Term Structure of Interest Rates & Bond Valuation

TAKEAWAYS:

• The term structure of interest rates describes how short-term vs. long-term rates relate, shaped by expectations, liquidity, and risk premiums.
• Bond valuation uses present value (PV) of coupon payments and principal to determine fair market price, adjusted for YTM and market conditions.
• Theories (expectations, liquidity preference, market segmentation) explain why yield curves slope upward, downward, or are flat.
• Key metrics like duration and convexity measure a bond’s interest-rate sensitivity and price volatility.
• Callable bonds and zero-coupon bonds introduce special valuation challenges (e.g., embedded options, reinvestment risk).
• Real-world links: NEPSE’s bond yields, Daraz’s supplier financing deals, and Ncell’s corporate bonds all rely on these concepts.

1. Introduction to the Term Structure of Interest Rates

The term structure shows how interest rates vary by bond maturity (e.g., 1-year vs. 10-year government bonds). It is plotted as a yield curve, where the x-axis is maturity (years) and the y-axis is yield (%). The shape of the curve (upward, downward, or flat) reflects market expectations, economic conditions, and risk premiums.

Why Does the Term Structure Matter?

• Investors compare short-term savings (e.g., Ncell’s fixed deposits) vs. long-term bonds (e.g., NEPSE’s government securities).
• Borrowers (e.g., Daraz for inventory loans) choose between floating-rate loans (linked to short-term rates) and fixed-rate bonds.
• Central banks (Nepal Rastra Bank) use term structure data to set monetary policy (e.g., raising short-term rates to flatten the curve).

Key Observations

• Normal yield curve: Upward-sloping (long-term rates > short-term rates), typical in growing economies.
• Inverted yield curve: Downward-sloping (short-term rates > long-term rates), often signals recession (e.g., pre-2008 financial crisis).
• Flat yield curve: Short- and long-term rates are nearly equal, indicating uncertainty.

FIGURE 1: Types of Yield Curves

flowchart TD
    A["Normal Curve\n(Upward-sloping)"] -->|"Long-term rates > Short-term rates"| B["Healthy economy, expected growth"]
    C["Inverted Curve\n(Downward-sloping)"] -->|"Short-term rates > Long-term rates"| D["Recession warning, liquidity crisis"]
    E["Flat Curve\n(Parallel)"] -->|"Rates equal"| F["Uncertainty, no clear growth signal"]

2. Theories Explaining the Term Structure

Three dominant theories explain why yield curves take their shape:

FIGURE 2: Liquidity Premium Theory

flowchart TD
    A["Short-term rate (r₁)"] -->|"+ Liquidity premium"| B["Long-term rate (r₂)"]
    C["Expected future short-term rate"] --> B
    B --> D["Long-term rate = r₁ + Liquidity premium + Expected r₁"]

3. Bond Valuation Basics

A bond’s price is the present value (PV) of its future cash flows (coupons + principal), discounted at its yield to maturity (YTM).

Key Terms

• Face value (FV): Rs. 1,000 (standard for Nepalese bonds).
• Coupon rate: Annual interest payment (e.g., 8% of FV = Rs. 80/year).
• YTM: The discount rate that makes the PV of cash flows equal to the bond’s market price.
• Maturity: Years until principal is repaid (e.g., 5-year bond).

Bond Valuation Formula

For a coupon bond: $$P=\sum _{t=1}^n\frac{C}{(1+YTM{)}^t}+\frac{FV}{(1+YTM{)}^n}$$ Where:

• $P$ = Market price of the bond
• $C$ = Annual coupon payment
• $n$ = Years to maturity
• $FV$ = Face value

WORKED EXAMPLE 1: Valuing a Nepalese Corporate Bond Scenario: Kathmandu Retail Ltd. issues a 5-year bond with:

• Face value (FV) = Rs. 1,000
• Coupon rate = 7% (paid semi-annually)
• Current market YTM = 6%

Step 1: Calculate semi-annual coupon payment $$C=\frac{7\%\times 1,000}{2}=\text{Rs. }35\text{ per period}$$

Step 2: Apply PV formula (n = 10 periods, YTM = 3% semi-annually) $$P=35\times \left(\frac{1-(1.03{)}^{-10}}{0.03}\right)+\frac{1,000}{(1.03{)}^{10}}$$ $$P=35\times 8.5302+744.09=\text{Rs. }1,060.58$$

Answer: The bond’s fair market price is Rs. 1,060.58.

FIGURE 3: Bond Cash Flow Timeline

timeline
    title: Kathmandu Retail Ltd. Bond Cash Flows
    section Semi-annual Payments
    0.5: Rs. 35
    1.0: Rs. 35
    1.5: Rs. 35
    2.0: Rs. 35
    2.5: Rs. 35
    3.0: Rs. 35
    3.5: Rs. 35
    4.0: Rs. 35
    4.5: Rs. 35
    5.0: Rs. 1,035 (Principal + Last Coupon)

4. Special Bond Types and Valuation Adjustments

(A) Zero-Coupon Bonds

• No periodic coupons; sold at a deep discount.
• Valuation: Only the principal’s PV matters. $$P=\frac{FV}{(1+YTM{)}^n}$$

Example: A 3-year zero-coupon bond with FV = Rs. 1,000 and YTM = 5%: $$P=\frac{1,000}{(1.05{)}^3}=\text{Rs. }863.84$$

(B) Callable Bonds

• Issuer can "call" (repay) the bond early at a set price (e.g., Rs. 1,050).
• Valuation challenge: The bond’s price depends on whether it will be called.
• Call risk: If rates fall, the issuer may call the bond, forcing investors to reinvest at lower rates.

Example: A bond with:

• FV = Rs. 1,000
• Call price = Rs. 1,050 in 2 years
• YTM = 6%
• Coupon = Rs. 50/year

The bond’s price is the lower of:

1. PV of cash flows if not called.
2. PV of cash flows if called (using call price).

FIGURE 4: Callable Bond Cash Flow Scenarios

flowchart TD
    A["Bond not called"] -->|"PV = Rs. X"| B["Price = Rs. X"]
    C["Bond called in 2 years"] -->|"PV = Rs. Y"| D["Price = Rs. Y"]
    E["Actual price"] -->|"min(X, Y)"| F["Rs. 950 (e.g.)"]

(C) Floating-Rate Bonds

• Coupon resets periodically (e.g., LIBOR + 2%).
• Valuation: Less sensitive to YTM changes; price stays close to par.

5. Duration and Convexity: Measuring Interest-Rate Risk

(A) Duration

• Modified Duration: Measures % change in bond price for a 1% change in YTM. $$\text{Modified Duration}=\frac{-1}{P}\times \frac{dP}{dY}$$
• Rule of thumb: For small YTM changes, %ΔPrice ≈ –Duration × ΔYTM.

Example: A bond with duration = 4.5 years will lose ~4.5% if YTM rises by 1%.

(B) Convexity

• Adjusts for the nonlinear relationship between price and YTM.
• Higher convexity = less price volatility.

FIGURE 5: Duration and Price Sensitivity

flowchart TD
    A["YTM ↑ 1%"] --> B["Price ↓ ≈ Duration × 1%"]
    C["YTM ↓ 1%"] --> D["Price ↑ ≈ Duration × 1% (but less due to convexity)"]

6. Bond Valuation with Inflation and Risk Premiums

(A) Real vs. Nominal Yields

• Nominal YTM = Real risk-free rate + Inflation premium + Risk premium.
• Real YTM = Nominal YTM – Inflation.

Example: If real risk-free rate = 2% and inflation = 4%, then: $$\text{Nominal YTM}=2\%+4\%+\text{Risk premium}=6\%+\text{Risk premium}$$

(B) Expected Inflation and Term Structure

• Investors demand higher yields for bonds with longer maturities to compensate for unexpected inflation.
• Fisher Equation: $$(1+\text{Nominal YTM})=(1+\text{Real YTM})\times (1+\text{Inflation})$$

7. Real-World Applications

## In the real world

1. NEPSE’s Bond Market

   • Idea: Term structure guides investors when buying/selling government securities (e.g., Nepal’s 5-year vs. 10-year bonds).
   • How: The yield curve helps NEPSE set benchmark rates for corporate bonds. For example, if the 10-year yield rises from 6% to 7%, new corporate bonds must offer higher coupons to attract buyers.

2. Daraz’s Supplier Financing

   • Idea: Bond valuation determines the cost of short-term loans to suppliers.
   • How: Daraz may issue commercial paper (short-term bonds) to fund inventory. The YTM on these bonds reflects market expectations of Daraz’s credit risk and liquidity.

3. Ncell’s Corporate Bonds

   • Idea: Duration and convexity help Ncell manage interest-rate risk on its debt.
   • How: If Ncell issues a 10-year bond with duration = 6.5, a 1% rise in rates could reduce the bond’s value by ~6.5%, forcing Ncell to hedge with interest-rate swaps.

WORKED EXAMPLE 2: Mortgage Loan Valuation (Nepal Context) Scenario: You take a 20-year mortgage for a Rs. 20,00,000 house in Kathmandu with:

• Down payment = 20% (Rs. 4,00,000)
• Loan amount = Rs. 16,00,000
• Annual interest rate = 6% (compounded semi-annually)

Step 1: Calculate semi-annual mortgage payment Using the mortgage formula: $$PMT=\frac{L\times r\times (1+r{)}^n}{(1+r{)}^n-1}$$ Where:

• $L=16,00,000$
• $r=0.06/2=0.03$ (semi-annual rate)
• $n=20\times 2=40$ periods

$$PMT=\frac{16,00,000\times 0.03\times (1.03{)}^{40}}{(1.03{)}^{40}-1}=\text{Rs. }12,000\text{ per period}$$

Step 2: Present Value of Payments The loan’s value is the PV of all payments: $$PV=PMT\times \left(\frac{1-(1+r{)}^{-n}}{r}\right)=12,000\times 25.4259=\text{Rs. }16,00,000$$

Answer: The mortgage’s fair value matches the loan amount, confirming the calculation.

FIGURE 6: Mortgage Amortization Schedule (First 3 Payments)

8. Exam Tips

1. Term Structure Theories:

   • Know the unbiased expectations, liquidity preference, and market segmentation theories.
   • Expectations theory implies long-term rates = average of expected short-term rates.
   • Liquidity preference adds a premium for long-term bonds.

2. Bond Valuation:

   • Always use the PV of cash flows formula. For semi-annual coupons, adjust $n$ and $r$ accordingly.
   • Callable bonds: Compare PV of cash flows with and without calling.

3. Duration and Convexity:

   • Duration measures interest-rate sensitivity. Higher duration = more risk.
   • Convexity adjusts for the nonlinearity of price changes.

4. Real-World Linkages:

   • Relate yield curves to Nepal’s economic conditions (e.g., inverted curve = recession warning).
   • Explain how NEPSE’s bond yields reflect investor expectations.

5. Numerical Problems:

   • For mortgage loans, use the mortgage formula and amortization schedule.
   • For zero-coupon bonds, only discount the principal.

6. Common Pitfalls:

   • Ignoring compounding: Always check if coupons are paid annually, semi-annually, or monthly.
   • Miscounting periods: For a 5-year bond with semi-annual coupons, $n=10$, not 5.
   • Forgetting call risk: If a bond is callable, consider the call price in valuation.

EXAM QUESTION TRACE: Past Paper Link Question: "Assume the real risk-free rate is 3%. Inflation is expected at 4% (Year 1), 5% (Year 2), 6% (Year 3), and 6.5% (Year 4). No maturity risk premium. What is the 4-year nominal yield?" Solution:

1. Use the Fisher equation for each year: $$(1+\text{Nominal YTM})=(1+\text{Real YTM})\times (1+\text{Inflation})$$
2. Calculate expected nominal yields for each year:
   • Year 1: $(1.03)(1.04)=1.0712$ → 7.12%
   • Year 2: $(1.03)(1.05)=1.0815$ → 8.15%
   • Year 3: $(1.03)(1.06)=1.0918$ → 9.18%
   • Year 4: $(1.03)(1.065)=1.09695$ → 9.695%
3. Unbiased expectations theory assumes the 4-year nominal yield is the average of these: $$\text{4-year YTM}=\frac{7.12\%+8.15\%+9.18\%+9.695\%}{4}=8.50\%$$ Answer: 8.50% (rounded).

FIGURE 7: Fisher Equation Worked Example

flowchart TD
    A["Real YTM = 3%"] --> B["Year 1 Inflation = 4%"]
    B --> C["Nominal YTM₁ = 7.12%"]
    D["Year 2 Inflation = 5%"] --> C
    E["Year 3 Inflation = 6%"] --> F["Nominal YTM₃ = 9.18%"]
    G["Year 4 Inflation = 6.5%"] --> F
    F --> H["Average = 8.50%"]