finance

Finance Lecture Notes: Time Value of Money, Present Value, Annuities, and Core Financial Concepts

  • Course context and agenda

    • Review of chapters 1–3 for upcoming quizzes; focus today on chapters 4–6.

    • Topics include cash flow, forms of business, balance sheet, income statement, assets, liabilities, equity, and moving toward time value of money concepts (present value, discounting, annuities).

    • Emphasis on staying current with the material and using the board to illustrate ideas.

  • Forms of business and finance roles

    • Forms mentioned: sole proprietorship, partnership, LLC, and corporations with stock, board of directors, CEO, shareholders.

    • Proprietorship vs proprietary (clarify: a sole proprietorship means the owner has ownership/control; ‘proprietor’ is the owner).

    • Legal forms require registration, licenses, and taxes; CFO role in finance is to present option scenarios to management and support decision-making.

    • The goal of many businesses is profitability and value creation; equity vs debt distinction is important (home equity vs debt concept).

    • Equity vs debt: equity represents ownership; debt represents IOUs; debt requires repayment with interest; equity ownership affects control and residual claims.

  • Key financial statements and concepts

    • Balance sheet components: assets, liabilities, and equity (equity and liabilities are part of the balance sheet; assets are what the company owns).

    • Income statement components: revenues, expenses, and net income; taxes reduce net income; EBITDA defined as earnings before interest, depreciation, amortization, and taxes.

    • Depreciation concept: asset wear and tear (e.g., a food truck or refrigerator wearing out) leading to setting aside funds for replacement.

    • Asset categories: current assets (convertible to cash within one year) vs long-term assets (e.g., property, equipment, vehicles).

    • Example asset list: cars, cash, property, equipment, refrigerators.

    • Liabilities example: debt such as a note to a brother-in-law; interest as a liability; principal as a liability.

    • Free cash flow (FCF): importance in evaluating how much cash a business truly has after sustaining capital expenditures; used in ratio analysis.

    • Industry differences in ratios: different industries (AI software vs natural gas company) have different cost structures; need apples-to-apples comparison; industry context matters when interpreting ratios (DuPont model covers ~18 ratios).

    • Accounts receivable (AR): defined as revenue recognized but not yet collected; an asset on the balance sheet.

    • Cash flow perspective: cash inflows and outflows matter for ongoing operations and liquidity.

  • Time value of money and the rule of 72 (Chapter 4 focus)

    • Core takeaway: money today is worth more than money tomorrow due to inflation, risk, and uncertainty (time value of money).

    • Discounting concept: bring future money back to present value using the discount rate; rising rates and risk alter present value.

    • Market-based rate determination: interest rates for different instruments (money market funds, mortgages, private credit) reflect risk and collateral; the market tends to determine a fair rate.

    • Rule of 72 (an estimation tool): money (or value) doubles when r × t ≈ 72, where r is the annual rate (in percent) and t is time in years. Examples:

    • At 1%: doubles in about 72 years.

    • At 5%: doubles in about 72 / 5 ≈ 14.4 years.

    • At 6%: doubles in about 72 / 6 = 12 years.

    • At 9%: doubles in about 72 / 9 = 8 years.

    • Inverse use: given a target doubling time, estimate the required rate r ≈ 72 / t.

    • Present value vs future value intuition: present value is higher today; future value depends on compounding; risk, inflation, and liquidity affect discounting.

    • Time value of money in practice: the shorter the time horizon, the higher confidence in value today; higher risk or uncertainty lowers present value for a given future amount.

    • Quick variability discussion: the same dollar grows differently under different rates; higher rate accelerates growth (e.g., 10% vs 5%).

    • Example intuition with inflation and risk: mortgage and asset-backed securities have collateral; higher risk requires higher return; liquidity affects current value (e.g., hotels with low liquidity require higher yields).

  • Present value and future value formulas (single sums)

    • Present value (PV) of a future amount FV:
      PV=racFV(1+r)nPV = rac{FV}{(1 + r)^n}

    • Future value (FV) of a present amount PV:
      FV=PVimes(1+r)nFV = PV imes (1 + r)^n

    • Illustrative example: selling you a guaranteed $5{,}000 in ten years. With rate r, the value today would be
      PV=rac5000(1+r)10.PV = rac{5000}{(1 + r)^{10}}.

    • A concrete example: $1{,}000 today growing at 5% for 1 year yields
      FV=1000imes(1+0.05)1=1050.FV = 1000 imes (1 + 0.05)^1 = 1050.

    • At 7 years, $1{,}000 grows to
      FV=1000imes(1.05)7ext(roughly1407ext).FV = 1000 imes (1.05)^7 ext{ (roughly } 1407 ext{).}

    • Present value of a future $1{,}000 in 7 years at 6%:
      PV=rac1000(1+0.06)7ext(roughly665ext).PV = rac{1000}{(1 + 0.06)^7} ext{ (roughly } 665 ext{).}

  • Annuities: series of equal payments over time

    • Definition: An annuity is a series of payments (not a single payment). Examples include mortgage payments and regular retirement payments.

    • Present value of an annuity (ordinary annuity, payments at end of each period):
      PVextannuity=PMTimesrac1−(1+r)−nrPV_{ ext{annuity}} = PMT imes rac{1 - (1 + r)^{-n}}{r}

    • Future value of an annuity (payments at end of each period):
      FVextannuity=PMTimesrac(1+r)n−1rFV_{ ext{annuity}} = PMT imes rac{(1 + r)^n - 1}{r}

    • Example 1: $1{,}200$ per year for 5 years at 6%:
      PV=1200imesrac1−(1.06)−50.06≈1200imes4.2123≈5,055.PV = 1200 imes rac{1 - (1.06)^{-5}}{0.06} \approx 1200 imes 4.2123 \approx 5{,}055.

    • Example 2: $100$ per month for 5 years (monthly payments) at 6% annual rate

    • Convert to monthly rate: $i = 0.06/12 = 0.005$; $n = 60$ months.

    • Present value:
      PV=100imesrac1−(1+0.005)−600.005≈100imes51.98≈5,198.PV = 100 imes rac{1 - (1 + 0.005)^{-60}}{0.005} \approx 100 imes 51.98 \approx 5{,}198.

    • Observation: monthly payments often have a higher present value than the same total annual amount paid as a lump sum or annual payments, due to the earlier receipt of funds.

    • Key takeaway: the present value of a stream of payments is higher when payments are received sooner (earlier cash flows carry more value).

  • Timeline method and problem-solving approach

    • Build a timeline to visualize when cash flows occur (present vs future) and to apply the correct discount or compounding factors.

    • Steps to solve PV/FV problems:

    • Identify time horizon (n), rate (r), and cash flow amounts (PMT or FV).

    • Choose the appropriate formula (FV of a lump sum, PV of a lump sum, FV of annuity, PV of annuity).

    • Compute using the factor (1 + r)^n or its reciprocal for discounting.

    • Practical use: for a lottery payout vs a lump-sum, use annuity formulas to determine equivalent present value under a given rate.

  • Monthly vs annual distributions and practical implications

    • Monthly distributions vs annual distributions for the same total amount yield different present values due to timing of cash flows.

    • Example contrast (conceptual): receiving $1,200 per year for 5 years versus $100 per month for 5 years; the monthly option typically has a higher PV due to earlier receipt in each year.

    • In financial planning or debt scenarios (mortgage), lenders generally prefer more frequent payments (monthly) to maintain cash flow and liquidity management.

  • Real-world examples and discussion points

    • Art as an investment example: scarcity can drive higher compounding-like growth; an art collection can appreciate at high rates under certain market conditions, illustrating alternative assets’ return profiles.

    • Inflation and commodity pricing anecdotes: long-term price levels can rise (e.g., bread, gas, etc.), reinforcing why money today is more valuable than money tomorrow.

    • Risk and liquidity considerations: the denomination and marketability of an asset affect its present value; higher risk typically requires higher expected returns.

    • Capabilities of a timeline and present value approach in everyday decisions: use these methods to evaluate debt, investments, or retirement planning.

  • Quick reference: core formulas to memorize

    • Present value of a future amount:
      PV=racFV(1+r)nPV = rac{FV}{(1 + r)^n}

    • Future value of a present amount:
      FV=PVimes(1+r)nFV = PV imes (1 + r)^n

    • Present value of an annuity (end-of-period payments):
      PVextannuity=PMTimesrac1−(1+r)−nrPV_{ ext{annuity}} = PMT imes rac{1 - (1 + r)^{-n}}{r}

    • Future value of an annuity (end-of-period payments):
      FVextannuity=PMTimesrac(1+r)n−1rFV_{ ext{annuity}} = PMT imes rac{(1 + r)^n - 1}{r}

    • Rule of 72 (order of magnitude guidance):

    • Doubling time t (years) ≈
      text(years) ≈ 72r%t ext{ (years)} \,\approx \, \frac{72}{r\%}
      where r is the annual rate in percent; equivalently, if r is in decimal, t ≈ 72 / (100 r).

    • Present value of a single sum vs annuity distinction: present value is a single sum discounted; annuity PV sums multiple discounted cash flows.

  • Closing note on exam strategy and integration

    • Focus on present value and future value concepts across problems.

    • Use the timeline method to translate word problems into PV/FV equations.

    • Practice with both lump-sum and annuity scenarios to build intuition about timing, rate, and duration.

    • Relate discount rates to market realities (risk, liquidity, collateral) to determine appropriate r for a given problem.

  • End-of-lecture prompts and student engagement