year withdrawals): PV ≈ $11,978; FV path shown in the slide.
• PV of 5-year annuity due of $3,000 payments at 8%: PV ≈ $11,978 (same as starting point given the table’s structure); detailed steps show ending values and balances.
• PV of a perpetuity with $100 payments at 4%: $100 / 0.04 = $2,500.

Examples of real-world implications and takeaways

• Start saving early: delaying saving reduces the accrual of compound interest; retirement outcomes can be significantly impacted.
• Early savers benefit from a longer horizon to compound, even with modest monthly contributions.
• Matching EAR across investments with different compounding schedules is essential for fair comparison.
• Amortization demonstrates how early payments reduce principal and subsequent interest charges over time, affecting total interest and payoff date.

Notable formulas to memorize (LaTeX-ready)

• FV of lump sum: FV = PV\times (1+i)^n
• PV of lump sum: PV = \frac{FV}{(1+i)^n}
• FV of ordinary annuity: FV_{annuity} = PMT \times \frac{(1+i)^n - 1}{i}
• PV of ordinary annuity: PV_{annuity} = PMT \times \frac{1 - (1+i)^{-n}}{i}
• FV of annuity due: FV_{due} = PMT \times \frac{(1+i)^n - 1}{i} \times (1+i)
• PV of annuity due: PV_{due} = PMT \times \frac{1 - (1+i)^{-n}}{i} \times (1+i)
• PV of perpetuity: PV_{perp} = \frac{PMT}{i}
• EAR (nominal INOM compounded M times per year): EAR = \left(1 + \frac{INOM}{M}\right)^M - 1
• Continuous compounding (continuous rate r): EAR_{cont} = e^{r} - 1
• Simple interest: I = P r t
• Rule of 72 (doubling time): \text{Years} \approx \frac{72}{r}$$
• Amortization balance steps (illustrative): Beg Bal, Interest = Beg Bal × i, Principal = PMT − Interest, End Bal = Beg Bal − Principal.

Note on sources and context

• The notes above summarize the key concepts, formulas, examples, and problem-solving steps presented across the transcript slides (Pages 1–96).
• The focus is on understanding how TVM underpins financial decision-making, including savings, lending, investing, and retirement planning, with practical examples and calculator/Excel methods.