How to Solve Accumulating and Discounting Questions in CM1? Simple Explanation (Force of Interest)

How to Solve Accumulating and Discounting Questions in CM1? Simple Explanation

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Timestamps
0:00 – Chapter overview. Accumulating vs discounting
1:10 – Constant interest and continuous compounding
2:05 – Meaning of force of interest δ
3:05 – Variable force δ(t) and integration idea
4:10 – Link between i, δ, v, and VT
5:20 – VT definition and reduction to Vⁿ
6:30 – Present value using VT
7:45 – Why V3 fails when time does not start at 0
9:10 – Accumulated value using 1 / VT
11:10 – Forward and backward movement rules
13:00 – Why V5 / V3 appears naturally
15:00 – Constant vs variable interest comparison
16:30 – Multi-cash-flow example logic
18:40 – Transition to continuous payments
20:00 – Meaning of rate of payment ρ(t)
22:30 – Present value of continuous payment streams
25:00 – Variable payment with constant interest
27:10 – Variable payment with variable interest
29:30 – Accumulated value of continuous payments
31:10 – Exam strategy and final guidance


This video explains the Accumulating and Discounting chapter from CS1 using force of interest. You learn how to move money forward and backward in time when interest is constant and when interest changes continuously.

The focus is on how students actually think during exams and how marks are lost due to symbol misuse.

What you already know and how this chapter extends it

You already know discounting using Vⁿ.
You already know accumulation using (1 + i)ⁿ.
You already know the link between i and δ.

This chapter extends these ideas to situations where interest acts continuously and may vary with time.

Key ideas built step by step

• Accumulating vs discounting using interest rate i
• Transition from i to force of interest δ
• Continuous compounding using e^(δt)
• Meaning of constant force vs time-dependent force δ(t)

Force of interest intuition

• Constant δ means steady growth at every instant
• Variable δ(t) means growth rate changes with time
• Accumulation becomes e^(∫δ(t)dt)
• Discounting becomes e^(−∫δ(t)dt)

VT notation. Core exam focus.

• VT means present value of 1 due at time t
• VT only works when time starts at 0
• VT replaces Vⁿ when δ(t) is given
• VT reduces to Vⁿ when δ is constant

Common student mistakes corrected

• Writing V3 when time does not start at 0
• Treating V5 / V3 as V2 under variable δ(t)
• Forgetting zero must be common to use VT
• Using powers instead of integrals

Accumulated value using VT

• Present value uses VT
• Accumulated value uses 1 / VT
• Forward movement divides by VT
• Backward movement multiplies by VT

Key exam rule

Interval length alone does not matter when δ(t) varies.
The location of the interval matters.

Worked structure for mixed time problems

• Always connect through time 0
• Move backward using VT
• Move forward using 1 / VT
• Combine movements as ratios
Example logic covered

• Accumulated value of payments at different times
• Present value of mixed cash flows
• Correct handling of time gaps
• Why V5 / V3 appears naturally

Transition to Part 2 of the chapter

Part 1
• Single payment
• Time-varying interest
• VT and accumulation logic

Part 2
• Continuous payment streams
• Introduction of rate of payment ρ(t)

Continuous payment stream explained

• Payments arrive continuously
• Payment over small interval is ρ(t)dt
• Each payment is discounted separately
• Integration replaces summation

Key formula for present value of continuous payments

Present Value
∫ from a to b of ρ(t) × discount factor dt

Key formula for accumulated value of continuous payments

Accumulated Value
∫ from a to b of ρ(t) × accumulation factor dt

Three exam-relevant cases explained

Constant payment, constant interest

Variable payment, constant interest

variable interest

What makes questions difficult

• Two layers of integration
• Splitting time intervals correctly
• Choosing correct discount direction
• Avoiding invalid symbols

Exam strategy advice

• Draw the timeline first
• Mark time 0 clearly
• Decide present or accumulated value
• Write integral limits carefully
• Substitute δ(t) only after limits are fixed

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