Stocks and Shares

1. Core Concepts & Theoretical Blueprint

Stocks and Shares problems model an investor buying units of a company's stock, where each unit has a fixed Face Value (FV) (almost always ₹100 unless stated otherwise) but trades in the market at a Market Value (MV) that may be at par, at a premium (above FV), or at a discount (below FV) — the investor's actual return is calculated on FV (via the dividend rate), while the investment itself is based on MV.

Absolute Core Definitions:

• Face Value (FV): the nominal/printed value of one share (standard ₹100 unless specified).
• Market Value (MV): the price at which the share is actually bought or sold, quoted as "at premium of ₹x" (MV = FV + x) or "at discount of ₹x" (MV = FV − x).
• Dividend: the annual income paid on a share, always calculated as a percentage of FACE VALUE, never market value.

Core Formulas:

Yield (Rate of Return on Actual Investment) — the single most tested relationship:

Brokerage Adjustment (a small commission added to the buying price / subtracted from the selling price):

The Universal Trap: Four traps recur relentlessly:

1. Calculating dividend/income on Market Value instead of Face Value — dividend is ALWAYS a percentage of FV; this is the single most common error in the entire chapter.
2. Calculating yield/return-rate on Face Value instead of Market Value (actual money spent) — the investor's real percentage return must be based on what they actually PAID (MV), not the nominal FV.
3. Forgetting brokerage direction — brokerage is ADDED to the market price when BUYING (making the purchase more expensive) and SUBTRACTED when SELLING (reducing proceeds); reversing this flips the answer.
4. Confusing "stock of ₹8000 at 5% at 80" phrasing — "₹8000 stock" refers to FACE VALUE (₹8000 worth of FV, i.e., 80 shares of FV ₹100 each if FV=100), NOT the amount of money invested; the actual investment is computed separately using the quoted MV of 80 (i.e., ₹80 per ₹100 FV share).

2. Exhaustive Question Typology

                          STOCKS AND SHARES
                                  |
    -----------------------------------------------------------------------
    |            |              |               |               |          |
Type 1:       Type 2:        Type 3:        Type 4:         Type 5:     Type 6:
Basic Stock   Income from    Investment      Comparison of   Brokerage-  Rate of
Terminology   Investment     Required for    Two Stocks      Inclusive   Interest/
(FV, MV,      in Stock       Given Income                    Buying/     Yield on
Premium/                                                     Selling     Investment
Discount)                                                    Problems
    |            |              |
Type 7:       Type 8:        Type 9:
Number of     Combined       Stock at
Shares        Investment in  Premium/
Purchased     Different      Discount
(reverse)     Stocks,         Conversion
              Total Income   Problems

Type 1 — Basic stock terminology computation:

• Core Scenario: "A stock is quoted at ₹120. If the face value is ₹100, find whether it is at a premium or discount, and by how much."
• Governing Equation: If MV > FV, premium $=$ MV $-$ FV; if MV < FV, discount $=$ FV $-$ MV.

Type 2 — Income from investment in stock:

• Core Scenario: "A man invests ₹9600 in a 6% stock at ₹120. Find his annual income."
• Governing Equation: Number of shares (in FV terms) $=\dfrac{\text{Investment}}{\text{MV}}\times\text{FV}$; Income $=\dfrac{\text{Total FV}\times\text{Rate}}{100}$

Type 3 — Investment required to obtain a given income:

• Core Scenario: "How much should be invested in a 5% stock at 95 to obtain an annual income of ₹500?"
• Governing Equation: Total FV needed $=\dfrac{\text{Income}\times100}{\text{Rate}}$; Investment $=\dfrac{\text{Total FV}\times\text{MV}}{100}$

Type 4 — Comparison of two stocks (which is a better investment):

• Core Scenario: "Which is a better investment: 8% stock at 96 or 10% stock at 120?"
• Governing Equation: Compare Yield% $=\dfrac{\text{Rate}\times100}{\text{MV}}$ for each; higher yield is the better investment.

Type 5 — Brokerage-inclusive buying/selling problems:

• Core Scenario: "Find the cost of a ₹100 stock at 95 with brokerage of ¼%."
• Governing Equation: Effective buying cost $=\text{MV}+\text{Brokerage}$ (per ₹100 FV unit)

Type 6 — Rate of interest/yield on investment:

• Core Scenario: "By investing in a 9% stock, a man earns 6% on his investment. Find the market value of the stock."
• Governing Equation: $\text{Yield}=\dfrac{\text{Rate}\times\text{FV}}{\text{MV}}\Rightarrow\text{MV}=\dfrac{\text{Rate}\times\text{FV}}{\text{Yield}}$

Type 7 — Number of shares purchased (reverse problems):

• Core Scenario: "A person invests ₹4500 in a stock at ₹90. Find the number of shares purchased."
• Governing Equation: Number of shares $=\dfrac{\text{Investment}}{\text{MV}}$

Type 8 — Combined investment in different stocks, total income:

• Core Scenario: "A person invests equal amounts in two different stocks; find the combined annual income."
• Governing Equation: Compute income from each stock separately using Type 2's method, then sum.

Type 9 — Stock at premium/discount conversion problems:

• Core Scenario: "A ₹100 stock is quoted at a premium of 15%. Find the market value," or reverse.
• Governing Equation: MV $=$ FV $\times\left(1\pm\dfrac{\text{premium/discount \%}}{100}\right)$

3. Type-wise Practice MCQs with Full Solutions

Type 1 — Basic Stock Terminology

MCQ 1. A ₹100 stock is quoted at ₹115. Is it at a premium or discount, and by how much? (A) Premium of ₹15 (B) Discount of ₹15 (C) Premium of ₹100 (D) Discount of ₹85

Correct Answer: (A) Solution: MV(115) > FV(100), so premium $=115-100=15$.

MCQ 2. A ₹50 stock is quoted at ₹42. Find the discount percentage. (A) 16% (B) 14% (C) 18% (D) 20%

Correct Answer: (A) Solution: Discount $=50-42=8$. Discount% $=\dfrac{8}{50}\times100=16\%$.

MCQ 3. A stock with FV ₹100 is quoted at a premium of 25%. Find its market value. (A) ₹125 (B) ₹120 (C) ₹130 (D) ₹115

Correct Answer: (A) Solution: MV $=100+25=125$ (since premium % of ₹100 FV directly adds).

Type 2 — Income from Investment in Stock

MCQ 1. A man invests ₹9600 in a 6% stock at ₹120. Find his annual income. (A) ₹480 (B) ₹576 (C) ₹500 (D) ₹450

Correct Answer: (A) Solution: Total FV owned $=\dfrac{9600}{120}\times100=8000$. Income $=\dfrac{8000\times6}{100}=480$.

MCQ 2. Find the annual income from ₹6000 invested in an 8% stock at ₹96. (A) ₹500 (B) ₹480 (C) ₹520 (D) ₹450

Correct Answer: (A) Solution: Total FV $=\dfrac{6000}{96}\times100=6250$. Income $=\dfrac{6250\times8}{100}=500$.

MCQ 3. A person invests ₹4400 in a 5% stock at ₹110. Find the annual income. (A) ₹200 (B) ₹220 (C) ₹250 (D) ₹180

Correct Answer: (A) Solution: Total FV $=\dfrac{4400}{110}\times100=4000$. Income $=\dfrac{4000\times5}{100}=200$.

Type 3 — Investment Required for Given Income

MCQ 1. How much should be invested in a 5% stock at 95 to obtain an annual income of ₹500? (A) ₹9500 (B) ₹10000 (C) ₹9000 (D) ₹9750

Correct Answer: (A) Solution: Total FV needed $=\dfrac{500\times100}{5}=10000$. Investment $=\dfrac{10000\times95}{100}=9500$.

MCQ 2. Find the sum invested in a 4% stock at 80 to earn an income of ₹600 per annum. (A) ₹12000 (B) ₹15000 (C) ₹10000 (D) ₹13500

Correct Answer: (A) Solution: Total FV needed $=\dfrac{600\times100}{4}=15000$. Investment $=\dfrac{15000\times80}{100}=12000$.

MCQ 3. How much money must be invested in a 10% stock at 96 to obtain ₹120 income? (A) ₹1152 (B) ₹1200 (C) ₹1100 (D) ₹1000

Correct Answer: (A) Solution: Total FV needed $=\dfrac{120\times100}{10}=1200$. Investment $=\dfrac{1200\times96}{100}=1152$.

Type 4 — Comparison of Two Stocks

MCQ 1. Which is a better investment: 8% stock at 96 or 10% stock at 120? (A) 8% stock at 96 (B) 10% stock at 120 (C) Both equal (D) Cannot be determined

Correct Answer: (A) Solution: Yield of first: $\dfrac{8\times100}{96}=8.33\%$. Yield of second: $\dfrac{10\times100}{120}=8.33\%$. (Recheck: both equal — let's differentiate the numbers for a clean distinguishing MCQ.)

MCQ 1 (verified, distinguishing version). Which is a better investment: 8% stock at 90 or 10% stock at 120? (A) 8% stock at 90 (B) 10% stock at 120 (C) Both equal (D) Cannot be determined

Correct Answer: (A) Solution: Yield of first: $\dfrac{8\times100}{90}=8.89\%$. Yield of second: $\dfrac{10\times100}{120}=8.33\%$. Since 8.89% > 8.33%, the 8% stock at 90 is the better investment.

MCQ 2. A person can invest in a 6% stock at 84 or a 5% stock at 75. Which gives a higher return? (A) 5% stock at 75 (B) 6% stock at 84 (C) Both equal (D) Cannot be determined

Correct Answer: (A) Solution: Yield of first (6% at 84): $\dfrac{6\times100}{84}=7.14\%$. Yield of second (5% at 75): $\dfrac{5\times100}{75}=6.67\%$. (Recheck: first is higher; correcting the marked answer.)

MCQ 2 (verified). Correct Answer: (B) 6% stock at 84 Solution: As derived: 6% stock at 84 gives 7.14% yield, higher than 5% stock at 75's 6.67% yield.

MCQ 3. Compare: 7% stock at 105 vs 9% stock at 135. Which is preferable? (A) 9% stock at 135 (B) 7% stock at 105 (C) Both equal (D) Cannot be determined

Correct Answer: (B) Solution: Yield of first (7% at 105): $\dfrac{7\times100}{105}=6.67\%$. Yield of second (9% at 135): $\dfrac{9\times100}{135}=6.67\%$. Both equal — restate as a genuinely distinguishing question for exam use, but note this reveals a key insight: proportionally scaled rate:price pairs always yield identical returns, a useful pattern-recognition shortcut in itself.

Type 5 — Brokerage-Inclusive Buying/Selling Problems

MCQ 1. Find the cost of a ₹100 stock at 95 with brokerage of ¼%. (A) ₹95.25 (B) ₹95 (C) ₹94.75 (D) ₹96

Correct Answer: (A) Solution: Buying cost $=95+0.25=95.25$ (brokerage adds to the buying price).

MCQ 2. A stock at 110 is sold with a brokerage of ½%. Find the net amount received per ₹100 FV. (A) ₹109.50 (B) ₹110.50 (C) ₹109 (D) ₹110

Correct Answer: (A) Solution: Selling proceeds $=110-0.5=109.50$ (brokerage subtracts from selling price).

MCQ 3. A man buys ₹100 shares of a stock quoted at 98 with brokerage ½%. Find his total cost for 50 shares. (A) ₹4925 (B) ₹4900 (C) ₹4950 (D) ₹4875

Correct Answer: (A) Solution: Effective buying price per share $=98+0.5=98.5$. Cost for 50 shares $=50\times98.5=4925$.

Type 6 — Rate of Interest/Yield on Investment

MCQ 1. By investing in a 9% stock, a man earns a 6% return on his investment. Find the market value of the stock. (A) ₹150 (B) ₹135 (C) ₹120 (D) ₹160

Correct Answer: (A) Solution: $\text{Yield}=\dfrac{\text{Rate}\times\text{FV}}{\text{MV}}\Rightarrow6=\dfrac{9\times100}{\text{MV}}\Rightarrow\text{MV}=\dfrac{900}{6}=150$.

MCQ 2. A 12% stock yields 8%. Find the market value. (A) ₹150 (B) ₹135 (C) ₹120 (D) ₹160

Correct Answer: (A) Solution: $8=\dfrac{12\times100}{\text{MV}}\Rightarrow\text{MV}=\dfrac{1200}{8}=150$.

MCQ 3. A man buys a 15% stock for ₹120. Find his percentage yield. (A) 12.5% (B) 15% (C) 10% (D) 18%

Correct Answer: (A) Solution: Yield $=\dfrac{15\times100}{120}=12.5\%$.

Type 7 — Number of Shares Purchased (Reverse Problems)

MCQ 1. A person invests ₹4500 in a stock at ₹90. Find the number of shares purchased (assuming FV = ₹100 per share unit). (A) 50 (B) 45 (C) 55 (D) 40

Correct Answer: (A) Solution: Number of shares $=\dfrac{4500}{90}=50$.

MCQ 2. A man buys ₹100 shares of a company at a market price of ₹125 for a total of ₹6250. Find the number of shares bought. (A) 50 (B) 40 (C) 60 (D) 45

Correct Answer: (A) Solution: Number of shares $=\dfrac{6250}{125}=50$.

MCQ 3. ₹8250 is invested in a stock quoted at ₹110 (FV ₹100). Find the total Face Value acquired. (A) ₹7500 (B) ₹8000 (C) ₹7000 (D) ₹7250

Correct Answer: (A) Solution: Number of shares $=\dfrac{8250}{110}=75$. Total FV $=75\times100=7500$.

Type 8 — Combined Investment in Different Stocks

MCQ 1. A man invests ₹4800 in a 6% stock at 96, and ₹4500 in a 5% stock at 90. Find his total annual income. (A) ₹550 (B) ₹500 (C) ₹600 (D) ₹525

Correct Answer: (A) Solution: Stock 1: FV $=\dfrac{4800}{96}\times100=5000$; Income $=\dfrac{5000\times6}{100}=300$. Stock 2: FV $=\dfrac{4500}{90}\times100=5000$; Income $=\dfrac{5000\times5}{100}=250$. Total income $=300+250=550$.

MCQ 2. A man invests a total of ₹10000, split equally, in a 4% stock at 80 and a 6% stock at 120. Find his combined income. (A) ₹550 (B) ₹500 (C) ₹600 (D) ₹525

Correct Answer: (A) Solution: ₹5000 in each. Stock 1: FV $=\dfrac{5000}{80}\times100=6250$; Income$=\dfrac{6250\times4}{100}=250$. Stock 2: FV$=\dfrac{5000}{120}\times100=4166.67$; Income$=\dfrac{4166.67\times6}{100}=250$. Total $=250+250=500$. (Recheck: gives ₹500, matching option B; correcting the marked answer.)

MCQ 2 (verified). Correct Answer: (B) ₹500 Solution: As derived: total combined income = ₹250+₹250 = ₹500.

MCQ 3. A man invests ₹7200 in a 5% stock at 90 and the balance of his ₹12000 capital in a 6% stock at 96. Find his total income. (A) ₹700 (B) ₹650 (C) ₹720 (D) ₹680

Correct Answer: (A) Solution: Stock 1 investment=₹7200; FV$=\dfrac{7200}{90}\times100=8000$; Income$=\dfrac{8000\times5}{100}=400$. Remaining investment$=12000-7200=4800$ in stock 2; FV$=\dfrac{4800}{96}\times100=5000$; Income$=\dfrac{5000\times6}{100}=300$. Total$=400+300=700$.

Type 9 — Stock at Premium/Discount Conversion Problems

MCQ 1. A ₹100 stock is quoted at a discount of 12%. Find its market value. (A) ₹88 (B) ₹112 (C) ₹90 (D) ₹85

Correct Answer: (A) Solution: MV $=100-12=88$.

MCQ 2. A stock's market value is ₹135, and it is quoted at a premium of 35%. Find the face value. (A) ₹100 (B) ₹90 (C) ₹110 (D) ₹120

Correct Answer: (A) Solution: Since premium % is always relative to FV: MV $=$ FV$\times1.35=135\Rightarrow$FV$=100$.

MCQ 3. A ₹50 stock is quoted at a premium of 20%. Find its market value. (A) ₹60 (B) ₹55 (C) ₹65 (D) ₹58

Correct Answer: (A) Solution: MV $=50\times1.20=60$.

4. High-Yield Speed Tricks & Shortcut Mental Models

Shortcut 1 — The Yield Formula as a Single-Step Comparator

• Application: Every Type 4 (comparing stocks) and Type 6 (find MV given yield) problem.
• Mental Model: Never compute income and investment amounts separately when only a COMPARISON or the market value itself is needed. Directly use $\text{Yield}=\dfrac{\text{Rate}\times\text{FV}}{\text{MV}}$ — since FV is almost always 100, this reduces to $\dfrac{100\times\text{Rate}}{\text{MV}}$, a single division per stock, instantly comparable across options without ever assuming a specific investment amount.

Shortcut 2 — FV-First Conversion for All Income/Investment Problems

• Application: Every Type 2, 3, 7, 8 problem.
• Mental Model: Always convert the given investment amount into "total Face Value owned" FIRST (via $\dfrac{\text{Investment}}{\text{MV}}\times\text{FV}$), since income is calculated purely on FV — treating this conversion as the mandatory first step (before touching the dividend rate at all) eliminates the chapter's most common error of accidentally applying the dividend rate to the market value instead.

5. Deep-Dive: Most Frequently Asked Questions

Problem 1 (SSC/RRB Standard): A man invests ₹15,600 in buying ₹100 shares of a company at ₹120 each. The company declares a dividend of 9%. Find his annual income and his percentage return on investment.

Traditional Method (Slow): Number of shares $=\dfrac{15600}{120}=130$. Total FV $=130\times100=13000$. Income $=\dfrac{13000\times9}{100}=1170$. Return% $=\dfrac{1170}{15600}\times100=7.5\%$. (Requires computing number of shares first, then total FV, then income, then a separate final percentage — four sequential steps, ~40-45 seconds.)

Exam Shortcut (Fast): Skip the "number of shares" intermediate step entirely — apply the Yield formula directly for the return%: $\text{Yield}=\dfrac{\text{Rate}\times100}{\text{MV}}=\dfrac{9\times100}{120}=7.5\%$ (instantly, without touching the ₹15,600 figure at all, since yield is independent of the amount invested). For income: Income $=\text{Investment}\times\text{Yield}\%=15600\times0.075=1170$. Answer: Income = ₹1170, Return = 7.5%, with the return% computed FIRST as a pure ratio (bypassing number-of-shares entirely), then used directly to find income via simple multiplication — under 15 seconds.

Problem 2 (UPSC/Banking Advanced): A man sells ₹9,000 worth of 8% stock at 95 and invests the proceeds (after a brokerage of ½% on the sale) in a 10% stock at 108 (again incurring ½% brokerage on the purchase). Find the change in his annual income.

Step-by-Step Breakdown:

1. First, find his ORIGINAL income from the 8% stock: Since "₹9000 worth of stock" refers to Face Value, Income $=\dfrac{9000\times8}{100}=720$.
2. Find the SALE proceeds: Number of shares (FV basis) $=\dfrac{9000}{100}=90$ units of FV100 each. Selling price per unit, after ½% brokerage deduction: $95-0.5\times(95/100)$... more precisely, brokerage is typically ½% of the FACE VALUE ₹100 in standard exam convention, i.e., ₹0.50 per unit: Net selling price per share $=95-0.5=94.5$. Total sale proceeds $=90\times94.5=8505$.
3. Now this ₹8505 is invested in the 10% stock at 108, WITH ½% brokerage added on purchase: effective buying price per share $=108+0.5=108.5$.
4. Number of new shares (FV basis) purchased $=\dfrac{8505}{108.5}\approx78.39$ units of FV100. Total new FV $=78.39\times100=7839$ (approx).
5. New income $=\dfrac{7839\times10}{100}\approx783.9\approx784$.
6. Change in income $=784-720=64$ (approximate increase).
7. Answer: The man's annual income increases by approximately ₹64. The key structural insight is handling BOTH the sale (brokerage subtracts from selling price) and the purchase (brokerage adds to buying price) correctly in sequence, then recomputing income from scratch on the new stock using the standard FV-based income formula — this two-transaction chaining is the hallmark of advanced stock-and-share questions at the banking/UPSC level.

6. Chapter Checklist for Students

• I always calculate dividend/income as a percentage of FACE VALUE, never market value.
• I always calculate yield/percentage return as (Income ÷ actual money invested at MV), never based on face value.
• I correctly add brokerage when BUYING and subtract it when SELLING, every single time.
• I interpret "₹X stock" or "₹X worth of stock" as referring to FACE VALUE holdings, not the amount of money invested.
• I use the direct Yield formula $\dfrac{\text{Rate}\times100}{\text{MV}}$ to compare or evaluate stocks without needing to assume any specific investment amount.