Unit 7 · 15 Marks
Topics & Key Formulas
Six topics examined across MCQs, short answers, long answers, and case studies. Memorise the formulas below — they appear in multiple question types every year.
1. Perpetuity
Present value of an infinite series of equal periodic payments.
\(\text{PV} = \dfrac{A}{r}\)
A = annual payment, r = rate per period
A = annual payment, r = rate per period
2. Sinking Funds
Periodic deposits to accumulate a target future amount at compound interest.
\(A = \dfrac{S \cdot r}{(1+r)^n - 1}\)
S = target amount, r = rate, n = periods
S = target amount, r = rate, n = periods
3. EMI & Amortization
Fixed monthly loan repayments; split into interest and principal components.
\(\text{EMI} = \dfrac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1}\)
r = monthly rate = annual rate ÷ 12
r = monthly rate = annual rate ÷ 12
4. CAGR
Compounded Annual Growth Rate — smooths growth for investment comparisons.
\(\text{CAGR} = \left(\dfrac{FV}{PV}\right)^{1/n} - 1\)
FV = final value, PV = initial value
FV = final value, PV = initial value
5. Depreciation
Straight-line (constant) and reducing-balance (WDV) methods.
SLM: \(D = \dfrac{\text{Cost} - \text{Salvage}}{n}\)
RBM: \(\text{BV} = \text{Cost} \times (1-r)^n\)
6. Valuation of Bonds
Price = PV of coupon stream + PV of face value redemption.
\(P = C \cdot \dfrac{1-(1+y)^{-n}}{y} + \dfrac{F}{(1+y)^n}\)
C = coupon, y = yield, F = face value
C = coupon, y = yield, F = face value
7
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Practice MCQs — Unit 7
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Q1Perpetuity
What is the present value of a perpetuity of ₹1,000 per year at 8% per annum?
Formula: \(\text{PV} = \dfrac{A}{r}\)
\(\text{PV} = \dfrac{1{,}000}{0.08} =\)₹12,500
\(\text{PV} = \dfrac{1{,}000}{0.08} =\)₹12,500
Q2Sinking Fund
A company wants to accumulate ₹1,00,000 in 5 years at 10% p.a. compounded annually. What is the approximate annual deposit required?
\(A = \dfrac{S \cdot r}{(1+r)^n - 1} = \dfrac{1{,}00{,}000 \times 0.10}{(1.10)^5 - 1}\)
\((1.10)^5 = 1.61051\)
\(A = \dfrac{10{,}000}{0.61051} \approx\)₹16,380
\((1.10)^5 = 1.61051\)
\(A = \dfrac{10{,}000}{0.61051} \approx\)₹16,380
Q3EMI & Amortization
What is the EMI formula for a loan of \(P\) at monthly rate \(r\) for \(n\) months?
Standard EMI formula from the PV of an annuity: \(\text{EMI} = \dfrac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1}\) where \(r\) = monthly rate (annual rate ÷ 12).
Q4EMI & Amortization
For a loan at 12% per annum, what monthly interest rate should be used in the EMI formula?
Monthly rate \(= \dfrac{12\%}{12} = 1\% = 0.01\). Always convert the annual rate to monthly before substituting into the EMI formula.
Q5CAGR
An investment grows from ₹10,000 to ₹15,000 in 3 years. What is the approximate CAGR?
\(\text{CAGR} = \left(\dfrac{15{,}000}{10{,}000}\right)^{1/3} - 1 = (1.5)^{0.333} - 1 \approx \mathbf{14.5\%}\)
Q6CAGR
What is the formula for CAGR?
\(\text{CAGR} = \left(\dfrac{\text{Ending}}{\text{Beginning}}\right)^{1/n} - 1\). Option (a) = total return; option (c) = simple average. CAGR accounts for compounding.
Q7Depreciation
A machine costs ₹1,00,000 with salvage value ₹10,000 after 10 years. What is the annual depreciation under the straight-line method?
\(D = \dfrac{1{,}00{,}000 - 10{,}000}{10} = \dfrac{90{,}000}{10} =\)₹9,000
Q8Depreciation
Using the reducing-balance method at 10% p.a., what is the book value of a ₹50,000 asset after 2 years?
\(\text{BV} = 50{,}000 \times (0.90)^2 = 50{,}000 \times 0.81 =\)₹40,500
Q9Depreciation
An asset depreciates from ₹80,000 to ₹64,000 in one year under the reducing-balance method. What is the depreciation rate?
\(64{,}000 = 80{,}000 \times (1-r) \Rightarrow r = 0.20 = \mathbf{20\%}\)
Q10Bond Valuation
A bond with face value ₹1,000 pays an 8% annual coupon. If the required return is 10%, at what does the bond trade?
Coupon rate (8%) < Required return (10%) → Discount.
Rule: Coupon > Yield → Premium | Coupon < Yield → Discount | Coupon = Yield → Par
Rule: Coupon > Yield → Premium | Coupon < Yield → Discount | Coupon = Yield → Par
Q11Bond Valuation
A bond with face value ₹10,000 and coupon rate 6% pays what annual interest?
Annual Coupon \(= 10{,}000 \times 0.06 =\)₹600
Q12Perpetuity
What is the present value of a perpetuity of ₹500 at 5% per annum?
\(\text{PV} = \dfrac{500}{0.05} =\)₹10,000
Q13EMI & Amortization
In an amortization schedule, what happens to the components of each EMI as time progresses?
Interest is charged on the outstanding principal. As principal reduces each month, the interest falls. Since total EMI is fixed, the principal repayment rises correspondingly.
Q14Sinking Fund
Annual deposits of ₹10,000 for 4 years at 8% compounded annually accumulate to approximately what amount?
\(FV = 10{,}000 \times \dfrac{(1.08)^4 - 1}{0.08} = 10{,}000 \times 4.5061 =\)₹45,061
Q15Bond Valuation
How is the current yield of a bond calculated?
\(\text{Current Yield} = \dfrac{\text{Annual Coupon Payment}}{\text{Current Market Price}}\). Shows annual income return based on what you pay today.
Q16Depreciation
Compared to the straight-line method, what does the reducing-balance method result in?
RBM applies a fixed rate to the declining book value. Early on, book value is highest, so depreciation is largest. SLM charges a constant amount regardless of book value.
Q17CAGR
What is CAGR most useful for?
CAGR smooths out year-to-year volatility into a single steady-growth equivalent rate, ideal for comparing two investments over different time periods.
Q18EMI & Amortization
What does each EMI payment consist of?
Every EMI = Interest portion + Principal portion. The total EMI stays constant, but the interest falls and principal rises as the outstanding balance decreases.
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Q1Perpetuity
Find the present value of a perpetuity that pays ₹2,000 at the end of each year if the interest rate is 5% per annum.
Formula: \(\text{PV} = \dfrac{A}{r}\)
Annual payment \(A =\)₹2,000, rate \(r = 5\% = 0.05\)
\(\text{PV} = \dfrac{2{,}000}{0.05} =\)₹40,000
Present Value = ₹40,000
Q2Sinking Fund
A company needs ₹5,00,000 in 4 years to replace machinery. Interest rate is 8% p.a. compounded annually. Find the annual deposit required.
Formula: \(A = \dfrac{S \cdot r}{(1+r)^n - 1}\)
Given: \(S =\)₹5,00,000, \(r = 0.08\), \(n = 4\). \((1.08)^4 = 1.36049\)
\(A = \dfrac{5{,}00{,}000 \times 0.08}{0.36049} = \dfrac{40{,}000}{0.36049}\)
Annual Deposit ≈ ₹1,10,978
Q3EMI
Calculate the EMI for a loan of ₹2,00,000 at 12% per annum for 2 years (24 months).
\(P =\)₹2,00,000, \(r = 0.01\), \(n = 24\). \((1.01)^{24} = 1.2697\)
\(\text{EMI} = \dfrac{2{,}00{,}000 \times 0.01 \times 1.2697}{1.2697 - 1} = \dfrac{2{,}539.4}{0.2697}\)
EMI ≈ ₹9,414 per month
Q4CAGR
An investment of ₹50,000 grows to ₹80,000 in 5 years. Calculate the CAGR.
\(\text{CAGR} = \left(\dfrac{80{,}000}{50{,}000}\right)^{1/5} - 1 = (1.6)^{0.2} - 1 = 1.0986 - 1\)
CAGR ≈ 9.86%
Q5Depreciation
A machine costs ₹2,00,000 with salvage value ₹20,000 after 8 years. Calculate (i) annual depreciation (SLM), (ii) book value after 3 years.
(i) Annual Depreciation
\(D = \dfrac{2{,}00{,}000 - 20{,}000}{8} = \dfrac{1{,}80{,}000}{8} =\)₹22,500
(ii) Book Value after 3 Years
Total depreciation \(= 22{,}500 \times 3 =\)₹67,500. Book Value \(= 2{,}00{,}000 - 67{,}500\)
(i) Annual Depreciation = ₹22,500 | (ii) Book Value = ₹1,32,500
Q6Bond Valuation
A bond has face value ₹1,000, coupon rate 10%, 3 years to maturity. Required return = 12%. State whether the bond trades at premium or discount and explain why.
Annual Coupon \(= 1{,}000 \times 10\% =\)₹100
Coupon Rate (10%) < Required Return (12%)
When coupon rate < market yield, investors require a price reduction to earn the higher yield.
Bond trades at DISCOUNT. Market Price < ₹1,000 because coupon rate (10%) < required return (12%).
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LA Q1Sinking Fund
A company wants to accumulate ₹10,00,000 in 6 years for machinery replacement at 10% p.a. compounded annually. (a) Calculate the annual deposit. (b) Prepare a sinking fund schedule for the first 3 years.
(a) Annual Deposit
\(A = \dfrac{10{,}00{,}000 \times 0.10}{(1.10)^6 - 1}\). \((1.10)^6 = 1.77156\). \(A = \dfrac{1{,}00{,}000}{0.77156} \approx\)₹1,29,607
(b) Sinking Fund Schedule
Year 1: Deposit ₹1,29,607 | Interest ₹0 | Balance ₹1,29,607
Year 2: Deposit ₹1,29,607 | Interest ₹12,961 | Balance ₹2,72,175
Year 3: Deposit ₹1,29,607 | Interest ₹27,218 | Balance ₹4,29,000
(a) Annual Deposit = ₹1,29,607 | (b) Balance after 3 years ≈ ₹4,29,000
LA Q2EMI & Amortization
A person borrows ₹5,00,000 at 12% p.a. for 3 years (36 months). (a) Calculate the monthly EMI. (b) Prepare an amortization schedule for the first 3 months.
(a) EMI Calculation
\(P =\)₹5,00,000, \(r = 0.01\), \(n = 36\). \((1.01)^{36} = 1.4308\). EMI \(= \dfrac{5{,}00{,}000 \times 0.01 \times 1.4308}{0.4308} \approx\)₹16,607
(b) Amortization Schedule
Month 1: Interest ₹5,000 | Principal ₹11,607 | Balance ₹4,88,393
Month 2: Interest ₹4,884 | Principal ₹11,723 | Balance ₹4,76,670
Month 3: Interest ₹4,767 | Principal ₹11,840 | Balance ₹4,64,830
(a) Monthly EMI = ₹16,607 | Note: interest falls while principal repayment rises each month.
LA Q3Depreciation
A machine costs ₹3,00,000 with a salvage value of ₹30,000 after 5 years. Compare the depreciation and book values under (a) the straight-line method, and (b) the reducing-balance method at 25% p.a., for all 5 years.
(a) Straight-Line Method
\(D = \dfrac{3{,}00{,}000 - 30{,}000}{5} =\)₹54,000/year. Year 5 BV = ₹30,000 ✓
(b) Reducing-Balance Method (r = 25%)
\(\text{BV}_n = 3{,}00{,}000 \times (0.75)^n\). Year 1 Dep ₹75,000; Year 5 BV ₹71,191
SLM: constant ₹54,000/year, reaches exact salvage. RBM: higher early depreciation (₹75,000 Year 1); final BV ₹71,191 > SLM salvage.
LA Q4CAGR
A portfolio is valued at ₹1,00,000 at the start (Y0), and grows to ₹1,15,000 at the end of year 1, ₹1,20,000 at the end of year 2, ₹1,45,000 at the end of year 3, and ₹1,60,000 at the end of year 4. Calculate (a) the CAGR over 4 years, (b) the year-on-year growth rates, and (c) compare the CAGR with the simple average growth rate.
(a) \(\text{CAGR} = (1.6)^{0.25} - 1 = 1.1246 - 1 = \mathbf{12.46\%}\)
(b) Y1: 15% | Y2: 4.35% | Y3: 20.83% | Y4: 10.34%. Simple avg = 12.63%
(a) CAGR = 12.46% | (b) Annual: 15%, 4.35%, 20.83%, 10.34% | (c) CAGR (12.46%) < simple average (12.63%) due to compounding.
LA Q5Bond Valuation
A bond has a face value of ₹1,000, a coupon rate of 10%, and 4 years to maturity, with a required return of 8%. Calculate (a) the present value of the coupons, (b) the present value of the face value, (c) the bond price, and (d) state whether the bond trades at a premium or a discount.
Coupon \(C =\)₹100, yield \(y = 0.08\), \(n = 4\)
(a) \(PV_C = 100 \times \dfrac{1-(1.08)^{-4}}{0.08} = 100 \times 3.3121 =\)₹331.21
(b) \(PV_F = \dfrac{1{,}000}{(1.08)^4} = \dfrac{1{,}000}{1.3605} =\)₹735.03
(a) ₹331.21 | (b) ₹735.03 | (c) Bond Price = ₹1,066.24 | (d) PREMIUM — coupon (10%) > yield (8%).
LA Q6Perpetuity
Compare (a) a perpetuity paying ₹10,000 per year at 8%, (b) a 20-year annuity, and (c) a 50-year annuity, both with the same payment and rate as the perpetuity. Comment on the relationship between their present values.
(a) Perpetuity: \(\dfrac{10{,}000}{0.08} =\)₹1,25,000
(b) 20-yr annuity: \(10{,}000 \times 9.8181 =\)₹98,181 (78.5% of perpetuity)
(c) 50-yr annuity: \(10{,}000 \times 12.2335 =\)₹1,22,335 (97.9% of perpetuity)
As annuity period grows, PV converges to perpetuity value. Cash flows far in the future contribute almost nothing due to discounting.
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Board-Pattern Questions
Case Study Based Questions
Real-world application problems. Read the context carefully, then click Show Answer under each part.
Rajesh wants to buy a house worth ₹50,00,000. He has ₹10,00,000 as a down payment and borrows the remaining amount. The bank offers a home loan at 9% per annum for 20 years (240 months).
(i)EMI
What is the loan amount Rajesh needs to borrow?
Loan = ₹50,00,000 − ₹10,00,000 = ₹40,00,000
(ii)
What is the monthly interest rate for this loan?
Monthly rate \(= \dfrac{9\%}{12} = 0.75\% = 0.0075\)
(iii)
What is the approximate monthly EMI?
\((1.0075)^{240} = 6.0092\). EMI \(= \dfrac{40{,}00{,}000 \times 0.0075 \times 6.0092}{5.0092} \approx\)₹35,984
(iv)
In the first month’s EMI, what is the approximate interest component?
Interest = \(40{,}00{,}000 \times 0.0075 =\)₹30,000. Principal = ₹5,984.
A company invested ₹20,00,000 in a business venture. After 4 years, the investment grew to ₹28,00,000. Simultaneously, the company purchased machinery worth ₹15,00,000 that depreciates at 15% per annum (RBM).
(i)CAGR
What is the approximate CAGR of the investment?
\(\text{CAGR} = (1.4)^{0.25} - 1 = 1.0878 - 1 \approx \mathbf{8.8\%}\)
(ii)Depreciation
What is the book value of the machinery after 1 year?
\(15{,}00{,}000 \times 0.85 =\)₹12,75,000
(iii)
What is the approximate book value of the machinery after 4 years?
\(15{,}00{,}000 \times (0.85)^4 = 15{,}00{,}000 \times 0.52201 \approx\)₹7,82,063
(iv)
What is the combined asset position (investment plus machinery book value) after 4 years?
₹28,00,000 + ₹7,82,063 = ₹35,82,063
Option A: Corporate bond, face value ₹10,000, coupon rate 12%, 5 years to maturity, market yield 10%. Option B: Perpetual bond paying ₹1,200 annually at market yield 10%.
(i)Bonds
What is the annual coupon payment from Option A?
Annual Coupon \(= 10{,}000 \times 12\% =\)₹1,200
(ii)
At what does the Option A bond trade?
Coupon rate (12%) > Market yield (10%) → PREMIUM
(iii)Perpetuity
What is the present value of Option B as a perpetuity?
\(\text{PV} = \dfrac{1{,}200}{0.10} =\)₹12,000
(iv)
If the market yield increases to 12%, what will happen to the value of Option B?
New PV \(= \dfrac{1{,}200}{0.12} =\)₹10,000. Bond/perpetuity value and market yield move in opposite directions.
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Exam Tips — Unit 7
How to score full marks in Unit 7 — mistakes to avoid and strategies that work.
✓ Tip 1
Always convert annual rate to monthly before applying the EMI formula. The most common error is using 12% directly as \(r\) instead of dividing by 12 to get 1% (= 0.01). Write \(r = \text{annual rate} \div 12\) as your first step and show it explicitly — it earns a method mark.
🔒 5 more exam tips — bond premium/discount rule, CAGR vs simple average, Perpetuity, Sinking Funds & Depreciation — are in the AI Question Bank.
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Frequently Asked Questions
Unit 7: Financial Mathematics carries 15 marks in the CBSE Class 12 Applied Mathematics board examination. Questions can appear as MCQs, short answers, long answers, or case studies from any of the 6 topics.
Unit 7 covers six topics: Perpetuity, Sinking Funds, EMI and Amortization, CAGR, Depreciation (SLM and RBM), and Bond Valuation.
\[\text{EMI} = \frac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1}\] where \(r\) = monthly rate (annual rate ÷ 12). The most common mistake is forgetting to divide the annual rate by 12.
SLM: \(D = (\text{Cost} - \text{Salvage}) \div n\) — constant every year; always reaches the exact salvage value. RBM: \(\text{BV}_n = \text{Cost} \times (1-r)^n\) — higher depreciation in early years.
Premium when coupon rate > yield. Discount when coupon rate < yield. Par when both rates are equal. Bond price and yield move in opposite directions.
Simple average adds annual rates and divides — ignores compounding. CAGR accounts for compounding and represents the true compound growth rate. CAGR is preferred for investment comparisons.
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