4.2 Bond Valuation and Yield Concepts

4.2 Bond Valuation and Yield Concepts

Fixed-income securities represent the cornerstone of the Philippine capital markets. In terms of outstanding nominal volume, the Philippine debt market—anchored by sovereign Treasury Bills (T-Bills) and Fixed-Rate Treasury Notes (FXTNs) issued by the Bureau of the Treasury (BTr) and traded over the Philippine Dealing & Exchange Corp. (PDEx)—substantially exceeds total domestic equity capitalization. Understanding bond valuation, yield dynamics, and interest rate risk metrics is critical for SEC Phase 1 candidates.

---

Bond Anatomy and Valuation Mechanics

A bond is a contractual debt instrument whereby the issuer (borrower) agrees to pay the bondholder (investor) periodic interest payments (coupons) and repay the principal balance (par value or face value) on a specified future date (maturity date).

Core Bond Parameters

• Par / Face Value ($M$): The nominal principal stated on the debt certificate, conventionally set at Php 1,000, Php 10,000, or Php 100,000 in Philippine corporate and sovereign issuances.
• Coupon Rate ($c$): The nominal annual interest rate pledged by the issuer, which dictates the annual cash interest paid per bond ($C = c \times M$).
• Payment Frequency: In the Philippine market, fixed-rate sovereign bonds (FXTNs and Retail Treasury Bonds - RTBs) and corporate bonds conventionally pay coupons semi-annually ($m = 2$).
• Maturity ($t$): The remaining lifespan until the par value is redeemed.
• Yield to Maturity ($y$ or YTM): The required market discount rate, reflecting current investor return demands for bonds of comparable credit quality and maturity.

The Bond Pricing Formula

A bond's theoretical intrinsic value is calculated by discounting all contractual future cash flows to the present at the market's required rate of return. A coupon-bearing bond is valued as the combination of an ordinary annuity (the periodic coupon payments) and a single lump sum (the par value at maturity): $P = \sum_{t=1}^{m \times T} \frac{C / m}{\left(1 + \frac{y}{m}\right)^t} + \frac{M}{\left(1 + \frac{y}{m}\right)^{m \times T}}$

Expanded in closed-form annuity notation: $P = \left[ \frac{C}{m} \times \left( \frac{1 - \left(1 + \frac{y}{m}\right)^{-m \times T}}{\frac{y}{m}} \right) \right] + \left[ \frac{M}{\left(1 + \frac{y}{m}\right)^{m \times T}} \right]$

Where:

• $P$ = Bond market price
• $C / m$ = Semi-annual coupon cash flow
• $y / m$ = Semi-annual market discount rate
• $m \times T$ = Total number of semi-annual coupon periods
• $M$ = Par value at maturity

Practical Valuation Example

Consider a 5-year Retail Treasury Bond (RTB) with a par value of Php 100,000 and a 6.00% annual coupon paid semi-annually ($C/2 = \text{Php } 3,000$; total periods $n = 10$):

• Scenario A: Market Yield Equals Coupon ($y = 6.00\%$):
  • Semi-annual rate = $3.00\%$
  • PV of coupons = $\text{Php } 3,000 \times \left[\frac{1 - (1.03)^{-10}}{0.03}\right] = \text{Php } 3,000 \times 8.530203 = \text{Php } 25,591$
  • PV of par = $\text{Php } 100,000 / (1.03)^{10} = \text{Php } 74,409$
  • Total Price = Php 25,591 + Php 74,409 = Php 100,000 (Par)
• Scenario B: Market Yield Rises to 8.00% ($y = 8.00\%$):
  • Semi-annual rate = $4.00\%$
  • PV of coupons = $\text{Php } 3,000 \times \left[\frac{1 - (1.04)^{-10}}{0.04}\right] = \text{Php } 3,000 \times 8.110896 = \text{Php } 24,333$
  • PV of par = $\text{Php } 100,000 / (1.04)^{10} = \text{Php } 67,556$
  • Total Price = Php 24,333 + Php 67,556 = Php 91,889 (Discount)
• Scenario C: Market Yield Falls to 4.00% ($y = 4.00\%$):
  • Semi-annual rate = $2.00\%$
  • PV of coupons = $\text{Php } 3,000 \times \left[\frac{1 - (1.02)^{-10}}{0.02}\right] = \text{Php } 3,000 \times 8.982585 = \text{Php } 26,948$
  • PV of par = $\text{Php } 100,000 / (1.02)^{10} = \text{Php } 82,035$
  • Total Price = Php 26,948 + Php 82,035 = Php 108,983 (Premium)

---

The Inverse Price-Yield Relationship and the Yield Seesaw

The fundamental law of fixed income is that bond prices and interest rates move in opposite directions:

• When market interest rates (yields) rise, outstanding bond prices decline.
• When market interest rates (yields) fall, outstanding bond prices increase.

          Market Yield Rises                     Market Yield Falls
                 ▲                                       ▼
                 │                                       │
        ┌────────┴────────┐                     ┌────────┴────────┐
        │                 │                     │                 │
   [Bond Price]      [Bond Yield]          [Bond Price]      [Bond Yield]
        ▼                 ▲                     ▲                 ▼

The Yield Hierarchy Across Trading Statuses

Securities analysts utilize three distinct yield measurements:

1. Nominal Yield (Coupon Rate): Stated annual coupon divided by par value.
2. Current Yield ($CY$): Annual coupon cash flow divided by the bond's current secondary market price: $CY = \frac{\text{Annual Coupon Payment}}{\text{Current Market Price}} = \frac{C}{P}$
3. Yield to Maturity ($YTM$): The total annualized internal rate of return earned by an investor who buys the bond at its current market price, receives all scheduled coupons, and holds the debt security to maturity, reinvesting all intermediate coupons at the YTM rate.

Depending on whether a bond trades at a discount, par, or premium, these yield metrics maintain a strict mathematical relationship known as the Yield Seesaw:

Yield to Call (YTC) and Yield to Worst (YTW)

Many Philippine corporate debentures include a call provision, granting the issuer the legal option to redeem the bonds prior to maturity at a specified call price. Issuers exercise call options when market interest rates drop, refinancing expensive debt at lower prevailing rates.

• Yield to Call ($YTC$): Replaces maturity date ($T$) with the call date and par value ($M$) with the call price.
• For a premium bond, the bond is likely to be called early, making $YTC < YTM$.
• Yield to Worst ($YTW$): The lowest potential yield generated among all possible redemption dates. Under prudent fiduciary standards, bond dealers quote Yield to Worst: for premium callable bonds, YTW is Yield to Call; for discount bonds, YTW is Yield to Maturity.

---

Duration and Convexity: Measuring Interest Rate Risk

Interest rate risk represents the vulnerability of a bond's market value to changes in interest rates. Two identical-maturity bonds can experience dramatically different price volatility depending on their coupon structures.

Macaulay Duration

Developed by Frederick Macaulay, Macaulay Duration ($D_{\text{mac}}$) measures the weighted average time (in years) an investor must hold a bond until the present value of its cash flows equals the price paid for the bond: $D_{\text{mac}} = \frac{\sum_{t=1}^n \frac{t \times CF_t}{(1 + y)^t}}{P}$

Key Characteristics of Duration:

• For a zero-coupon bond, there are no intermediate cash flows; therefore, its Macaulay duration is exactly equal to its maturity ($D_{\text{mac}} = T$).
• For a coupon-bearing bond, Macaulay duration is strictly less than its maturity ($D_{\text{mac}} < T$) because coupons return capital early.
• Higher coupon rates produce shorter duration (more cash flow received early).
• Longer maturities generally produce longer duration.

Modified Duration

Modified Duration ($D_{\text{mod}}$) directly quantifies a bond's percentage price sensitivity to a 100-basis-point (1.0%) shift in yield: $D_{\text{mod}} = \frac{D_{\text{mac}}}{1 + \frac{y}{m}}$

The estimated percentage change in bond price ($\Delta P / P$) resulting from a change in yield ($\Delta y$) is: $\% \Delta P \approx -D_{\text{mod}} \times \Delta y$

Example: A Philippine corporate bond portfolio has a modified duration of 5.4 years. If the BSP raises policy rates and market benchmark yields increase by 75 basis points ($\Delta y = +0.0075$): $\% \Delta P \approx -5.4 \times (+0.0075) = -0.0405 = -4.05\%$ The portfolio's value will decline by approximately 4.05%.

Convexity

Modified duration is a first-derivative linear approximation of the price-yield relationship. In reality, the price-yield curve is non-linear—it is convex to the origin.

Price ▲
      │     Actual Bond Price Curve (Convex)
      │    . '
      │   /  
      │  /   . ' Tangent Line (Duration Approximation)
      │ / . '
      │/ ' 
      └────────────────────────► Yield

The Advantage of Positive Convexity:

• When yields decline, the actual price rises by more than duration predicts.
• When yields rise, the actual price falls by less than duration predicts.
• All standard option-free bonds exhibit positive convexity. The combined price change incorporating convexity is: $\% \Delta P \approx \left( -D_{\text{mod}} \times \Delta y \right) + \left( \frac{1}{2} \times \text{Convexity} \times (\Delta y)^2 \right)$

---

The Term Structure of Interest Rates and Yield Curves

The term structure of interest rates plots the relationship between bond yields and different maturities for default-free sovereign securities of identical credit quality. In the Philippines, this is represented by the PHP BVAL Reference Rates compiled by PDEx from secondary trading of Republic of the Philippines Treasury securities.

Four Yield Curve Configurations

1. Normal (Upward-Sloping): Short-term yields are lower than long-term yields. This is the standard configuration during economic expansions, reflecting compensation for maturity risk and expectations of positive future economic growth.
2. Inverted (Downward-Sloping): Short-term yields exceed long-term yields. An inverted yield curve is a historically reliable harbinger of macroeconomic contraction and recession. It occurs when investors anticipate heavy central bank rate cuts in response to economic weakness.
3. Flat: Yields across short, medium, and long maturities are virtually identical. Typically observed during transitional inflection points between expansion and slowdown.
4. Humped (Bell-Shaped): Intermediate-term yields (e.g., 3-to-5 years) are higher than both short-term and long-term yields, reflecting medium-term policy uncertainty.

Theoretical Explanations of Term Structure

Three prominent theories explain yield curve behavior on the SEC Phase 1 exam:

• Pure Expectations Theory: Posits that long-term interest rates are solely determined by market expectations of future short-term rates. Under this theory, an upward-sloping curve indicates markets expect short rates to rise; an inverted curve indicates markets expect short rates to fall.
• Liquidity Preference Theory: Proposes that investors prefer short-term liquidity and demand an additional yield premium (liquidity premium) to induce them to lock up capital in longer tenors. Because the liquidity premium increases with maturity, this theory explains why yield curves are naturally upward-sloping.
• Market Segmentation / Preferred Habitat Theory: Asserts that borrowers and lenders are restricted by regulation or business models to specific maturity segments (e.g., commercial banks in short tenors, life insurers and pension funds like SSS and GSIS in long tenors). Yields in each segment are determined independently by supply and demand within that maturity bucket.

---

Practical Exam Traps & Regulatory Context