CM1 Foundation · Theory of Interest Rates
Define the force of interest and connect it to continuous compounding.
Recommended before this lesson — you can still read and practice below:
As compounding frequency increases, the effective rate approaches a limit. The force of interest (delta) is the infinitesimal rate that describes this limit and underpins continuous compounding.
This lesson defines , connects it to the effective rate , and shows why actuaries use when rates vary continuously in time.
Let be the accumulation function with . The at time is
When is constant,
Rs. at grows to under continuous compounding at force .
Recall from Lesson 3 that . As (compounding every instant), the limit is where is the nominal rate convertible continuously. That limit defines :
Problem. Force of interest . Find the effective annual rate (percentage to two decimals).
Solution.
Note slightly exceeds 6% because for .
If is not constant, accumulation over is
This integral form is the continuous analogue of multiplying over sub-periods (Lesson 6). CM1 often uses piecewise-constant ; the integral then becomes a sum of .
Problem. Rs. 15,000 is invested with constant force of interest per annum for 3 years. Find the accumulated value.
Solution.
You should be able to (1) state for constant rates, (2) accumulate with , and (3) recognise as the continuous-compounding limit of p-thly nominal rates. Practice and mastery checks below test that.
For constant force of interest , the accumulation factor from to is:
Effective annual rate . Find to four decimal places.
. Find to four decimal places.
Present value at constant over years uses factor:
as force of interest satisfies times:
effective. Find to four decimal places.
. Find to four decimal places.
As , approaches:
. Accumulation factor for 3 years to four decimal places.
effective. Find to four decimal places, then to four decimal places.
Rs. 1,000 at constant from to . Value at (nearest rupee).
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
The relationship holds when:
links force of interest to:
Constant implies interest is compounded:
Locked — complete prior skills first
Rs. 10,000 at constant p.a. for 2 years. Accumulated value to the nearest rupee?
Rs. 5,000 at for 5 years. Value (nearest rupee).
. to four decimal places.
Locked — complete prior skills first
As compounding frequency with nominal rate held fixed, the accumulation factor over one year approaches:
Continuous compounding limit of as with fixed nominal is where equals:
Rs. 2,000 at for 2 years. Accumulated value (nearest rupee).
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