8.1 Fixed Income Instruments, Pricing, Yield Measures & Term Structure of Interest Rates

8.1 Fixed Income Instruments, Pricing, Yield Measures & Term Structure of Interest Rates

Fixed income securities form the defensive anchor and liability-matching backbone of institutional portfolios. For investment management consultants, mastering fixed income valuation requires moving beyond simple yield quotes to understand cash flow discounting mechanics, day-count settlement conventions, spot curve bootstrapping, and the macroeconomic forces shaping the term structure of interest rates.

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1. Bond Pricing Mechanics & Fundamental Valuation Principles

The Cash Flow Discounting Framework

A standard fixed-rate bond represents a contractual promise by the issuer to pay periodic coupon payments ($C$) over $N$ periods and return the principal or par value ($M$) at maturity. The fundamental economic value (price $P$) of an option-free bond equals the sum of its discounted future cash flows:

$P = \sum_{t=1}^N \frac{C}{(1+y)^t} + \frac{M}{(1+y)^N}$

In the U.S. market, corporate, municipal, and Treasury bonds pay coupons semi-annually. The semi-annual valuation model adjusts the annual coupon ($C$), annual required yield ($y$), and maturity in years ($N$):

$P = \sum_{t=1}^{2N} \frac{C / 2}{\left(1 + \frac{y}{2}\right)^t} + \frac{M}{\left(1 + \frac{y}{2}\right)^{2N}}$

The Inverse Price-Yield Relationship and Convexity

Bond prices and interest rates share a fundamental inverse relationship: as market required yields rise, the present value of fixed future cash flows declines, lowering the bond's market price. Conversely, falling yields increase present value, driving prices higher.

Because the discount function $\frac{1}{(1+y)^t}$ is a non-linear convex function of $y$, the price-yield curve is curved (convex to the origin):

• For a given change in yield ($\Delta y$), the percentage price increase when yields fall exceeds the percentage price decrease when yields rise by an identical magnitude.
• This structural property—positive convexity—benefits the bondholder, as upside price gains exceed downside price losses for symmetric yield shifts.

  Bond Price
      ^
      |     * (Price rises more when yield falls)
      |      \
      |       \  <--- Convex Price-Yield Curve
  Par |--------*---------
      |         \ 
      |          \* (Price falls less when yield rises)
      +-------------------------> Yield to Maturity

The Coupon-Yield-Price Hierarchy

The relationship between a bond's stated coupon rate ($c$), current yield ($CY$), and Yield to Maturity ($YTM$) directly reflects whether the bond trades at par, at a discount, or at a premium.

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2. Clean vs. Dirty Pricing & Accrued Interest Conventions

When bonds trade between coupon payment dates, the buyer must compensate the seller for the interest accrued during the fraction of the coupon period the seller owned the security.

Flat (Clean) vs. Full (Dirty) Price

• Clean Price (Flat Price): The quoted market price of the bond excluding accrued interest. Quoted prices in financial media and dealer screens are clean prices to prevent artificial daily price drift between coupon dates.
• Dirty Price (Full / Invoice Price): The actual gross settlement cash paid by the buyer to the seller on trade settlement date ($T+1$).

$\text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest (AI)}$

$\text{Accrued Interest} = \text{Coupon Amount per Period} \times \left( \frac{\text{Days Accrued from Prior Coupon Date}}{\text{Total Days in Current Coupon Period}} \right)$

Day-Count Conventions

Different fixed income sectors apply standardized market conventions for counting accrued days:

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3. Comprehensive Yield Measures & Return Frameworks

Institutional allocators utilize multiple distinct yield metrics depending on cash flow structures and investment objectives.

1. Current Yield ($CY$)

Current yield measures the annual coupon income relative to the bond's current clean market price:

$CY = \frac{\text{Annual Coupon Payment}}{\text{Clean Market Price}}$

Analytical Limitation: Current yield ignores the time value of money, reinvestment income on interim coupons, and the capital gain or loss realized when the bond matures at par.

2. Yield to Maturity (YTM)

Yield to Maturity is the internal rate of return (IRR) that equates the present value of all scheduled future coupon and principal cash flows to the bond's dirty price. YTM is standardly quoted as an annualized Bond Equivalent Yield (BEY) by doubling the semi-annual IRR:

$\text{BEY} = 2 \times y_{\text{semi-annual}}$

The YTM Reinvestment Assumption: YTM accurately represents the investor's realized annual return only if:

1. The bond is held to final maturity with zero default; and
2. Every interim coupon payment is reinvested immediately at an interest rate exactly equal to the calculated YTM.

If market reinvestment rates fall below YTM over the holding period, realized total return will lag the initial YTM—an effect known as reinvestment risk.

3. Yield to Call (YTC) & Yield to Worst (YTW)

For callable debt, the issuer holds the right to redeem the bond prior to maturity at a specified call price ($P_{\text{call}}$) on designated call dates.

$\text{Dirty Price} = \sum_{t=1}^{k} \frac{C/2}{\left(1 + \frac{y_c}{2}\right)^t} + \frac{P_{\text{call}}}{\left(1 + \frac{y_c}{2}\right)^k}$

• Yield to Call (YTC): The annualized IRR calculated assuming the bond is called at the first or next permissible call date.
• Yield to Worst (YTW): The minimum calculated yield among YTM and all possible call/put/refunding scenarios. Institutional fixed income mandates require callable bonds to be quoted on a Yield-to-Worst basis to enforce conservative return expectations.
• Rule of Thumb: When a callable bond trades at a substantial premium (above call price), call risk is high and $\text{YTC} < \text{YTM}$, making YTC the Yield to Worst. When trading at a discount, $\text{YTM} < \text{YTC}$, making YTM the Yield to Worst.

4. Realized Compound Yield (Horizon Total Return)

Realized compound yield evaluates total portfolio return over an explicit investment horizon ($H < N$). It incorporates explicit assumptions for:

1. The specific reinvestment rate ($r_{\text{reinvest}}$) earned on interim coupon cash flows;
2. The terminal sale price ($P_H$) at horizon end, determined by projected terminal yield ($y_H$).

$\text{Total Future Value (FV}_H\text{)} = \text{FV of Reinvested Coupons} + P_H$ $\text{Realized Compound Annual Return} = \left( \frac{\text{FV}_H}{P_0} \right)^{\frac{1}{H}} - 1$

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4. Spot Curves, Par Curves & Forward Rate Bootstrapping

Spot Rates ($z_t$) and Arbitrage-Free Valuation

A spot rate ($z_t$) is the annual yield to maturity on a pure zero-coupon default-free Treasury security maturing at exact time $t$. Under the Law of One Price, a coupon-bearing bond is economically equivalent to a portfolio of individual zero-coupon strips. Its theoretical arbitrage-free price must equal the sum of each discrete cash flow discounted by its corresponding maturity-matched spot rate:

$P = \sum_{t=1}^N \frac{C_t}{(1 + z_t)^t} + \frac{M}{(1 + z_N)^N}$

Discounting every cash flow at a single average YTM introduces pricing distortions whenever the yield curve is non-flat.

Par Yield Curves

A par curve represents the coupon rate at which benchmark coupon bonds would price exactly at par ($100$) across successive maturities. It reflects observed on-the-run Treasury auction yields.

Forward Rate Bootstrapping Mechanics

Because liquid zero-coupon Treasuries do not trade at every maturity, analysts bootstrap spot rates from on-the-run par coupon bonds, and derive implied forward rates from the spot curve.

A forward rate (${}_t f_k$) is the break-even interest rate contracted today for a loan originating at future time $t$ and maturing $k$ periods later. Under the no-arbitrage condition, investing in an $N$-period spot zero must yield the identical compounded wealth as investing in an $A$-period zero and rolling the proceeds into a forward contract for the remaining $(B-A)$ periods:

$(1 + z_B)^B = (1 + z_A)^A \times (1 + {}_A f_{B-A})^{B-A}$

$\implies {}_A f_{B-A} = \left[ \frac{(1 + z_B)^B}{(1 + z_A)^A} \right]^{\frac{1}{B-A}} - 1$

Bootstrapping Spot Rates and Implied Forward Rates: Step-by-Step

Consider the following annual zero-coupon spot rate curve:

• 1-Year Spot Rate ($z_1$): $4.00%$
• 2-Year Spot Rate ($z_2$): $5.00%$
• 3-Year Spot Rate ($z_3$): $5.75%$

1. Deriving the 1-Year Forward Rate 1 Year Forward (${}_1f_1$): $(1 + z_2)^2 = (1 + z_1)^1 \times (1 + {}_1f_1)^1$ $(1 + 0.05)^2 = (1 + 0.04)^1 \times (1 + {}_1f_1)$ $1.1025 = 1.04 \times (1 + {}_1f_1) \implies (1 + {}_1f_1) = \frac{1.1025}{1.04} = 1.060096$ ${}_1f_1 = 6.01\%$

2. Deriving the 1-Year Forward Rate 2 Years Forward (${}_2f_1$): $(1 + z_3)^3 = (1 + z_2)^2 \times (1 + {}_2f_1)^1$ $(1 + 0.0575)^3 = (1 + 0.05)^2 \times (1 + {}_2f_1)$ $1.182604 = 1.1025 \times (1 + {}_2f_1) \implies (1 + {}_2f_1) = \frac{1.182604}{1.1025} = 1.072657$ ${}_2f_1 = 7.27\%$

Key Takeaway: When the spot curve is upward-sloping ($z_1 < z_2 < z_3$), implied forward rates lie strictly above the spot curve (${}_2f_1 > {}_1f_1 > z_2 > z_1$).

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5. Theories of the Term Structure of Interest Rates

Four core theories explain why yield curves exhibit different slopes and how market expectations interact with risk premia.

+-----------------------------------------------------------------------------------------+
|                         TERM STRUCTURE OF INTEREST RATES THEORIES                       |
+-----------------------------+-----------------------------+-----------------------------+
| PURE EXPECTATIONS           | LIQUIDITY PREFERENCE        | MARKET SEGMENTATION         |
| - Forward = Expected Spot   | - Upward slope via Term     | - Supply / Demand in strict |
| - Investors risk-neutral    |   Premium L_t > 0           |   independent buckets       |
| - No maturity preference    | - Long rates require risk   | - No cross-maturity trading |
|                             |   compensation for duration | - ALM institutional mandates|
+-----------------------------+-----------------------------+-----------------------------+
| PREFERRED HABITAT: Modifies segmentation; participants cross buckets for yield premium  |
+-----------------------------------------------------------------------------------------+