Bond & Stock Valuation: Types, Formulas, Real-World Applications

TAKEAWAYS

• Bonds are loans to companies/governments; their value depends on coupon rate, market interest rate, and time to maturity—use the bond valuation formula to calculate present value.
• Stocks are ownership claims; their price is determined by dividend growth models (constant growth, zero growth) and investor expectations.
• A premium bond trades above par when its coupon rate > market rate; a discount bond trades below par when coupon rate < market rate.
• Real-world applications: Ncell’s bond issuances (fixed income), NEPSE’s stock valuations (dividend growth), and eSewa’s risk-adjusted returns (investor required rate).
• Key formulas:
  • Bond value = $\sum \frac{C}{(1+r{)}^t}+\frac{F}{(1+r{)}^n}$
  • Stock value (constant growth) = $\frac{D_1}{k-g}$
  • Zero-coupon bond = $\frac{F}{(1+r{)}^n}$
• Exam focus: Numerical problems (bond/stock valuation), comparisons (premium vs. discount), and justifying why stocks are riskier than bonds.

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1. Bond Valuation: The Basics

Bonds are debt instruments issued by governments or corporations to raise capital. Investors buy bonds to earn fixed interest payments (coupons) and repayment of principal at maturity. The market value of a bond depends on:

• Face value (par value): Rs 1,000 (standard in Nepal).
• Coupon rate: Annual interest rate (e.g., 10% of par = Rs 100/year).
• Market interest rate (required return): What investors demand today.
• Time to maturity: Years until the bond is repaid.

How Bonds Are Valued

The present value (PV) of a bond is the sum of:

1. Present value of coupons (annuity stream).
2. Present value of face value (lump sum at maturity).

Formula: $$P{V}_{\text{bond}}=\sum _{t=1}^n\frac{C}{(1+r{)}^t}+\frac{F}{(1+r{)}^n}$$ Where:

• $C$ = Annual coupon payment.
• $r$ = Market interest rate (per period).
• $F$ = Face value.
• $n$ = Years to maturity.

Types of Bonds

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2. Real-World Example: Ncell’s Bond Issuance

Ncell (Nepal’s largest telecom) issues 7-year bonds with:

• Par value (F): Rs 1,000
• Coupon rate: 9% (paid semiannually → Rs 45 every 6 months)
• Market rate (r): 12% (investors demand 12% return today)

Question: What is the bond’s value today?

Step-by-Step Calculation

1. Adjust for semiannual payments:

   • New coupon ($C$) = Rs 45
   • New rate per period ($r$) = 12%/2 = 6% = 0.06
   • Number of periods ($n$) = 7 × 2 = 14

2. Calculate PV of coupons (annuity): $$P{V}_{\text{coupons}}=45\times \left[\frac{1-(1.06{)}^{-14}}{0.06}\right]=45\times 9.207=\text{Rs 414.32}$$

3. Calculate PV of face value: $$P{V}_{\text{face}}=\frac{1000}{(1.06{)}^{14}}=\frac{1000}{2.393}=\text{Rs 418.00}$$

4. Total bond value: $$P{V}_{\text{bond}}=414.32+418.00=\text{Rs 832.32}$$ Interpretation: The bond trades at a discount (Rs 832 < Rs 1,000) because its 9% coupon is lower than the 12% market rate.

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3. Stock Valuation: Dividend Discount Models

Stocks represent ownership in a company. Their value depends on:

• Dividends: Cash payments to shareholders.
• Growth rate (g): Expected dividend growth (e.g., 5% annually).
• Required return (k): Investor’s minimum expected return.

Key Models

Example: Sagarmatha Company

• Current stock price ($P_0$) = Rs 360
• Last dividend ($D_0$) = Rs 24
• Growth rate ($g$) = 5%
• Required return ($k$) = 12%

Question: Is the stock over/under-valued?

Calculation

1. Next dividend ($D_1$): $$D_1=D_0\times (1+g)=24\times 1.05=\text{Rs 25.20}$$
2. Intrinsic value: $$P_0=\frac{25.20}{0.12-0.05}=\frac{25.20}{0.07}=\text{Rs 360}$$ Conclusion: The stock is fairly valued (market price = intrinsic value).

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4. Why Are Stocks Riskier Than Bonds?

Real-World Tie-In:

• NEPSE (Nepal Stock Exchange): Stocks like Nabil Bank or Global IME offer growth but can crash (e.g., 2020 COVID dip).
• Ncell Bonds: Safer fixed income vs. Nepal Investment Bank’s stocks (higher risk, higher reward).

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5. Premium vs. Discount Bonds: A Comparison

Visual: Premium vs. Discount Bond

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6. Zero-Coupon Bonds: The "Deep Discount" Case

Zero-coupon bonds pay no interest; investors buy them at a deep discount and earn return from the difference between purchase price and face value.

Example: Rs 1,000 zero-coupon bond, 7 years to maturity, market rate = 10%.

Calculation: $$PV=\frac{1000}{(1.0{)}^7}=\frac{1000}{1.9487}=\text{Rs 513.16}$$ Interpretation: Buy for Rs 513.16, get Rs 1,000 at maturity → 10% annual return.

Real-World Use:

• Nepal Rastra Bank (NRB) issues zero-coupon bonds for long-term projects (e.g., infrastructure).
• Treasury bills (T-bills) in the US/Europe (short-term zero-coupons).

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7. The Accounting Cycle of Bond/Stock Valuation

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8. Worked Example: Kathmandu Retail Shop’s Stock

Scenario: A small retail shop in Kathmandu pays Rs 10/share annually and expects dividends to grow at 4% forever. Investors require 10% return.

Question: What is the maximum price a rational investor should pay?

Solution:

1. Model: Constant growth (Gordon model).
2. Given:
   • $D_0=10$
   • $g=4\%$
   • $k=10\%$
3. Next dividend ($D_1$): $$D_1=10\times 1.04=\text{Rs 10.40}$$
4. Intrinsic value: $$P_0=\frac{10.40}{0.10-0.04}=\frac{10.40}{0.06}=\text{Rs 173.33}$$ Conclusion: The shop’s stock is overvalued if trading above Rs 173.33.

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In the Real World

1. Ncell Bonds (Fixed Income)

   • Idea Used: Bond valuation (coupon vs. market rate).
   • How: Ncell issues bonds to raise capital. Investors compare the bond’s coupon rate (e.g., 9%) to the current market rate (e.g., 12%) to decide if it’s a good investment. If the market rate rises, existing bonds become more attractive (price increases).

2. NEPSE Stocks (Dividend Growth Model)

   • Idea Used: Constant growth stock valuation.
   • How: Companies like Nabil Bank or Global IME are valued using dividend growth models. For example, if a stock pays Rs 20/dividend and grows at 5%, investors use $P_0=\frac{D_1}{k-g}$ to estimate fair price. A high growth rate (e.g., 10%) increases the stock’s value.

3. eSewa’s Risk-Adjusted Returns

   • Idea Used: Required return ($k$) and risk.
   • How: eSewa invests in various assets (stocks, bonds, real estate). The platform adjusts expected returns based on risk: high-risk stocks require higher $k$, while government bonds (low risk) require lower $k$. This aligns with the CAPM (Capital Asset Pricing Model) concept from this unit.

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Exam Tip

1. Numerical Problems (50% of marks):

   • Bonds: Always adjust for semiannual payments (divide rate and multiply periods).
   • Stocks: Use $D_1=D_0\times (1+g)$ and plug into $\frac{D_1}{k-g}$.
   • Zero-coupon: Directly use $PV=\frac{F}{(1+r{)}^n}$.

2. Theoretical Questions (30% of marks):

   • Premium vs. discount: Explain using coupon rate vs. market rate.
   • Stocks riskier than bonds: Cite no fixed income, bankruptcy priority, and volatility.
   • Growth rate ($g$): Must be less than required return ($k$); otherwise, the model fails.

3. Shortcuts for Speed:

   • Bond tables: Memorize PV annuity factors for common rates (e.g., 6%, 10%).
   • Stock formula: Write $P_0=\frac{D_1}{k-g}$ from memory.
   • Real-world tie-ins: Always relate to Ncell bonds, NEPSE stocks, or NTC/Daraz financing.

4. Common Mistakes to Avoid:

   • Forgetting to adjust for semiannual periods in bond problems.
   • Using $D_0$ instead of $D_1$ in stock valuation.
   • Ignoring the $g<k$ condition in growth models (leads to negative denominators).

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Final Checklist Before Exam