How to Calculate Accumulated and Discounted Value with δ(t) force of interest ?CM1 Actuarial Science

How to Calculate Accumulated and Discounted Value with δ(t)? When Force of Interest Depends on Time

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Detailed timestamps

0:00 – 1:30
Force of interest vs constant force of interest
Understanding δ(t) as a function of time

1:31 – 3:20
Graph of δ(t). Linear increase and constant sections
Functional value at boundary points

3:21 – 6:10
Piecewise functions, limits, and continuity intuition
Why a function cannot be one-to-many

6:11 – 9:30
Future value and present value using δ and δ(t)
Exponential form with integrals

9:31 – 12:40
Introduction to VT and VN notation
When VT is allowed and when it is not

12:41 – 16:30
Why V5 / V3 ≠ V2 under variable force of interest
Correct interpretation of discounting intervals

16:31 – 19:50
When constant δ or constant i simplifies everything
Link between δ, i, and vⁿ

19:51 – 23:20
Deriving VT for piecewise δ(t)
Case 1. t is less than 5
Case 2. t is great than or equal to 5

23:21 – 26:40
Applying VT to a past exam question
Present value of a sum due after many years

26:41 – 32:10
Finding equivalent nominal rate convertible monthly
Matching continuous and monthly compounding

32:11 – 35:30
Continuous payment streams
Meaning of ρ(t) and rate of payment

35:31 – 38:30
Three common continuous payment cases
Only payments vary
Only interest varies
Both vary

38:31 – 42:10
Payments over sub-intervals like 6 to 10
How to discount correctly to time 0
This video breaks down a high-scoring CS1 past exam question on force of interest with time-varying rates. You learn how to think, not memorize. Every step is linked to exam logic and common student mistakes.

What this class solves clearly

• Difference between force of interest δ and time-dependent force δ(t)
• How to read and interpret piecewise interest functions
• Why boundary points matter in actuarial questions
• When a function fails if definitions overlap
• How to sketch δ(t) graphs and interpret growth behavior

Core actuarial concepts explained step by step

• Accumulation and discounting using force of interest
• Correct use of exponential integrals
• Difference between
– e^(nδ)
– e^(∫δ(t)dt)
• Why formulas change when δ depends on time

VT notation. This is where most students fail.

• Meaning of VT as present value of 1 due at time t
• When VT notation is allowed
• Why VT requires time starting at 0
• Why V5 / V3 ≠ V2 when δ(t) varies
• Why interval length alone is misleading
• Physical intuition using changing interest environments

Piecewise force of interest. Exam critical.

• Case 1. t is less than 5
• Case 2. t greater than or equal to 5
• How to split integrals correctly
• How continuity and limits affect interpretation
• How to derive VT symbolically for both cases

Past exam problem walkthrough

• Deriving VT in closed form
• Applying VT to compute present value of a future sum
• Why VT for 3 years differs from VT for 7 years
• How examiners test logic, not calculation

Equivalence of interest rates

• Converting continuous force results to nominal rates
• Finding rate convertible monthly from continuous setup
• Matching equivalent present values across banks
• Why examiners ask this conversion

Continuous payment streams. Advanced CS1 topic.

• Meaning of ρ(t) as rate of payment
• Present value of continuous income
• How to discount infinitesimal payments
• Why integration replaces summation

Three continuous payment scenarios you must master

Constant payment, constant interest

Variable payment, constant interest

Variable payment, variable interest

Also covered

• Why VT symbols break when start time is not equal to 0
• Why you cannot shortcut discounting under δ(t)
• How to think like an examiner
• Where students lose marks despite correct formulas

Who this video is for

• CS1 students struggling with force of interest
• Students confused by VT and Vⁿ notation
• Anyone losing marks in continuous interest questions
• Students aiming for conceptual clarity, not shortcuts

How to study this video

• Pause and redraw graphs
• Rewrite integrals yourself
• Practice both cases t is less than 5 5 and t greater than or equal to 5
• Re-solve without symbols once
• Then re-solve using VT notation