Why Discount Rate and Force of Interest Feel Confusing? Why Interest & Discount Rates Feel Opposite

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00:00 Class plan and learning goal
01:30 What will be covered and why alignment matters
02:00 Core accumulation formula revisited
02:45 Nominal interest rate and effective interest rate
03:30 Discount rate formula introduced
04:30 Link between interest and discount
05:40 Simple interest vs compound interest recap
07:00 Timeline explanation of compounding
08:30 Accumulation under compound interest
09:40 Compounding more than once per year
11:00 Meaning of nominal rate convertible p times
12:30 Handling non-integer time periods
13:50 Choosing the correct formula based on given rate
15:00 Why rounding errors propagate
16:30 Why direct formulas save time and accuracy
17:30 Present value using v
18:30 Present value using discount rate d
19:40 Loan example to explain discount intuition
21:30 Why 10% interest is not 10% discount
23:00 Deriving relationship between i and d
24:30 Real-life discount example
26:00 Two viewpoints. Price vs discount
27:30 Extending discount to multiple periods
29:00 Nominal discount rate convertible p times
30:30 Example using discount rate convertible quarterly
33:00 Importance of the word “convertible”
35:00 Transition to force of interest
36:00 Increasing compounding frequency intuition
38:00 From discrete to continuous compounding
39:30 Drinking water analogy for continuity
41:00 Daily, hourly, moment-by-moment compounding
42:30 Introduction to force of interest delta
44:00 e power delta relation explained
45:30 Constant force of interest
47:00 Variable force of interest idea
48:30 Driving speed analogy for delta(t)
50:00 General accumulation formula using integration
52:00 How constant force becomes a special case
54:00 Comparing i, d, delta, and delta(t)
56:00 Final example using all three cases
58:30 What to focus on before annuities
01:00:10 Closing remarks

This session is a consolidation class designed to align all learners before moving ahead. The focus is clarity of interest rate concepts, not speed or heavy problem solving.

The class covers two areas.
Interest rates and discount rates.
Force of interest.

It begins by revisiting the link between nominal and effective interest rates using the core accumulation identity. Nominal rates convertible p times per year are connected to effective annual rates. A parallel identity for discount rates is introduced to show that interest and discount describe the same time value of money from opposite directions.

Interest rates move values forward in time.
Discount rates move values backward in time.

Simple and compound interest are compared using a number line. Simple interest grows linearly. Compound interest grows on accumulated amounts, creating interest on interest. This explains why compounding accelerates over time.

The compound interest formula is reinforced as the main tool. Different compounding frequencies only change the number of periods and the rate per period. The logic stays the same.

An important exam rule is emphasized. Always use the rate given in the question directly. Converting rates introduces rounding errors, which grow in long calculations such as annuities and bonds.

Present value is then explained in two equivalent ways.
Using the discount factor v.
Using the discount rate d.

A loan example shows why a 10 percent interest rate does not equal a 10 percent discount rate. Because discount is applied at the start of the period, the equivalent discount rate is lower, about 9.10 percent. The algebraic relationship between interest rate and discount rate is derived.


Discounting is extended to multiple compounding periods. Nominal discount rates convertible p times per year follow the same structure as nominal interest rates. The importance of reading the word “convertible” carefully in questions is stressed.

The session then introduces force of interest by increasing compounding frequency from annual to continuous. As intervals shrink, accumulation becomes smooth and exponential.

Force of interest is defined as the nominal rate of interest per annum convertible continuously. It represents instantaneous growth intensity.


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