ST226
Week 1 Complete Mastery Guide
Time Value of Money & Interest Rates
Complete Coverage with:
Detailed Theory & Explanations
Exam-Optimized Formula Sheets
Fully Worked Past Paper Questions
Common Mistakes & Examiner Traps
One-Page Quick Reference
Based on LSE Course Materials 2024–25
Course Notes Chapter 1
Lecture Slides Week 1
Past Exam Papers 2022–2024
London School of Economics
Department of Statistics
Academic Year 2025–26
,ST226 Week 1 Complete Guide 1
Contents
1 Week 1: Time Value of Money & Interest Rates 2
1.1 A. Core Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 The Time Value of Money Principle . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Accumulation and Discount Factors . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.4 Present Value and Accumulated Value Calculations . . . . . . . . . . . . 3
1.1.5 The Principle of Consistency . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.6 Interest Rate Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.7 Rate of Discount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.8 Nominal and E!ective Interest Rates . . . . . . . . . . . . . . . . . . . . . 6
1.1.9 Force of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.10 Cash Flows: Discrete Payments . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.11 Cash Flows: Continuous Payments . . . . . . . . . . . . . . . . . . . . . . 8
1.1.12 Combined Discrete and Continuous Payments . . . . . . . . . . . . . . . . 9
1.2 B. Formula Sheet (Exam-Optimised) . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 C. Typical Exam Question Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 D. Fully Worked Exam-Style Questions . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Question 1: Basic Accumulation (4 marks) . . . . . . . . . . . . . . . . . 14
1.4.2 Question 2: Present Value Calculation (5 marks) . . . . . . . . . . . . . . 14
1.4.3 Question 3: Interest Rate Conversions (6 marks) . . . . . . . . . . . . . . 15
1.4.4 Question 4: Time-Dependent Force of Interest (10 marks) . . . . . . . . . 16
1.4.5 Question 5: Show That (5 marks) . . . . . . . . . . . . . . . . . . . . . . 18
1.5 E. Common Mistakes (Examiner Traps) . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 F. One-Page Cheat Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Additional Practice Questions 25
2.1 Quick Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Solutions to Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
,ST226 Week 1 Complete Guide 2
1 Week 1: Time Value of Money & Interest Rates
1.1 A. Core Theory
1.1.1 The Time Value of Money Principle
Definition: Money available today is worth more than the same amount in the future due to
its earning potential through investment opportunities and the e!ects of inflation.
Key Concepts:
Present Value (PV): The current worth of a future sum of money or stream of cash
flows given a specified rate of return.
Future Value (FV/AV): The value of a current asset at a future date based on an
assumed rate of growth over time.
Discounting: The process of determining the present value of future cash flows.
Accumulation: The process of determining the future value of current money.
Financial Intuition:
Why is 100 today worth more than 100 in one year?
1. Investment opportunity: 100 today can be invested to earn interest
2. Inflation: 100 today buys more than 100 will buy next year
3. Risk: Future payments are uncertain; present money is certain
4. Consumption preference: People prefer consumption now rather than later
1.1.2 Accumulation and Discount Factors
The accumulation factor A(t1 , t2 ) represents the value at time t2 of an investment of 1 made
at time t1 (where t1 < t2 ).
Accumulation Factor
A(n) = A(0, n) = value at time n of 1 invested at time 0
For a constant e!ective annual interest rate i:
A(n) = (1 + i)n
Example: If i = 5%, then A(3) = (1.05)3 = 1.157625
This means 1 invested today grows to 1.1576 in 3 years.
The discount factor v gives the present value of a payment of 1 due at time n:
Discount Factor
1
v= = (1 + i)→1
1+i
1
v(n) = v n = = (1 + i)→n
A(n)
Key relationship: A(n) · v(n) = 1
, ST226 Week 1 Complete Guide 3
Example: If i = 5%, then:
1
v= = 0.952381
1.05
v 3 = (1.05)→3 = 0.863838
This means 1 due in 3 years is worth 0.8638 today.
Financial Interpretation:
accumulate
Time 0 →→→→→→→↑ Time n
A(n)=(1+i)n
1 →→→→→→→→↑ (1 + i)n
discount
vn ↓→→→→→ 1
1.1.3 Compound Interest
Compound Interest Formula
A capital C invested at time 0 at a constant annual e!ective interest rate i will have
accumulated value at time t:
F V = C(1 + i)t
The compound interest earned is:
Interest = C[(1 + i)t → 1]
Example: 500 invested for 6 years at 4.47% p.a.
F V = 500 ↔ (1.0447)6
= 500 ↔ 1.3 = 650
Interest = 650 → 500 = 150
Key Insight: Interest earns interest in subsequent periods.
Compound vs. Simple Interest:
Simple Interest Compound Interest
Formula F V = C(1 + it) F V = C(1 + i)t
Interest on interest? No Yes
Growth rate Linear Exponential
Used in ST226? No Yes (always)
In this course we ONLY use compound interest.
1.1.4 Present Value and Accumulated Value Calculations
PV and FV Relationships
Future Value:
F V = P V · A(n) = P V · (1 + i)n
Present Value:
P V = F V · v(n) = F V · (1 + i)→n
Summary Table:
Week 1 Complete Mastery Guide
Time Value of Money & Interest Rates
Complete Coverage with:
Detailed Theory & Explanations
Exam-Optimized Formula Sheets
Fully Worked Past Paper Questions
Common Mistakes & Examiner Traps
One-Page Quick Reference
Based on LSE Course Materials 2024–25
Course Notes Chapter 1
Lecture Slides Week 1
Past Exam Papers 2022–2024
London School of Economics
Department of Statistics
Academic Year 2025–26
,ST226 Week 1 Complete Guide 1
Contents
1 Week 1: Time Value of Money & Interest Rates 2
1.1 A. Core Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 The Time Value of Money Principle . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Accumulation and Discount Factors . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.4 Present Value and Accumulated Value Calculations . . . . . . . . . . . . 3
1.1.5 The Principle of Consistency . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.6 Interest Rate Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.7 Rate of Discount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.8 Nominal and E!ective Interest Rates . . . . . . . . . . . . . . . . . . . . . 6
1.1.9 Force of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.10 Cash Flows: Discrete Payments . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.11 Cash Flows: Continuous Payments . . . . . . . . . . . . . . . . . . . . . . 8
1.1.12 Combined Discrete and Continuous Payments . . . . . . . . . . . . . . . . 9
1.2 B. Formula Sheet (Exam-Optimised) . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 C. Typical Exam Question Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 D. Fully Worked Exam-Style Questions . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Question 1: Basic Accumulation (4 marks) . . . . . . . . . . . . . . . . . 14
1.4.2 Question 2: Present Value Calculation (5 marks) . . . . . . . . . . . . . . 14
1.4.3 Question 3: Interest Rate Conversions (6 marks) . . . . . . . . . . . . . . 15
1.4.4 Question 4: Time-Dependent Force of Interest (10 marks) . . . . . . . . . 16
1.4.5 Question 5: Show That (5 marks) . . . . . . . . . . . . . . . . . . . . . . 18
1.5 E. Common Mistakes (Examiner Traps) . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 F. One-Page Cheat Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Additional Practice Questions 25
2.1 Quick Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Solutions to Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
,ST226 Week 1 Complete Guide 2
1 Week 1: Time Value of Money & Interest Rates
1.1 A. Core Theory
1.1.1 The Time Value of Money Principle
Definition: Money available today is worth more than the same amount in the future due to
its earning potential through investment opportunities and the e!ects of inflation.
Key Concepts:
Present Value (PV): The current worth of a future sum of money or stream of cash
flows given a specified rate of return.
Future Value (FV/AV): The value of a current asset at a future date based on an
assumed rate of growth over time.
Discounting: The process of determining the present value of future cash flows.
Accumulation: The process of determining the future value of current money.
Financial Intuition:
Why is 100 today worth more than 100 in one year?
1. Investment opportunity: 100 today can be invested to earn interest
2. Inflation: 100 today buys more than 100 will buy next year
3. Risk: Future payments are uncertain; present money is certain
4. Consumption preference: People prefer consumption now rather than later
1.1.2 Accumulation and Discount Factors
The accumulation factor A(t1 , t2 ) represents the value at time t2 of an investment of 1 made
at time t1 (where t1 < t2 ).
Accumulation Factor
A(n) = A(0, n) = value at time n of 1 invested at time 0
For a constant e!ective annual interest rate i:
A(n) = (1 + i)n
Example: If i = 5%, then A(3) = (1.05)3 = 1.157625
This means 1 invested today grows to 1.1576 in 3 years.
The discount factor v gives the present value of a payment of 1 due at time n:
Discount Factor
1
v= = (1 + i)→1
1+i
1
v(n) = v n = = (1 + i)→n
A(n)
Key relationship: A(n) · v(n) = 1
, ST226 Week 1 Complete Guide 3
Example: If i = 5%, then:
1
v= = 0.952381
1.05
v 3 = (1.05)→3 = 0.863838
This means 1 due in 3 years is worth 0.8638 today.
Financial Interpretation:
accumulate
Time 0 →→→→→→→↑ Time n
A(n)=(1+i)n
1 →→→→→→→→↑ (1 + i)n
discount
vn ↓→→→→→ 1
1.1.3 Compound Interest
Compound Interest Formula
A capital C invested at time 0 at a constant annual e!ective interest rate i will have
accumulated value at time t:
F V = C(1 + i)t
The compound interest earned is:
Interest = C[(1 + i)t → 1]
Example: 500 invested for 6 years at 4.47% p.a.
F V = 500 ↔ (1.0447)6
= 500 ↔ 1.3 = 650
Interest = 650 → 500 = 150
Key Insight: Interest earns interest in subsequent periods.
Compound vs. Simple Interest:
Simple Interest Compound Interest
Formula F V = C(1 + it) F V = C(1 + i)t
Interest on interest? No Yes
Growth rate Linear Exponential
Used in ST226? No Yes (always)
In this course we ONLY use compound interest.
1.1.4 Present Value and Accumulated Value Calculations
PV and FV Relationships
Future Value:
F V = P V · A(n) = P V · (1 + i)n
Present Value:
P V = F V · v(n) = F V · (1 + i)→n
Summary Table:
