Financial Management: Foundations and Analysis

Duration rule — predicted percentage change in price

The duration rule predicts the percentage change in a bond's price for a small change in yield as:

$\%\Delta P \approx -D_{\text{mod}} \times \Delta y$

where (D_{\text{mod}}) is the modified duration and (\Delta y) is the change in yield (in decimal form).

Percentage error of the duration rule

The duration rule ignores convexity. The actual price change is found by re-pricing the bond at the new yield. The error is:

$\text{Percentage error} = \frac{\text{Duration-predicted change} - \text{Actual change}}{\text{Actual change}} \times 100\%$

Equivalently, the error arises from the second-order (convexity) term:

$\%\Delta P_{\text{actual}} \approx -D_{\text{mod}}\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2$

so the duration rule understates the price increase when yields fall and overstates the price decrease when yields rise. The magnitude of the error grows with the size of (\Delta y) and with the bond's convexity.

Note: The numerical value cannot be computed here because the question does not supply the bond's coupon rate, maturity, yield to maturity, or the size of the yield change. Substitute those figures into the formulas above — first compute (D_{\text{mod}} = D_{\text{Macaulay}}/(1+y)), then apply the duration rule, then re-price the bond to obtain the actual change and the percentage error.

Duration-with-convexity rule: predicted price

The duration rule gives only a straight-line estimate of a bond's price change. The duration-with-convexity rule adds a second-order (curvature) term:

$\frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2$

so the predicted price is

$P_{\text{new}} = P_0\left[1 - D_{\text{mod}}\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2\right]$

where $D_{\text{mod}}$ is modified duration, $C$ is convexity and $\Delta y$ is the change in yield (in decimals).

Illustration (assumed figures, since the question gives none): $P_0 = \text{Rs }1{,}000$, $D_{\text{mod}} = 7$, $C = 60$, yield rises by 1% ($\Delta y = 0.01$).

$\frac{\Delta P}{P} = -(7)(0.01) + \tfrac{1}{2}(60)(0.01)^2 = -0.07 + 0.003 = -6.7\%$

$P_{\text{new}} = 1000 \times (1 - 0.067) = \text{Rs }933$

Duration alone predicts Rs 930. The convexity term adds back Rs 3, so the rule predicts Rs 933, which is closer to the true price.

Why the percentage price changes differ under duration and convexity

1. Duration is a linear approximation. It measures only the slope of the price-yield curve at the current yield, so it is accurate only for small yield changes.
2. The true price-yield relationship is convex. The actual curve bends away from the tangent line, so for any yield change the real price is higher than the duration-only prediction. This holds whether yields rise or fall. A fall in yield raises the price by more than duration predicts, and a rise lowers it by less.
3. Convexity measures this curvature. The correction $\tfrac{1}{2}C(\Delta y)^2$ is always positive, negligible for small $\Delta y$, and grows with the square of the yield change. A 3% shift gives a correction nine times that of a 1% shift.
4. Hence the gap. The difference between the two predictions is exactly $\tfrac{1}{2}C(\Delta y)^2$. It widens as convexity and the size of the yield change increase.

Conclusion: Duration captures the linear effect and suits small yield changes. Convexity captures the non-linear effect and matters for large changes, so the combined rule gives a more accurate price.

Given

• Market expected return, (R_m = 14%)
• Risk-free rate, (R_f = 6%)
• Market risk premium (= R_m - R_f = 14% - 6% = 8%)

CAPM equation:

[ E(R_i)=R_f+\beta_i(R_m-R_f) ]

[ E(R_i)=6%+8%\beta_i ]

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a. Required return on each stock

So,

[ R_X=18%,\quad R_Y=22%,\quad R_Z=10% ]

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b. Undervalued, overvalued and worth investing

Compare expected return with required return.
Alpha (= \text{Expected return} - \text{Required return}).

• Undervalued stock: Z, because its expected return (12%) is greater than required return (10%).
• Overvalued stock: Y, because its expected return (20%) is less than required return (22%).
• Worth investing: Stock Z is worth investing because it gives positive alpha. Stock X is fairly priced, so it gives only normal return. Stock Y should be avoided or sold.

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c. SML graph and location of stocks

SML equation:

[ E(R_i)=6%+8%\beta_i ]

Important points:

• Risk-free asset: (\beta=0,\ E(R)=6%)
• Market portfolio: (\beta=1,\ E(R)=14%)
• Stock Z: (\beta=0.5,\ E(R)=12%). SML at (\beta=0.5) gives (10%). So Z lies above SML.
• Stock X: (\beta=1.5,\ E(R)=18%). SML at (\beta=1.5) gives (18%). So X lies on SML.
• Stock Y: (\beta=2.0,\ E(R)=20%). SML at (\beta=2.0) gives (22%). So Y lies below SML.

Rough SML graph:

E(R) %
22 |                         SML
   |                       /
20 |                      /  Y(2.0,20)
18 |                    X(1.5,18)
   |                  /
16 |                /
14 |              M(1.0,14)
12 |     Z(0.5,12)
   |    /
10 |  /
 8 | /
 6 |/ Rf(0,6)
   +-------------------------- β
     0.5  1.0  1.5  2.0

Thus:

• Z is above the SML → undervalued
• X is on the SML → fairly valued
• Y is below the SML → overvalued

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d. Risk premium for all stocks

Market risk premium:

[ R_m-R_f=14%-6%=8% ]

Required risk premium under CAPM:

[ \text{Risk premium}_i=\beta_i(R_m-R_f) ]

If expected excess return is asked instead:

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e. Difference between systematic and unsystematic risk

Therefore, CAPM rewards only systematic risk, because unsystematic risk can be diversified away.

(a) Index values on day 1 and day 2

Take day 0 as the base (index = 100).

Market capitalisation (price × shares):

(i) Value-weighted index $=\dfrac{\text{Current total market value}}{\text{Base total market value}}\times 100$

$\text{Day 1} = \frac{58{,}75{,}000}{54{,}00{,}000}\times100 = 108.80 \qquad \text{Day 2} = \frac{65{,}50{,}000}{54{,}00{,}000}\times100 = 121.30$

(ii) Price-weighted index (average of prices, base day 0 average $=510/3 = 170$)

(iii) Equally weighted index (equal amount invested in each stock on day 0, average of price relatives)

$\text{Day 1} = \frac{\tfrac{135}{160}+\tfrac{280}{240}+\tfrac{120}{110}}{3}\times100 = \frac{0.8438+1.1667+1.0909}{3}\times100 = 103.38$

$\text{Day 2} = \frac{\tfrac{150}{160}+\tfrac{280}{240}+\tfrac{150}{110}}{3}\times100 = \frac{0.9375+1.1667+1.3636}{3}\times100 = 115.59$

(b) Day 3 after the 2-for-1 split of B

B's shares double to 20,000 and its price halves, from Rs 280 to Rs 140 at the end of day 2.

Price-weighted: the divisor must change so the index does not move because of the split. After the split, the day 2 sum of prices is $150+140+150 = 440$. The new divisor $d$ satisfies

$\frac{440}{d} = \frac{580}{3} \;\Rightarrow\; d = 2.2759$

Day 3 sum of prices $=160+150+170 = 480$, so

$\text{Average} = \frac{480}{2.2759} = 210.91 \;\Rightarrow\; \text{Index} = \frac{210.91}{170}\times100 = 124.07$

Value-weighted: a split leaves total market value unchanged. Day 3 market values are A $=160\times5{,}000 = 8{,}00{,}000$, B $=150\times20{,}000 = 30{,}00{,}000$ and C $=170\times20{,}000 = 34{,}00{,}000$, a total of Rs 72,00,000.

$\text{Index} = \frac{72{,}00{,}000}{54{,}00{,}000}\times100 = 133.33$

(c) Factors in constructing a market index, and its uses

Factors to consider:

1. Sample of stocks. The index should be representative, covering major sectors (banks, hydropower, insurance and so on) and enough liquid stocks.
2. Weighting method. Choose between price weighting (high-priced stocks dominate and splits distort it), value weighting (large companies dominate, and it is the most widely used) and equal weighting.
3. Base period and base value. A suitable base date and value, usually 100 or 1,000.
4. Adjustments for corporate actions. Splits, bonus and right shares, new listings and delistings require a divisor or base adjustment.
5. Free-float and liquidity. Only publicly tradable shares should count, and thinly traded stocks should be treated with care.
6. Computation method. Arithmetic or geometric averaging, and frequency of calculation.
7. Periodic review. The constituents should be reviewed regularly.

Uses of a market index:

• It measures overall market performance and investor sentiment.
• It serves as a benchmark for evaluating portfolio and fund managers.
• It is a basis for index funds and derivative products.
• It is used to estimate market return and risk (beta) in CAPM.
• It acts as an indicator of the economy and helps in forecasting and research.

Stages of Business Cycle

The business cycle (also called the trade cycle) refers to the recurring, wave-like fluctuations in the level of economic activity — output, employment, income and prices — over a period of time. It is not a regular or periodic phenomenon, but it passes through five distinct stages:

1. Expansion (Boom / Upswing) In this stage, economic activity rises rapidly. Production, employment, income, investment and prices all increase. Consumer demand is high, business confidence is optimistic, bank credit expands and profit margins rise. The economy moves towards full employment. This phase continues until it reaches a peak, where resources become fully employed and further expansion becomes difficult.

2. Peak (Upper Turning Point) The peak is the highest point of economic activity. Output, employment and prices reach their maximum. Demand may exceed supply, causing inflationary pressures. However, at this point, costs begin to rise faster than prices, profit margins start shrinking, and the economy becomes vulnerable to a downturn. The peak marks the end of expansion and the beginning of recession.

3. Contraction (Recession / Downswing) In this stage, economic activity declines. Production, employment, income, investment and prices fall. Demand falls, inventories pile up, bank credit contracts and business profits decline. Investment falls due to pessimistic expectations. If the contraction is severe and prolonged, it becomes a depression, characterised by mass unemployment, idle capacity and falling prices. The economy reaches its lowest point, called the trough.

4. Trough (Lower Turning Point) The trough is the lowest point of the business cycle. Output, employment, income and prices are at their minimum. Unemployment is high, investment is very low and business confidence is at its worst. However, at this point, prices and costs become so low that demand begins to revive, and the economy starts moving towards recovery.

5. Recovery (Revival / Upswing) In this stage, the economy begins to improve. Investment increases as business confidence returns, employment and income rise, and demand picks up. Bank credit expands and prices start rising slowly. The economy gradually moves towards a new expansion phase, and the cycle repeats itself.

Conclusion Thus, the business cycle consists of five stages — expansion, peak, contraction (recession), trough and recovery. These stages are recurring but not regular in duration or intensity. Understanding them helps the government and central bank to adopt appropriate fiscal and monetary policies to stabilise the economy.

Recent Developments in the Securities Market of Nepal

The securities market of Nepal, regulated primarily by the Securities Board of Nepal (SEBON) and operated through the Nepal Stock Exchange (NEPSE), has undergone significant transformation in recent years. The major recent developments are:

1. Establishment of CDSC and Dematerialization – The CDS and Clearing Limited (CDSC) was established as the central depository, enabling full dematerialization of shares. Physical share certificates have been largely replaced by electronic records, making trading, transfer and settlement faster and safer.

2. Fully Automated Online Trading System – NEPSE shifted from floor trading to a fully automated screen-based trading system, allowing investors to place buy/sell orders online through broker members from anywhere in the country.

3. T+2 Settlement Cycle – The settlement period has been shortened to T+2 (Trade plus Two days), reducing counterparty risk and improving liquidity in the market.

4. Introduction of New Instruments – New tradable instruments such as debentures, mutual fund units, promoter shares, and closed-end funds have been introduced. SEBON has also permitted book-building for Initial Public Offerings (IPOs) of large companies, replacing the earlier fixed-price system for big issues.

5. Online IPO Application (Mero Share) – The Mero Share platform, operated through CDSC, allows investors to apply for IPOs, rights shares and FPOs online, and to view their demat accounts, greatly increasing retail participation.

6. Growth of Mutual Funds and Institutional Investors – Several mutual fund schemes and Depository Participants (DPs) have emerged, along with growing participation of institutional investors such as insurance companies and Citizen Investment Trust.

7. Strengthening of SEBON's Regulatory Framework – SEBON has issued stricter rules on disclosure, insider trading, market manipulation, and corporate governance. The Securities Act and related regulations have been amended to align with international practices.

8. Digitalization and Use of Technology – E-subscription, online KYC, digital payment for share purchases, and electronic reporting by listed companies have modernized market operations.

9. Expansion of Market Infrastructure – Increase in the number of broker members, DPs, and listed companies, along with the opening of NEPSE branches and wider access outside the Kathmandu Valley.

10. Market Capitalization Growth – The market capitalization of NEPSE has grown substantially relative to GDP, and the number of listed companies and daily turnover have increased, reflecting the market's expanding role in capital formation.

Conclusion: These developments have made Nepal's securities market more transparent, efficient, inclusive and technology-driven, contributing to better resource mobilization and investor protection. However, challenges such as low financial literacy, market volatility and weak enforcement still remain.

(a) Intrinsic value

Plowback ratio $b = 0.60$, so the payout ratio is $0.40$.

Sustainable growth rate: $g = ROE \times b = 10\% \times 0.60 = 6\%$

Next year's earnings and dividend (the current dividend has just been paid, so the next dividend falls a year ahead): $E_1 = 12 \times 1.06 = \text{Rs }12.72, \qquad D_1 = 12.72 \times 0.40 = \text{Rs }5.09$

Required return by CAPM: $k = 5\% + 1.2\,(13\% - 5\%) = 14.6\%$

Constant-growth model: $P_0 = \frac{D_1}{k-g} = \frac{5.09}{0.146 - 0.06} \approx \text{Rs }59.16$

(b) Present value of growth opportunities

Value of the firm with no growth (all earnings paid out): $\frac{E_1}{k} = \frac{12.72}{0.146} \approx \text{Rs }87.12$

$PVGO = P_0 - \frac{E_1}{k} = 59.16 - 87.12 \approx -\text{Rs }27.96$

PVGO is negative because ROE (10%) is below the required return (14.6%). Plowing back 60% of earnings into projects earning less than the cost of equity destroys value. The firm would be worth more if it paid out all earnings as dividends.

Behavioural Finance

Concept

Behavioural finance is a field of finance that studies how psychological factors, cognitive biases, and emotional influences affect the financial decisions of investors, managers, and markets. It emerged as a critique of traditional (standard) finance, which assumes that investors are fully rational, have complete information, and always aim to maximise expected utility.

Behavioural finance argues that real investors are not perfectly rational; they are "normal" human beings whose judgements are systematically distorted by biases. Hence, market prices and corporate decisions often deviate from what rational models predict.

Key Ideas

• Bounded rationality: Investors have limited time, information-processing capacity, and computational ability, so they use mental shortcuts (heuristics) rather than full optimisation.
• Cognitive biases: Systematic errors in thinking such as overconfidence, anchoring, representativeness, availability bias, and confirmation bias.
• Emotional and social factors: Fear, greed, regret aversion, herd behaviour, and social proof influence buying and selling decisions.
• Limits to arbitrage: Even when mispricing exists, rational arbitrageurs cannot always correct it due to risk, transaction costs, and constraints, so anomalies persist.

Main Biases Studied

• Overconfidence — investors overestimate their knowledge and trading skill, leading to excessive trading.
• Anchoring — fixating on an initial reference point (e.g., purchase price) when judging value.
• Loss aversion — losses hurt more than equivalent gains please, so investors hold losers too long and sell winners too early (disposition effect).
• Herding — following the crowd instead of independent analysis, causing bubbles and crashes.
• Mental accounting — treating money differently depending on its source or intended use.
• Representativeness — judging the future by recent patterns (e.g., extrapolating past returns).

Prospect Theory

The foundation of behavioural finance is Prospect Theory (Kahneman and Tversky, 1979). It shows that people evaluate outcomes as gains or losses relative to a reference point, are loss averse, and weigh losses more heavily than gains. This explains why investors behave differently in gain versus loss situations.

Significance

• Explains market anomalies such as bubbles, crashes, momentum, and the equity premium puzzle.
• Helps investors recognise their own biases and improve decision-making.
• Guides financial advisers, regulators, and corporations in designing better products, disclosures, and policies (e.g., automatic enrolment in retirement plans).
• Complements traditional finance by adding a realistic psychological dimension to financial analysis.

Conclusion

Behavioural finance is the study of how human psychology shapes financial behaviour and market outcomes. It does not reject rationality altogether but recognises that rationality is bounded and that emotions and biases systematically affect decisions — making it an essential bridge between finance and psychology.

Given:

• Securities (assets) = Rs 750 million
• Liabilities = Rs 25 million
• Shares outstanding = 55 million
• Discount from NAV = 12%

a. Net Asset Value (NAV) per share

[ \text{Net assets} = \text{Total assets} - \text{Liabilities} ]

[ = 750 - 25 = \text{Rs } 725 \text{ million} ]

[ \text{NAV per share} = \frac{\text{Net assets}}{\text{Number of shares outstanding}} ]

[ = \frac{725}{55} = \text{Rs } 13.1818 \approx \text{Rs } 13.18 ]

NAV per share = Rs 13.18

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b. Current market price per share

Since the fund is trading at a 12% discount from NAV:

[ \text{Market price} = \text{NAV per share} \times (1 - \text{Discount rate}) ]

[ = 13.1818 \times (1 - 0.12) ]

[ = 13.1818 \times 0.88 = \text{Rs } 11.60 ]

Current market price per share = Rs 11.60

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c. Two possible reasons for trading at a discount to NAV

1. Lack of liquidity and no redemption facility: Closed-end fund shares cannot be redeemed directly with the fund at NAV; investors must sell them in the secondary market. This reduces demand and can push the market price below NAV.

2. Management fees and expenses: Investors may discount the fund’s value because a part of the fund’s assets is used to pay management fees, operating expenses, and other costs, reducing the effective return to shareholders.

Other acceptable reasons: poor past performance, tax liability on unrealized capital gains, limited investor awareness/demand, or general market sentiment.

CAPM vs. APT

Conclusion: CAPM is a simple one-factor equilibrium model whose validity rests on restrictive assumptions, whereas APT is a more general multi-factor model based on the no-arbitrage condition. APT is theoretically more flexible, but CAPM remains more popular in practice because it needs only one easily estimated input, beta.