CM1 Foundation · Theory of Interest Rates
Derive the compound accumulation formula from first principles.
Recommended before this lesson — you can still read and practice below:
When money earns interest and that interest itself earns interest, we have compound interest. If Rs. 1 is invested at effective annual rate , the balance after one year is . After two years the balance is not but , because the interest from year 1 also grows in year 2.
This lesson derives the compound accumulation formula from the same TVOM machinery as Lesson 1 — no new axioms, only repeated application of .
Let be capital at . Under compound interest at constant effective rate per year, each year multiplies the balance by . After years (with a non-negative integer first):
We call the compound accumulation factor from to .
Given , the rate is recovered from one period:
If Rs. 1,000 grows to Rs. 1,080 in one year, then and p.a. effective.
Problem. Rs. 25,000 is invested at p.a. effective for 4 years. Find the accumulated value.
Solution.
Plotting against gives exponential growth when : equal time steps multiply the balance by the same factor , so the curve bends upward. Compare this mentally to simple interest , which is linear in .
Problem. Rs. 10,000 at p.a. effective for 3 years. Find compound and simple accumulation (simple: interest only on principal).
Solution. Compound: .
Simple: .
Compound exceeds simple because year-2 and year-3 interest earn interest.
You should be able to (1) write , (2) extract from a one-year growth factor, and (3) explain why compound balances grow exponentially when . Practice and mastery checks below test that.
Rs. 5,000 at effective annual rate for 3 years. Find to the nearest rupee.
Rs. 1,000 grows to Rs. 1,125 in exactly one year. What is as a percentage (one decimal place)?
Under constant effective rate , compound accumulation plotted against is:
Rs. 3,200 at for 5 years. (nearest rupee).
Rs. 10,000 at for 10 years. Accumulated value (nearest rupee).
Rs. 4,000 grows to Rs. 4,480 in one year. as percentage (one decimal).
Under compound interest over years, the growth factor is:
Each period, compound interest is earned on:
Constant effective gives vs that is:
Rs. 2,000 grows to Rs. 2,420 in 2 years. Effective annual as percentage (nearest integer).
Rs. 1,000 at for 3 years under compound interest:
Pass each skill by answering correctly on 2 of 3 checks (fewer if fewer are available). Practice questions above are rehearsal — only mastery checks count toward progress. Retry wrong answers until you succeed.
0/2 verified
Rs. 8,000 at p.a. effective for 2 years. Accumulated value to the nearest rupee?
Rs. 6,500 at for 6 years. Value at (nearest rupee).
implies equals:
Locked — complete prior skills first
If Rs. 2,000 becomes Rs. 2,180 after one year at effective rate , then equals:
Rs. 7,500 becomes Rs. 8,137.50 in one year. as percentage (one decimal).
One year's compound growth factor at effective rate is:
Locked — complete prior skills first
For the same principal and positive , compound interest over 5 years gives a balance that is:
Slope of vs for compound growth is:
Compound vs for is:
Lessons 1–10 are free. Founding members get the complete module — paid lessons, practice, and atom-level mastery checks — with lifetime access.