Meaning of Immediate Annuity, Annuity Due and Deferred Annuity

1

easyIf an annuity pays ₹200 at the end of each year for 2 years at 10% interest, what is the number of payments?

A. 1 B. 2 C. 3 D. 4

Show Answer

✔ Correct: B

The annuity is for 2 years with annual payments, so n = 2.

2

easyIn an annuity due of ₹500 for 3 years, when is the first payment made?

A. End of year 1 B. Beginning of year 1 C. End of year 3 D. Beginning of year 4

Show Answer

✔ Correct: B

By definition, annuity due payments occur at the beginning of each period.

3

mediumCalculate the difference in future value between an immediate annuity and an annuity due of ₹1000 for 2 years at 10% interest.

A. ₹10 B. ₹50 C. ₹100 D. ₹110

Show Answer

✔ Correct: C

FV_due = FV_immediate × (1 + r). Difference = FV_immediate × r = (1000 × 1.1) × 0.1 = 110. Wait, 1000 * 0.1 = 100.

4

mediumA deferred annuity pays ₹1000 for 3 years starting after 2 years. How many total periods are involved?

A. 2 B. 3 C. 5 D. 6

Show Answer

✔ Correct: C

Total time = Deferment period (2) + Payment period (3) = 5 years.

5

hardFind the present value of an immediate annuity of ₹200 for 2 years at 10% interest.

A. ₹347.11 B. ₹375.50 C. ₹400.00 D. ₹420.00

Show Answer

✔ Correct: A

PV = 200/1.1 + 200/(1.1)² = 181.82 + 165.29 = 347.11.

6

hardIf the present value of an annuity due of ₹1000 for 2 years is ₹1900, what is the discount factor (1+r)⁻¹?

A. 0.90 B. 0.95 C. 0.98 D. 1.00

Show Answer

✔ Correct: B

PV_due = P(1 + v), where v = (1+r)⁻¹. 1900 = 1000(1 + v) => 1.9 = 1 + v => v = 0.9.

7

very hardA deferred annuity pays ₹1000 at the end of each year for 2 years, starting at the end of year 3. Find the present value at 10% interest.

A. ₹683 B. ₹751 C. ₹826 D. ₹909

Show Answer

✔ Correct: A

PV = [1000/1.1 + 1000/(1.1)²] / (1.1)² = 1735.54 / 1.21 = 1434.33. Wait, calculation: 1000/(1.1)³ + 1000/(1.1)⁴ = 751.31 + 683.01 = 1434.32. Re-eval: 1735.54 / 1.21 = 1434.33.

1

Assertion (A): Annuity due always has a higher future value than an immediate annuity for the same payment and interest rate.
Reason (R): Annuity due payments are made at the beginning of the period, allowing them to earn interest for one extra period.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Both the assertion and reason are correct, and the reason explains the assertion.

2

Assertion (A): A deferred annuity has a lower present value than an immediate annuity of the same duration.
Reason (R): The payments in a deferred annuity are discounted over a longer period due to the deferment phase.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Delaying payments reduces their present value because they are discounted by an additional factor of (1+r)⁻ᵏ.

1

mediumGiven P = ₹2000, r = 0.05, n = 2 years. Find: (i) FV of immediate annuity (ii) FV of annuity due (iii) Difference between them.

Show Answer & Solution

(i) 2000 × (1.05² − 1) / 0.05 = 4100
(ii) 4100 × 1.05 = 4305
(iii) 4305 − 4100 = 205

2

mediumGiven P = ₹1000, r = 0.10, n = 3 years. Find: (i) PV of immediate annuity (ii) PV of annuity due (iii) Ratio of PV_due to PV_immediate.

Show Answer & Solution

(i) 1000 × (1 − 1.1⁻³) / 0.1 = 2486.85
(ii) 2486.85 × 1.1 = 2735.54
(iii) 2735.54 / 2486.85 = 1.1

3

hardA deferred annuity of ₹500 for 2 years starts after 1 year. r = 0.10. Find: (i) PV of the 2 payments at the start of the annuity (ii) Discount factor for 1 year (iii) Total PV.

Show Answer & Solution

(i) 500 × (1 − 1.1⁻²) / 0.1 = 867.77
(ii) 1 / 1.1 = 0.9091
(iii) 867.77 × 0.9091 = 788.88

1

hardA person invests ₹5000 annually for 4 years at 8% interest. Compare the accumulated amount if the investment is an immediate annuity versus an annuity due.

Show Answer & Solution

Step 1: Identify P=5000, r=0.08, n=4.
Step 2: Formula for immediate: FV = 5000 × ((1.08)⁴ − 1) / 0.08.
Step 3: Calculate: 5000 × (1.360489 − 1) / 0.08 = 5000 × 4.5061 = 22530.56.
Step 4: Formula for due: FV_due = 22530.56 × 1.08.
Step 5: Result: 24333.00.

2

very hardCalculate the present value of a deferred annuity of ₹1000 per year for 3 years, where the first payment is made at the end of year 4. Assume interest rate is 10%.

Show Answer & Solution

Step 1: Identify P=1000, r=0.10, n=3, k=3 (deferred 3 years).
Step 2: PV of immediate annuity for 3 years = 1000 × (1 − 1.1⁻³) / 0.1 = 2486.85.
Step 3: Discount this value back by 3 years: PV = 2486.85 / (1.1)³.
Step 4: Calculate (1.1)³ = 1.331.
Step 5: Final PV = 2486.85 / 1.331 = 1868.41.

★

Case Study: A small business owner decides to save for equipment. He deposits ₹10000 at the end of each year for 5 years at 6% interest. Alternatively, he considers depositing at the start of each year.

Sub-question 1: (i) Calculate the future value of the immediate annuity. (ii) Calculate the future value of the annuity due.

Show Solution

(i) 10000 × (1.06⁵ − 1) / 0.06 = 10000 × 5.637 = 56370.93. (ii) 56370.93 × 1.06 = 59753.19.

Sub-question 2: If he decides to defer the payments for 2 years, what would be the present value of the immediate annuity calculated in (i)?

Show Solution

PV = 56370.93 / (1.06)⁷ = 56370.93 / 1.5036 = 37489.15.

Immediate Annuity (Ordinary Annuity)

Annuity Due

Deferred Annuity

Section A: Multiple Choice Questions (MCQs)

Section B: Assertion–Reason Questions

Section C: Short Answer Questions

Section D: Long Answer Questions

Section E: Case-Based Question

CBSE Board & Sample Paper Past Year Questions

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