Financial Context of Business III: Discounting and Investment Appraisal

Introduction

  • Investment appraisal techniques are crucial for deciding where to 'take' the business.

  • Investment decisions impact the success and growth of the business.

The Investment Decision-Making Process

  • Origination of proposals: Generating various alternatives.

  • Project screening: Assessing projects based on long-term aims.

  • Analysis and acceptance: Detailed financial analysis and qualitative considerations.

  • Monitor and review: Tracking progress and comparing against budgets.

Time Value of Money

  • Money received today is worth more than the same amount in the future.

  • Reasons:

    • Potential for earning interest/cost of finance.

    • Impact of inflation.

    • Effect of risk.

  • Discounted cash flow (DCF) techniques account for the time value of money.

Potential for Earning Interest

  • Cash received sooner can be invested to earn interest.

  • $1 now is more valuable than $1 in the future because of potential investment returns.

Impact of Inflation

  • Inflation erodes purchasing power; money received today buys more than the same amount later.

Risk

  • Earlier cash flows are more certain and considered more valuable.

Interest

Simple Interest

  • Interest is paid only on the original principal.

Illustration 1 - Simple Interest

  • Investing P = $200 for n=3n = 3 years at an annual interest rate of r=5r = 5%:

    • Annual interest: 200 × 0.05 = $10

    • Total interest after 3 years: 3 × 10 = $30

    • Final sum: 200 + 30 = $230

Formula

  • If PP is invested at a fixed interest rate of rr per annum, the interest earned each year is r×Pr × P.

  • After nn years, the total value VV is:

    • V=P+r×P×nV = P + r × P × n

    • V=P(1+r×n)V = P(1 + r × n)

Illustration 2 - Simple Interest with Non-Annual Time Periods

  • Investing P = $2,000 in a deposit account at r=0.1r = 0.1% per month for n=24n = 24 months:

    • V = 2,000 × (1 + 0.001 × 24) = 2,000 × 1.024 = $2,048

Compound Interest

  • Interest is paid on the original principal plus any accrued interest.

Illustration 3 - Compound Interest

  • Investing P = $200 for n=3n = 3 years at r=5r = 5% compound interest:

    • Year 1: Interest = 55% × $200 = $10, Total = 200 + 10 = $210

    • Year 2: Interest = 55% × $210 = $10.50, Total = 210 + 10.50 = $220.50

    • Year 3: Interest = 55% × $220.50 = $11.025, Total = 220.50 + 11.025 = $231.525

  • Each year, the sum grows by a factor of 1.05.

    • Short cut: 200 × (1.05) × (1.05) × (1.05) = 200 × (1.05)^3 = $231.525

Formula

  • If PP is invested for nn years at an annual interest rate rr, compounded annually, the future value VV is:

    • V=P(1+r)nV = P(1 + r)^n

Time Value of Money as an Annual Interest Rate

  • Expressed as:

    • Discount rate

    • Required return

    • Cost of capital

Example

  • Required return rate of 10% per annum:

    • Investing $100 now results in 100 × 1.10 = $110 in one year.

    • $100 now is equivalent to $110 in one year.

    • $110 in a year is worth \$110 / 1.10 = $100 today (90.9% of its actual value).

Discounted Cash Flows

  • Cash flows at different times need to be converted to a common point in time (usually the present day).

  • Process: Discounting.

Discounting

  • Converting future cash flows into present values.

Illustration 6 - Discounting

  • (a) Find the present value of $200 payable in 2 years at an investment rate of 7% per annum.

    • 200=P(1+0.07)2200 = P(1 + 0.07)^2

    • P = \frac{$200}{1.1449} = $174.69

    • Paying $174.69 now is equivalent to paying $200 in 2 years at a 7% interest rate.

  • (b) Find the present value of $350 receivable in 3 years at an investment rate of 6% per annum.

    • 350=P(1+0.06)3350 = P(1 + 0.06)^3

    • P = \frac{$350}{1.191016} = $293.87

Discounting a Single Sum

  • Present value (PV) is the cash equivalent now of money receivable/payable in the future.

  • Formula:

    • P=V(1+r)n=V×(1+r)−nP = \frac{V}{(1 + r)^n} = V × (1 + r)^{-n}

  • (1+r)−n(1 + r)^{-n} is the discount factor (DF).

Example

  • If r=10r = 10% and n=5n = 5, then DF=(1.1)−5=0.621DF = (1.1)^{-5} = 0.621

  • The DF can be found in PV tables.

Net Present Value (NPV)

  • The total of individual present values.

  • Represents the net gain or loss on a project, considering the timing of cash flows and the time value of money.

NPV Criteria

  • If NPV > 0: Project is financially viable.

  • If NPV = 0: Project breaks even.

  • If NPV < 0: Project is not financially viable.

  • Choose the project with the highest NPV when considering mutually exclusive projects.

  • NPV indicates the impact of the project on shareholder wealth.

Illustration 7 - Net Present Value

  • A machine costs $10,000 and generates $2,500 per annum for 5 years, then is scrapped for $500. Interest rate is 5%.

Annuities

  • Arrangement where cash is received or paid in constant annual amounts.

  • Payments can be until death or a guaranteed minimum term is reached.

  • Payments can be deferred.

  • NPV is relevant when comparing annuities with different time periods.

Formula for NPV of a $1 Annuity

  • Annuity factor = 1−(1+r)−nr\frac{1 - (1 + r)^{-n}}{r}

Present Value of an Annuity Cash Flow

  • PV = future cash flow × annuity factor

  • Cumulative present value tables can be used to find annuity factors.

Illustration 8 - Annuities

  • Machine costs $10,000, contributes $2,500 per annum for 5 years, and has a scrap value of $500. Interest rate is 5%.

Perpetuities

  • Annuity where payments continue forever.

  • Useful for ensuring continued payments to descendants or good causes.

  • Constant payments decrease in value due to inflation.

Present Value of a Perpetuity Cash Flow

  • PV = future cash flow × perpetuity factor

  • Perpetuity factor = 1r\frac{1}{r}

Test Your Understanding

  • Bob wins $12,000 per annum for life, payable to descendants at a 6% interest rate.

Internal Rate of Return

  • If NPV is positive, the project is more profitable than investing at the discount rate.

  • If NPV is negative, the project is less profitable than a simple investment at the discount rate.

  • NPV falls as the discount rate increases.

  • The internal rate of return (IRR) is the discount rate at which the NPV is zero.

Investment Appraisal Methods

  1. Accept if NPV > 0.

  2. Accept if actual discount rate < project IRR.

Calculating IRR

  • IRR is the discount rate at which NPV is zero.

  1. Find a discount rate where the NPV is slightly positive.

  2. Find a discount rate where the NPV is slightly negative.

  3. Use linear interpolation to find the point where NPV is zero.

Terminal Values and Sinking Funds

  • Instead of discounting cash flows to the present day, they are compounded to the end of the project.

Illustration 11 - Terminal Values

  • Investment of $3,000 initially, then $1,800 at the end of the first, second, and third years, and $600 at the end of the fourth year. Interest rate is 6.5%.

Calculation

  • $3,000 invested for 5 years grows to: 3,000 × 1.065^5 = $4,110.26

  • Three sums of $1,800 invested for 4, 3, and 2 years grow to: 1,800 × (1.065^4 + 1.065^3 + 1.065^2) = $6,531.55

  • $600 invested for 1 year grows to: 600 × 1.065 = $639

  • Total value at the end of 5 years is: 4,110.26 + 6,531.55 + 639 = $11,280.81

Sinking Fund

  • Constant amount invested each year to reach a specified future value.

Illustration 12 - Sinking Funds

  • A company needs $50,000 in 6 years and makes six annual investments starting immediately at 5.5%.

Timeline Diagram
Calculation

  • 50,000=P(1.055)6+P(1.055)5+P(1.055)4+P(1.055)3+P(1.055)2+P(1.055)50,000 = P(1.055)^6 + P(1.055)^5 + P(1.055)^4 + P(1.055)^3 + P(1.055)^2 + P(1.055)

  • 50,000=P(1.0556+1.0555+1.0554+1.0553+1.0552+1.055)50,000 = P(1.055^6 + 1.055^5 + 1.055^4 + 1.055^3 + 1.055^2 + 1.055)

  • 50,000=P×7.26750,000 = P × 7.267

  • P = \frac{$50,000}{7.267} = $6,880