Actuarial Mathematics for Modelling · Present value and accumulated value of cashflow streams

Accumulation and Present Value of Single Payments

Updated 11 October 2026 · Fact-checked

The accumulated value of a payment is what it grows to at a later date. The present value is what a later payment is worth today. Under compound interest, accumulate with (1 + i)^t and discount with v^t = (1 + i)^(-t). Under simple interest, accumulate with 1 + it.

Understand Accumulation and Present Value of Single Payments

Money has a time value. ₹1,000 today is worth more than ₹1,000 a year from now, because you can invest it and earn interest. Every cashflow question in CM1 starts from this idea.

To accumulate a payment, you move it forward in time. You ask: what will this amount be worth at a later date? The accumulation factor is the number you multiply by. For a payment C made at time 0, the accumulated value at time t is C × A(t).

To find a present value, you move a payment backward in time. You ask: what amount today is equivalent to a payment due later? The discount factor is the number you multiply by. It is the reciprocal of the accumulation factor over the same period.

The type of interest decides the factor. Under simple interest, interest is earned only on the original amount, so growth is linear: 1 + it. Under compound interest, interest is earned on interest too, so growth is exponential: (1 + i)^t. Compound interest is the default in the IAI syllabus unless the question says otherwise.

The two methods give the same answer at t = 1 but differ at other times. For t > 1 compound gives more than simple. For 0 < t < 1 simple gives more than compound. Also, present values under simple interest depend on the date you move back from, so state the basis clearly.

Key rules to remember

Compound interest accumulation: A(t) = (1 + i)^t, so accumulated value = C × (1 + i)^t; i is the effective annual rate and t is in years. The rate and the time unit must match.
Compound interest discount factor: v = 1 ÷ (1 + i); PV = C × v^t = C × (1 + i)^(-t); PV is the value at time 0 of a payment C due at time t.
Simple interest accumulation: A(t) = 1 + it, so accumulated value = C × (1 + it); Interest is earned only on the original amount C.
Simple interest present value: PV = C ÷ (1 + it); This is the usual convention: the present value is the amount that accumulates to C under simple interest over time t.
Accumulating or discounting between two dates: Value at time s = C × (1 + i)^(s - t) for a payment C at time t; Under compound interest this works for any s, whether before or after t. A negative power discounts.
Interest earned over a period: Interest = C × [A(t) - 1]; Simple: C × it. Compound: C × [(1 + i)^t - 1].

How to solve Accumulation and Present Value of Single Payments questions

Use this method for any question on accumulating or discounting a single payment.

1Read the question and note the type of interest: simple or compound. If it is not stated, assume compound.
2Write down the rate and check what it is: effective annual rate, or a nominal rate that you must convert first.
3Draw a quick timeline. Mark the payment date and the date at which you need the value.
4Work out the time gap t between the two dates, in the same unit as the rate (usually years).
5Decide the direction. Moving forward in time means accumulate. Moving back means discount.
6Write the formula with numbers, for example 5,000 × (1.06)^3, before using the calculator.
7Calculate, keep at least four decimal places in factors, and round only the final answer.
8Check the answer is sensible: accumulated values should be larger than the payment, and present values smaller (for positive interest).

Quickest way: Factor shortcut with a timeline

When to use it: Use this in multiple-choice questions and in the first part of written questions where one payment is moved once.

Sketch a one-line timeline with the payment and the target date.
Count the years between them.
Pick the factor: (1 + i)^t to go forward, (1 + i)^(-t) to go back.
Enter 1 + i, press the power key, then multiply by the payment.
Estimate to check: 8% for 10 years roughly doubles the amount, so the answer should be about twice the payment.

Common mistakes in Accumulation and Present Value of Single Payments

Using simple interest formula when compound interest is intended, or the reverse.
Students use 1 + it out of habit from school problems.
Fix: Look for the words 'simple' or 'compound' in the question. If neither appears, use compound and state that assumption.
Writing 1 + i as 1.i, for example using 1.6 for i = 6%.
Rushing and misplacing the decimal point.
Fix: Convert the rate to a decimal first: 6% = 0.06, so 1 + i = 1.06.
Discounting by multiplying by (1 + i)^t or accumulating by dividing by it.
Confusing the direction of movement on the timeline.
Fix: Ask whether the target date is later or earlier. Later means the value goes up. Earlier means it goes down.
Using the wrong time gap, such as counting from time 1 to time 5 as 5 years.
Students use the payment date instead of the difference between dates.
Fix: Subtract: t = 5 - 1 = 4 years. Always use the gap between the two dates.
Applying a nominal rate directly in (1 + i)^t.
The rate is quoted as 'per annum' but compounded more often.
Fix: Convert first. If the rate is i^(m) compounded m times a year, use (1 + i^(m) ÷ m)^(mt).
Rounding the factor to two decimals before multiplying.
Wanting tidy numbers.
Fix: Keep the full calculator value until the end. Rounding early can change the answer enough to lose a mark.

Worked examples

Example 1

A payment of ₹50,000 is made today. Find its accumulated value after 4 years (a) at 6% per annum compound interest, and (b) at 6% per annum simple interest.

Show the solution

Compound: accumulation factor = (1.06)^4.
(1.06)^2 = 1.1236, so (1.06)^4 = 1.1236 × 1.1236 = 1.26248 (to 5 decimal places).
Accumulated value = 50,000 × 1.26248 = ₹63,124 (to the nearest rupee).
Simple: accumulation factor = 1 + 0.06 × 4 = 1.24.
Accumulated value = 50,000 × 1.24 = ₹62,000.
Check: compound gives more than simple because t > 1.

Answer: (a) ₹63,124 under compound interest. (b) ₹62,000 under simple interest.

Example 2

You will receive ₹2,00,000 in 5 years' time. Find its present value (a) at 8% per annum effective, and (b) if the effective rate is 8% for the first 2 years and 10% for the next 3 years.

Show the solution

(a) Discount factor = (1.08)^(-5).
(1.08)^5 = 1.46933 (to 5 decimal places), so (1.08)^(-5) = 1 ÷ 1.46933 = 0.68058.
PV = 2,00,000 × 0.68058 = ₹1,36,117 (to the nearest rupee).
(b) Discount from time 5 back to time 2 at 10%: (1.10)^(-3) = 1 ÷ 1.331 = 0.75131.
Then discount from time 2 back to time 0 at 8%: (1.08)^(-2) = 1 ÷ 1.1664 = 0.85734.
Combined factor = 0.75131 × 0.85734 = 0.64413.
PV = 2,00,000 × 0.64413 = ₹1,28,826 (to the nearest rupee).
Check: part (b) is lower because the later rate is higher.

Answer: (a) ₹1,36,117. (b) ₹1,28,826.

Exam tips

In multiple-choice questions, estimate first. Direction and rough size of the answer often eliminate two options at once.
In written questions, show the formula, the substituted numbers and the factor before the final figure. Method marks are awarded even if the arithmetic slips.
State your assumption when the type of interest is not given: write 'assuming compound interest'.
When the rate changes over time, discount in stages from the later date back to the earlier date, one rate at a time.
Watch for time in months. Convert to years (for example 18 months = 1.5 years) before using an annual rate.

Practice questions from Present value and accumulated value of cashflow streams

Accumulation and Present Value of Single Payments in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Accumulation and Present Value of Single Payments: frequently asked questions

What is the difference between accumulated value and present value?

Accumulated value is the worth of a payment at a later date after interest has been added. Present value is the worth today of a payment due later. They are the same relationship viewed in opposite directions.

How do I calculate present value using a discount factor?

Find the discount factor v = 1 ÷ (1 + i), raise it to the power t, and multiply by the payment. For ₹10,000 due in 3 years at 5%, PV = 10,000 × (1.05)^(-3).

What is the difference between simple and compound interest for present value?

Under compound interest, PV = C × (1 + i)^(-t). Under simple interest, PV = C ÷ (1 + it). The two agree at t = 1 but differ at other times, and compound grows faster for t > 1.

Which interest type should I assume if the question does not say?

Assume compound interest, as it is the standard in actuarial work. State this assumption in your answer so the examiner can see your reasoning.