Comprehensive guide and practice questions for CBSE Class 11 Applied Mathematics curriculum (2026-27).
Comprehensive Concepts & Principles: Valuation of Annuities
Valuation of annuities is the mathematical evaluation of the equivalent lump-sum worth of a series of periodic payments at a specific focal date. In CBSE Class 11 Applied Mathematics, valuation centers on two fundamental metrics: Future Value (Accumulated Sum) and Present Value (Capitalized Lump Sum) across ordinary annuities, annuities due, and perpetuities.
Key Valuation Formulations:
Future Value of an Ordinary Annuity ($FVA$): The accumulated terminal worth at the end of $n$ periods:
$$FVA = R \left[ \frac{(1 + i)^n - 1}{i} \right] = R \cdot s_{\overline{n}|i}$$
Present Value of an Ordinary Annuity ($PVA$): The single lump-sum equivalent today required to fund $n$ future periodic withdrawals:
$$PVA = R \left[ \frac{1 - (1 + i)^{-n}}{i} \right] = R \cdot a_{\overline{n}|i}$$
Future Value of an Annuity Due ($FVA_{\text{due}}$): Since each payment earns interest for 1 additional interval:
$$FVA_{\text{due}} = R \left[ \frac{(1 + i)^n - 1}{i} \right](1 + i) = FVA \times (1 + i)$$
Present Value of an Annuity Due ($PVA_{\text{due}}$): $$PVA_{\text{due}} = R \left[ \frac{1 - (1 + i)^{-n}}{i} \right](1 + i) = R + R \left[ \frac{1 - (1 + i)^{-(n-1)}}{i} \right]$$
Present Value of a Perpetuity ($PV_{\infty}$): As $n \to \infty$, $(1 + i)^{-n} \to 0$:
$$PV_{\infty} = \frac{R}{i}$$
Present Value of a Perpetuity Due: $$PV_{\infty, \text{due}} = R + \frac{R}{i} = R\left(1 + \frac{1}{i}\right)$$
Present Value of a Deferred Perpetuity (deferred by $k$ periods): $$PV_{\infty, \text{def}} = \frac{R}{i}(1 + i)^{-k}$$
Timeline & Cash Flow Equivalence
Every valuation problem can be conceptualized on a geometric progression cash flow timeline. The relationship between Future Value and Present Value over term $n$ is governed by the compound interest discount/accumulation factor:
An individual deposits ₹10,000 at the end of each year in a savings scheme offering 8% per annum compound interest for 5 years. Calculate the total accumulated amount at maturity. [Given: $(1.08)^5 = 1.4693$]
Find the present value of an ordinary annuity of ₹15,000 payable at the end of each half-year for 3 years at 10% per annum compounded semi-annually. [Given: $(1.05)^{-6} = 0.7462$]
Show Step-by-Step Solution
• Periodic payment $R = ₹15,000$
• Periodic rate $i = 0.10 / 2 = 0.05$
• Number of periods $n = 3 \times 2 = 6$
$$PVA = R \left[ \frac{1 - (1 + i)^{-n}}{i} \right] = 15,000 \left[ \frac{1 - (1.05)^{-6}}{0.05} \right]$$
$$PVA = 15,000 \left[ \frac{1 - 0.7462}{0.05} \right] = 15,000 \times \frac{0.2538}{0.05} = 15,000 \times 5.076 = \mathbf{₹76,140}$$ Answer: The present value is ₹76,140.
Example 3 (Present Value of Perpetuity):
A philanthropist wishes to establish a perpetual annual scholarship of ₹60,000 to an engineering college. If the endowment fund earns 7.5% per annum interest, how much money must be donated today?
Graded textbook problems on Valuation of Ordinary Annuities, Annuities Due, and Perpetuities based on CBSE Class 11 Applied Mathematics.
Problem 1Future Value Calculation
A company deposits ₹25,000 at the end of each quarter into a sinking fund earning 8% per annum compounded quarterly. Find the accumulated amount at the end of 4 years. [Use: $(1.02)^{16} = 1.3728$]
Find the present value of an ordinary annuity of ₹8,000 payable annually for 6 years at 9% per annum compounded annually. [Use: $(1.09)^{-6} = 0.5963$]
A person deposits ₹12,000 at the beginning of every year for 5 years in an account yielding 10% per annum compound interest. Calculate the future value at the end of 5 years. [Use: $(1.10)^5 = 1.6105$]
Show Step-by-Step Solution
• Since deposits are made at the beginning, this is an Annuity Due.
• $R = ₹12,000$, $i = 0.10$, $n = 5$.
$$FVA_{\text{due}} = R \left[ \frac{(1 + i)^n - 1}{i} \right](1 + i)$$
$$FVA_{\text{due}} = 12,000 \left[ \frac{1.6105 - 1}{0.10} \right] \times 1.10 = 12,000 \times 6.1051 \times 1.10 = \mathbf{₹80,587.32}$$ Answer: The future value of the annuity due is ₹80,587.32.
Problem 4Valuation of Perpetuity
What sum of money deposited in a trust paying 6% annual interest compounded quarterly will yield a scholarship of ₹15,000 at the end of every quarter forever?
Show Step-by-Step Solution
• Periodic payment $R = ₹15,000$
• Periodic interest rate $i = \frac{0.06}{4} = 0.015$
$$PV_{\infty} = \frac{R}{i} = \frac{15,000}{0.015} = \mathbf{₹10,00,000}$$ Answer: A deposit of ₹10,00,000 must be made today.
Problem 5Capitalized Cost Application
The initial cost of installing an industrial cooling system is ₹5,00,000. It requires ₹40,000 at the end of each year for indefinite maintenance. If the interest rate is 8% p.a., find the capitalized cost of the equipment.
Show Step-by-Step Solution
• Capitalized Cost = Initial Cost + Present Value of Indefinite Maintenance Perpetuity.
$$PV_{\text{maint}} = \frac{R}{i} = \frac{40,000}{0.08} = ₹5,00,000$$
$$\text{Capitalized Cost} = 5,00,000 + 5,00,000 = \mathbf{₹10,00,000}$$ Answer: The capitalized cost is ₹10,00,000.
Comprehensive Practice Worksheet: Valuation of Annuities
Examination standard questions classified into 1-Mark, 2-Mark, and 4-Mark sections according to CBSE marking guidelines.
Q2: Express the relationship between the Future Value of an Annuity Due ($FVA_{\text{due}}$) and the Future Value of an Ordinary Annuity ($FVA$) of identical parameters.
Q3: A machine can be purchased for ₹3,00,000 cash down or for an immediate payment of ₹60,000 followed by 4 equal annual installments of ₹75,000. If the interest rate is 9% per annum compounded annually, calculate the present value of the installment plan and determine which option is cheaper for the buyer. [Given: $(1.09)^{-4} = 0.7084$]
Show Answer & Scheme
• Present value of 4 installments:
$$PVA = 75,000 \left[ \frac{1 - 0.7084}{0.09} \right] = 75,000 \times 3.24 = ₹2,43,000 \quad [1 \text{ Mark}]$$
• Total PV of Installment Plan = $60,000 + 2,43,000 = ₹3,03,000$.
• Cash Down Price = ₹3,00,000. Decision: Cash down payment is cheaper by ₹3,000. [1 Mark]
Q4: Prove algebraically that $\frac{1}{PVA} - \frac{1}{FVA} = \frac{i}{R}$.
Q5: A person plans to accumulate a retirement corpus of ₹50,00,000 over a career spanning 20 years by making equal monthly deposits at the end of each month into an index fund yielding 12% per annum compounded monthly.
(i) Calculate the required monthly deposit $R$. [Given: $(1.01)^{240} = 10.8926$]
(ii) Find the total amount of money directly contributed by the person, and hence compute the total compound interest earned over the 20-year period.
Show Step-by-Step Solution & Marking Scheme
Part (i) Monthly Deposit Calculation [2 Marks]:
• Target Future Value $FVA = ₹50,00,000$
• Periodic rate $i = 0.12/12 = 0.01$
• Number of months $n = 20 \times 12 = 240$
$$FVA = R \left[ \frac{(1+i)^n - 1}{i} \right] \implies 50,00,000 = R \left[ \frac{(1.01)^{240} - 1}{0.01} \right]$$
$$50,00,000 = R \left[ \frac{10.8926 - 1}{0.01} \right] = R \times 989.26$$
$$R = \frac{50,00,000}{989.26} \approx \mathbf{₹5,054.28} \text{ per month}$$
Part (ii) Direct Contribution & Interest [2 Marks]:
• Total Principal Contributed = $5,054.28 \times 240 = \mathbf{₹12,13,027.20}$.
• Total Compound Interest Earned = $50,00,000 - 12,13,027.20 = \mathbf{₹37,86,972.80}$. Interest accounts for over 75.7% of the total retirement fund!
High-Yield Quick Revision: Valuation of Annuities
Master reference summary of formulas, valuation rules, and algebraic identities for Chapter 13.3.
Master Formula Sheet
Annuity Type
Present Value ($PV$)
Future Value ($FV$)
Ordinary Annuity
$PVA = R \left[ \frac{1 - (1+i)^{-n}}{i} \right]$
$FVA = R \left[ \frac{(1+i)^n - 1}{i} \right]$
Annuity Due
$PVA_{\text{due}} = PVA \times (1 + i)$
$FVA_{\text{due}} = FVA \times (1 + i)$
Perpetuity
$PV = \frac{R}{i}$
Undefined (Infinite)
Perpetuity Due
$PV = R + \frac{R}{i}$
Undefined (Infinite)
Critical Exam Shortcuts
Connecting PVA and FVA: $FVA = PVA \cdot (1+i)^n$. If you have already calculated the present value, multiply by $(1+i)^n$ to get the future value without recalculating the bracket from scratch!
Reciprocal Identity: $\frac{1}{PVA} - \frac{1}{FVA} = \frac{i}{R}$. Useful for quick cross-checking in board exam verification questions.
Negative Exponents in PV: $(1+i)^{-n} = \frac{1}{(1+i)^n}$. Ensure proper order of operations when calculating $1 - (1+i)^{-n}$.
10-Question Diagnostic Board Mock Examination: Valuation of Annuities
Standard timed test: 4 Objective MCQs / Assertion-Reason with instant grading, plus 6 multi-mark subjective problems with CBSE marking rubrics.
PART 1: Objective Section (4 Marks)
Q1: The present value of an ordinary perpetuity of ₹18,000 payable annually at 9% per annum is:
Solution: (B) $PV = R / i = 18,000 / 0.09 = ₹2,00,000$.
Q2: If the present value of an ordinary annuity is ₹45,000 and the periodic interest rate is 5%, what is the present value of an annuity due with identical payments, rate, and term?
Q4 (Assertion-Reason): Assertion (A): The present value of a perpetuity is finite even though the payments continue infinitely. Reason (R): Because money has time value, cash flows in the distant future discount to near zero, causing the infinite geometric series $\sum_{k=1}^\infty (1+i)^{-k}$ to converge to $\frac{1}{i}$.
Solution: (A) Both statements are true and Reason provides the exact infinite GP convergence rationale ($|r| = \frac{1}{1+i} < 1$).
PART 2: Subjective Section (Q5 - Q10)
Q52 Marks
Find the amount of an ordinary annuity of ₹5,000 payable half-yearly for 4 years at 8% per annum compounded half-yearly. [Given: $(1.04)^8 = 1.3686$]
Calculate the present value of an annuity due of ₹20,000 payable at the beginning of each year for 3 years at 10% p.a. compounded annually. [Given: $(1.10)^{-2} = 0.8264$]
A hospital receives an offer of a perpetual annual donation of ₹1,20,000, with the first installment starting 4 years from now. If the interest rate is 6% per annum compounded annually, compute the present value of this deferred perpetuity. [Given: $(1.06)^{-3} = 0.8396$]
Show Step-by-Step Solution & Marking Scheme
• The perpetuity begins at $t = 4$. In an immediate perpetuity, payment occurs at $t = 1$.
• Deferment period $k = 4 - 1 = 3$ years [1 Mark].
• Value of perpetuity at $t = 3$:
$$V_3 = \frac{R}{i} = \frac{1,20,000}{0.06} = ₹20,00,000 \quad [1 \text{ Mark}]$$
• Discounting back to $t=0$:
$$PV = V_3 \times (1 + i)^{-3} = 20,00,000 \times 0.8396 = \mathbf{₹16,79,200} \quad [1 \text{ Mark}]$$
Q83 Marks
Show that the present value of an ordinary annuity $P$ and its future value $A$ satisfy the equation $A = P(1 + i)^n$ using their algebraic summation formulas.
Show Step-by-Step Solution & Marking Scheme
• Recall: $P = R \left[ \frac{1 - (1+i)^{-n}}{i} \right] = \frac{R}{i} \left[ 1 - \frac{1}{(1+i)^n} \right] = \frac{R}{i} \left[ \frac{(1+i)^n - 1}{(1+i)^n} \right]$ [1.5 Marks].
• Multiplying both sides by $(1+i)^n$:
$$P(1+i)^n = \frac{R}{i} \left[ (1+i)^n - 1 \right] = R \left[ \frac{(1+i)^n - 1}{i} \right]$$
• The right-hand side is precisely the formula for Future Value $A$.
• Hence, $A = P(1+i)^n$. [1.5 Marks]
Q9 & Q10Case Study (4 Marks Total)
Case Context: Corporate Pension Fund Portfolio
A corporate pension fund must pay a retiring employee an ordinary annuity of ₹30,000 at the end of each month for 15 years. The fund's asset portfolio earns a guaranteed annual return of 6% compounded monthly ($i = 0.06/12 = 0.005$). [Given: $(1.005)^{-180} = 0.40748$, $(1.005)^{180} = 2.4541$]
Q9 (2 Marks): Calculate the exact lump sum corpus that the corporation must allocate today into the fund to guarantee these payments. Q10 (2 Marks): What is the total sum of money the pensioner will receive over the 15-year period, and how much of that total is funded by investment interest earned by the corpus?
Solution to Q10 [2 Marks]:
• Total Payout to Pensioner = $30,000 \times 180 = \mathbf{₹54,00,000}$.
• Interest Earned = Total Payout - Initial Corpus = $54,00,000 - 35,55,120 = \mathbf{₹18,44,880}$.
• Over ₹18.44 Lakhs of the retirement benefits are generated directly through compound interest.