Week 1: Principles of Finance & Decision Rules Section B Core

Master TVM, NPV, IRR, and capital budgeting decisions — foundational skills for the 70-mark problem-solving section.

      🎯 Exam Relevance: These concepts underpin Section B's valuation questions. You must be able to: calculate EAR, work with annuities, compute NPV/IRR, and apply decision rules under capital constraints. The exam rewards showing formula → substitution → calculation → interpretation.
    

1. Effective Annual Rate (EAR)

EAR converts any stated rate to its true annual equivalent, enabling apples-to-apples comparison of different compounding frequencies.

EAR = (1 + r/m)^m − 1

where: r = stated annual rate, m = compounding periods per year

💡 Exam Application:

Loans: Choose the lower EAR (you pay less interest)
Investments/CDs: Choose the higher EAR (you earn more interest)

📝 Example: Tutorial Problem 1

Loan A: 7.45% compounded daily

EAR = (1 + 0.0745/365)³⁶⁵ − 1 = 7.73%

Loan B: 7.5% compounded semi-annually

EAR = (1 + 0.075/2)² − 1 = 7.64%

Decision: For a loan, choose B (lower effective rate = less interest paid).
For a CD, choose A (higher effective rate = more interest earned).

2. Annuities: Present & Future Value

Annuities are series of equal payments. Master both PV (for loans, valuations) and FV (for savings goals).

Present Value of Ordinary Annuity:
PV = C/r × [1 − 1/(1+r)^N]

Future Value of Ordinary Annuity:
FV = C/r × [(1+r)^N − 1]

⚠️ Timing Matters:

Ordinary annuity: Payments at END of each period
Annuity due: Payments at START of each period (multiply PV by (1+r))

📝 Example: Tutorial Problem 2 (MBA Savings)

Goal: $25,000 in years 2016 and 2017. Uncle makes 5 equal payments from 2010-2014. Rate = 5%.

Find PV of tuition at year 2015

PV = 25,000/0.05 × [1 − 1/(1.05)²] = $46,485.26

Discount back to year 2009

PV₂₀₀₉ = 46,485.26 / (1.05)⁶ = $34,688.02

Solve for annual payment

34,688.02 = C/0.05 × [1 − 1/(1.05)⁵]

C = 34,688.02 / 4.3295 = $8,012

2010

−$8,012

2011

−$8,012

2012

−$8,012

2013

−$8,012

2014

−$8,012

2015

—

2016

+$25,000

2017

+$25,000

3. Loan Amortization

Understand how each payment splits between interest and principal, and how to find remaining balance.

Payment (PMT):
PMT = Loan × r / [1 − (1+r)^−N]

Remaining Balance after payment k:
Balance_k = PMT/r × [1 − (1+r)^−(N−k)]

📝 Example: Tutorial Problem 3 (Car Purchase)

Scenario: $20,000 car, $5,000 down, borrow $15,000 at 1% EAR over 5 years. Alternative: pay $17,000 cash. Your bank pays 8%.

Calculate annual payment

PMT = 15,000 × 0.01 / [1 − (1.01)⁻⁵] = $3,090.60

Balance after year 3

Balance₃ = 3,090.60/0.01 × [1 − (1.01)⁻²] = $6,089.70

Year	Begin Balance	Payment	Interest	Principal	End Balance
1	$15,000	$3,090.60	$150.00	$2,940.60	$12,059.40
2	$12,059.40	$3,090.60	$120.59	$2,970.00	$9,089.40
3	$9,089.40	$3,090.60	$90.89	$2,999.70	$6,089.70
4	$6,089.70	$3,090.60	$60.90	$3,029.70	$3,060.00
5	$3,060.00	$3,090.60	$30.60	$3,060.00	$0.00

Compare options (use YOUR opportunity cost of 8%)

PV of loan option = $5,000 + $3,090.60 × [1 − (1.08)⁻⁵]/0.08 = $17,339.86

PV of cash option = $17,000

Decision: Pay cash ($17,000) — it's cheaper in PV terms than the loan ($17,339.86).
Key insight: The low 1% loan rate seems attractive, but we evaluate using OUR discount rate (8%).

4. NPV & IRR Decision Rules

Net Present Value:
NPV = −C₀ + Σ C_t/(1+r)^t

Internal Rate of Return:
IRR: the rate r where NPV = 0

Decision Rules:

Method	Accept If	Pros	Cons
NPV	NPV > 0	Directly measures value creation; additive	Requires known discount rate
IRR	IRR > cost of capital	Easy to communicate (%)	Multiple IRRs possible; misleading for mutually exclusive projects

⚠️ Multiple IRRs Warning: When cash flows change sign more than once (e.g., negative, positive, negative), there can be multiple IRRs. In such cases, rely on NPV.

📝 Example: Tutorial Problem 4 (Innovation Co.)

Project: −$5M upfront, +$1M/year for 10 years, −$100K/year support cost in perpetuity.

NPV(r) = −5 + 1/r × [1 − (1+r)⁻¹⁰] − 0.1/r

Discount Rate	NPV	Decision
2.745784%	≈ $0	IRR #1 (breakeven)
5.438761%	$721,162	Accept ✓
10.879183%	≈ $0	IRR #2 (breakeven)

              Interpretation: Two IRRs exist because cash flows flip signs twice (initial outflow, 10-year inflows, then perpetual outflow). NPV > 0 only when the discount rate is between the two IRRs (2.75% to 10.88%).
            

5. Profitability Index (PI) for Capital Rationing

PI = NPV / Initial Investment

Interpretation: NPV per dollar invested — higher is better when capital is constrained

When to Use PI vs NPV:

Unlimited capital: Rank by NPV (take all NPV > 0)
Capital rationing: Rank by PI to maximize total NPV given budget constraint

📝 Example: Tutorial Problem 5

Budget: $1,000,000. Choose from projects A-F.

Project	Cost	NPV	PI	Rank (NPV)	Rank (PI)
A	$200K	$100K	0.50	3	2
B	$500K	$120K	0.24	2	6
C	$400K	$300K	0.75	1	1
D	$200K	$75K	0.38	4	4
E	$100K	$30K	0.30	6	5
F	$100K	$40K	0.40	5	3

Ranking by NPV (Incorrect for rationing):

Select: C ($400K) + B ($500K) + F ($100K) = $1,000,000

Total NPV = $300K + $120K + $40K = $460,000

Ranking by PI (Correct for rationing):

Select: C ($400K) + A ($200K) + F ($100K) + D ($200K) + E ($100K) = $1,000,000

Total NPV = $300K + $100K + $40K + $75K + $30K = $545,000

PI method creates $85,000 more value! This happens because PI prioritizes efficiency (NPV per dollar), allowing more projects to fit within the budget.

Practice Problems

Work through these problems showing all steps. Click to reveal solutions.

Problem 1: EAR Comparison (Medium)

Question: You're comparing two savings accounts:

Account A: 4.8% compounded monthly
Account B: 4.85% compounded quarterly

Which account should you choose?

Problem 2: Retirement Savings (Hard)

Question: You want to retire in 30 years and withdraw $60,000 per year for 25 years of retirement (first withdrawal at year 31). If the interest rate is 6% throughout, how much must you save at the end of each year for the next 30 years?

Problem 3: NPV with Multiple Cash Flow Types (Hard)

Question: A project requires $2 million today. It generates:

$400,000/year for years 1-5
$300,000/year for years 6-10
A salvage value of $500,000 at year 10

If the cost of capital is 10%, should you accept the project?

Problem 4: Capital Rationing with PI (Medium)

Question: You have $500,000 to invest. Choose the optimal combination:

Project	Cost	NPV
W	$150,000	$45,000
X	$250,000	$60,000
Y	$200,000	$70,000
Z	$300,000	$90,000

Project	Cost	NPV	PI	Rank
W	$150K	$45K	0.30	3
X	$250K	$60K	0.24	4
Y	$200K	$70K	0.35	1
Z	$300K	$90K	0.30	2 (tie)

Problem 5: Loan vs Cash Decision (Exam-Style)

Question: A machine costs £50,000. You can either:

Option A: Pay cash today
Option B: Put £10,000 down, finance the rest at 3% over 4 years (annual payments)

Your opportunity cost of capital is 9%. Which option should you choose? Show the amortization schedule for Option B.

Year	Balance	Payment	Interest	Principal
1	£40,000	£10,767	£1,200	£9,567
2	£30,433	£10,767	£913	£9,854
3	£20,579	£10,767	£617	£10,150
4	£10,429	£10,767	£313	£10,429

Quick Quiz: Test Your Understanding

Answer these conceptual and computational questions. Score yourself!

Financial Calculators

Use these tools to check your work and build intuition.

1. EAR Calculator

Stated Annual Rate (%)

Compounding Frequency

2. Annuity PV Calculator

Payment (C)

Interest Rate (%)

Number of Periods (N)

3. Loan Payment Calculator

Loan Amount

Annual Rate (%)

Years

4. NPV Calculator

Initial Investment (negative)

Discount Rate (%)

Cash Flows (comma-separated, year 1 onwards)

Enter annual cash flows separated by commas. The last value can repeat with notation: "900000*10" for 10 years of $900,000

5. Profitability Index Calculator

NPV

Initial Investment

📋 Week 1 Exam Checklist

Before moving on, make sure you can:

☐ Convert between stated rates and EAR for any compounding frequency
☐ Calculate PV and FV of ordinary annuities
☐ Build a loan amortization schedule and find remaining balance
☐ Compare financing options using the correct discount rate (your opportunity cost)
☐ Compute NPV and make accept/reject decisions
☐ Recognize when multiple IRRs may exist
☐ Apply PI for capital rationing problems
☐ Always show: Formula → Substitution → Calculation → Interpretation

Week 1: Principles of Finance & Decision Rules Section B Core

1. Effective Annual Rate (EAR)

💡 Exam Application:

2. Annuities: Present & Future Value

Find PV of tuition at year 2015

Discount back to year 2009

Solve for annual payment

3. Loan Amortization

Calculate annual payment

Balance after year 3

Compare options (use YOUR opportunity cost of 8%)

4. NPV & IRR Decision Rules

Decision Rules:

5. Profitability Index (PI) for Capital Rationing

Ranking by NPV (Incorrect for rationing):

Ranking by PI (Correct for rationing):

Practice Problems

Calculate EAR for Account A

Calculate EAR for Account B

Find PV of retirement withdrawals at year 30

Find annual savings to accumulate $767,004

PV of years 1-5 cash flows

PV of years 6-10 cash flows

PV of salvage value

Calculate NPV

Calculate PI for each project

Select by PI ranking within budget

Calculate loan payment

Amortization Schedule

Compare using YOUR discount rate (9%)

Quick Quiz: Test Your Understanding

Financial Calculators

1. EAR Calculator

2. Annuity PV Calculator

3. Loan Payment Calculator

4. NPV Calculator

5. Profitability Index Calculator

📋 Week 1 Exam Checklist