CM1 Foundation · Theory of Interest Rates
Work with piecewise-constant and time-varying rates.
Recommended before this lesson — you can still read and practice below:
So far, interest has been constant. Many contracts specify rates that change over time — for example, a loan at 5% for two years followed by 6% thereafter. This lesson accumulates and discounts when the rate is piecewise constant or otherwise time-varying.
Divide the timeline into intervals with effective rate over each interval. The from to is the product of single-period factors:
When each is a whole number of years with rate , this simplifies to multiplying for each full year at that rate.
For varying rates, (1) mark interval boundaries on the timeline, (2) write the product of raised to the length of each interval, (3) multiply or divide as needed for the valuation direction.
Problem. Rs. 10,000 is invested for 2 years at effective, then the balance remains invested for 1 more year at . Find the accumulated value after 3 years.
Solution.
Each rate applies only to its own interval.
Problem. Rs. 20,000 is due at . Rate is 4% effective for years 0–1 and 5% for years 1–3. Find PV at .
Solution. Discount year by year (or use the product):
When varies continuously, Lesson 4 gave . Piecewise-constant on each interval contributes to the exponent — the continuous version of the product above.
You should be able to (1) build accumulation products over intervals with different rates, (2) discount by inverting those products, and (3) identify which rate applies to which time segment on a timeline. Practice and mastery checks below test that.
When the effective rate changes from to at , accumulation from to uses:
Rs. 5,000 for 1 year at then 1 year at effective. Accumulated value after 2 years (nearest rupee)?
Rs. 1,000 due at . Year 1 rate , year 2 rate effective. PV at (nearest rupee)?
Rates then over consecutive years give 2-year factor:
Piecewise constant rates mean each applies to:
Rs. 2,000: 1 yr at 4%, then 1 yr at 6%. Value after 2 yrs (nearest rupee).
Rs. 1,000 for 3 yrs: 5%, 5%, 8% effective yearly. Value (nearest rupee).
Rs. 2,000 at . Rates 3% then 4% for yrs 1-2. PV at (nearest rupee).
Discount factor over varying rates is:
Rs. 6,000: 1 yr at 4%, 2 yrs at 5%. Value at (nearest rupee).
Rs. 3,000 at . Rates 5%, 4%, 6% in years 1-3. PV 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 accumulation factor with piecewise-constant rates is:
Order of rate intervals matters for:
with varying rates is found by:
Locked — complete prior skills first
Rs. 8,000 for 2 years at then 1 year at effective. Value at (nearest rupee)?
Rs. 3,000: 2 yrs at 5%, 1 yr at 4%. Value at (nearest rupee).
Rs. 4,000: yr1 3%, yr2 3%, yr3 7%. Value at (nearest rupee).
Locked — complete prior skills first
To discount from to with varying rates, you:
Rs. 5,000 at . Yr1 6%, yr2 5%. PV at (nearest rupee).
To discount to with varying rates, divide by:
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