Understanding Bond Fundamentals

Bond Fundamentals

• Face or par value is typically $1,000 per bond in the U.S. market
• Face value represents the principal amount repaid at maturity and calculates coupon payments
• For instance, a bond possessing a face value of $1,000 returns $1,000 at maturity to the bondholder, irrespective of purchase price

Coupon Details

• Coupon rate refers to the annual interest rate based on the bond's face value
• Coupon payment is the actual dollar amount is disbursed periodically
• Payment frequency in the U.S. is typically semi-annual, but can be annual, quarterly, or monthly
• For example: A $1,000 bond, carrying a 6% annual coupon rate on semi-annual payments, disburses $30 every six months ($1,000 * 6% / 2)

Bond Maturity

• Term to maturity defines the duration until the bond's principal is repaid
• Short-term bonds are those maturing in under five years
• Intermediate-term bonds mature between 5 to 12 years
• Long-term bonds mature after more than 12 years
• The maturity date remains fixed upon the bond's issue

Yield to Maturity (YTM)

• YTM signifies the predicted total return on a bond held until maturity
• YTM takes into account coupon income, capital gains/losses (if bought at a discount/premium), and the money's time value
• It functions as the discount rate in present value calculations

Bond Market Terminology

• Premium bonds trade above par value (>$1,000) where the coupon rate exceeds the market rate
• Discount bonds trade below par value (<$1,000) where the coupon rate is less than the market rate
• Par bonds trade at par value (= $1,000) where coupon rate equals the market interest rate
• Current yield is the annual coupon payment divided by the current market price
• A bond indenture is a legal document outlining all terms/conditions
• A call provision enables the issuer to redeem bonds before maturity, generally at a premium
• A put provision grants bondholders the right to sell bonds back to the issuer at a specified price

Different Bond Types

• Government bonds are issued by national governments such as U.S. Treasury bonds, bills, and notes
• Municipal bonds are issued by states, cities, or local governments, often tax-exempt
• Corporate bonds are issued by corporations, which may have higher yields but carry bigger risk
• Agency bonds are issued by government-sponsored enterprises
• Foreign bonds are issued in a domestic market by foreign entities
• Convertible bonds can be converted into a specified number of common shares

Bond Valuation

• Bond price equals the present value of all future cash flows expressed as: $$\text{Bond Price} = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^n}$$
• C = coupon payment for each period
• F = face value
• r = required yield per period (YTM per period)
• n = number of periods until maturity
• C = annual coupon payment / 2 for semi-annual payments
• r = annual yield / 2 for semi-annual payments
• n = years to maturity * 2 for semi-annual payments

Calculating Bond Price (Example)

• For a 10-year bond possessing a $1,000 face value and an 8% annual coupon rate paid semi-annually, with a 6% market yield the components are:
• Face value (F) = $1,000
• Annual coupon = $1,000 * 0.08 = $80
• Semi-annual coupon (C) = $80 / 2 = $40
• Semi-annual yield (r) = 0.06 / 2 = 0.03 (3%)
• Number of periods (n) = 10 * 2 = 20
• The present value of the coupon payments is expressed as: $$PV(\text{coupons}) = \sum_{t=1}^{20} \frac{$40}{(1+0.03)^t} = $595.10$$
• The present value of the face value is expressed as: $$PV(\text{face value}) = \frac{$1,000}{(1+0.03)^{20}} = $553.70$$
• Sum up the present values, as per, $$\text{Bond Price} = $595.10 + $553.70 = $1,148.80$$
• Since it's coupon rate (8%) exceeds market yield (6%) this bond is priced at a premium ($1,148.80 > $1,000)

Bond Pricing (Alternative Formula)

• For bonds entailing regular coupon payments, a shortcut formula is expressed as: $$\text{Bond Price} = C \times \left[ \frac{1-\frac{1}{(1+r)^n}}{r} \right] + \frac{F}{(1+r)^n}$$
• This formula simplifies calculation appreciably

Understanding YTM

• YTM mirrors the internal rate of return (IRR) on a bond investment
• The bond is kept until maturity
• Coupon payments are reinvested at the YTM rate
• The issuer fulfills all payments on schedule

YTM Calculation Methods

• Trial and Error
  • Initiate with a YTM guess
  • Calculate the bond price through the YTM
  • Adjust the YTM upwards/downwards based on if the calculated price is above/below the actual price
  • Keep repeating until the calculated price aligns with the real price
• Financial Calculator Method:
  • N = number of periods
  • PV = present bond price (inputted as negative)
  • PMT = periodic coupon payment
  • FV = face value
  • Solve for I/Y (interest/yield)
• Bond Pricing Rearrangement
  • Set the bond pricing equation equal to the current market price
  • Solve for r (YTM)
  • Numerical methods must be used as there's no direct algebraic solution

YTM vs Current Yield vs Coupon Rate

• Coupon Rate = (Annual coupon payment) / (Face value)
• Current Yield = (Annual coupon payment) / (Current bond price)
• YTM = Comprehensive return inclusive of coupon income and capital gain/loss
• When a bond is trades at a premium (price is greater than face value): Coupon rate > Current yield > YTM
• When a bond trades at discount (price is less than the face value): Coupon rate < Current yield < YTM
• When a bond trades at par (price equals face value): Coupon rate = Current yield = YTM

YTM Approximation

• When precise calculation tools aren't available, this approximation can help determine the YTM: $$\text{YTM} \approx \frac{C + \frac{F-P}{n}}{(F+P)/2}$$
  • C = annual coupon payment
  • F = face value
  • P = current price
  • n = years to maturity
• This approximation is most effective for bonds that trade around par value

Zero Coupon Bonds: Characteristics

• Do not provide periodic interest payments
• Issued at a substantial discount to their face value
• Its return is derived entirely from the difference between purchase price and face value
• They are subject to "phantom income" taxation in taxable accounts, in which interest accrues annually absent reception until maturity
• Examples encompass U.S. Treasury STRIPS and zero-coupon corporate bonds

Zero Coupon Bond Valuation

• The core formula in valuing zero-coupon bonds is articulated as: $$P = \frac{F}{(1+r)^n}$$
  • P = current value
  • F = Bond's face value
  • r = yield per period
  • n = number of periods

Example: Zero Coupon Bond Calculation

• A zero-coupon bond has a face value of $1,000 and matures in 15 years, with 7% market yield that is compounded semi-annually.
• Convert to semi-annual compounding: rate (r) = 0.07 / 2 = 0.035 per six-month period and number of periods (n) = 15 * 2 = 30 periods
• Calculate the price with: $$P = \frac{$1,000}{(1+0.035)^{30}} = $356.30$$

Interest Rate sensitivity

• Zero-coupon bonds exhibit the highest interest rate sensitivity (duration) among fixed-income because:
  • The entirety of their value originates in a singular payment at maturity
  • There aren't any intermediate cash flows to reinvest at potentially higher rates
• Changes in rates impacts the whole investment
• Good for:
  • Matching with known future obligations
  • Speculating on interest rate decreases
  • Eliminating reinvestment risk

Factors that Impact Bond Price Sensitivity

• Maturity Effect
  • Greater price sensitivity and longer maturities are correlated
  • The longer a bond's maturity is, the bigger affect rate chances have
  • 1% hike in yield lowers the price of a 30-year bond around 3x more than a 5-year bond
• Coupon Rate Effect
  • Lesser coupon rates correspond to greater price sensitivity
  • The more value in lower coupon bonds is concentrated in the final payment
  • A 3% coupon bond will change in price more dramatically than an 8% coupon bond with the same maturity
• Initial Yield Level Effect
  • Price sensitivity is greater where initial yields are lower
  • Price changes are non-linear (convex) with respect to yield changes
  • Ex: A bond entailing a 3% initial yield undergoes a larger price change than a bond with a 8% initial yield

Bond Duration

• Duration directly measures a bond's price sensitivity and interest rate changes
• Macaulay Duration measures weighted average time until all cash flows are received and is measured in years
• Macaulay Duration Formula: $$D = \frac{\sum_{t=1}^{n} \frac{t \times CF_t}{(1+r)^t}}{\text{Bond Price}}$$
  • CF_t is cash flow in period t

Bond Modified Duration

• Measures approximate percentage change in price when there's a 1% change in yield
• The formula is: $$\text{Modified Duration} = \frac{\text{Macaulay Duration}}{1+r}$$
• Approx. Percentage Price Change: $$% \Delta \text{Price} \approx -\text{Modified Duration} \times \Delta \text{Yield}$$

Duration Properties

• Higher duration and longer maturity are correlated
• Higher duration and lower coupon rate are correlated
• Higher duration and lower market yield are correlated
• Zero coupon bonds have duration that is equal to maturity

Bond Convexity

• Duration approximation functions best when yield changes are small but loses accuracy when changes are larger because of the convex relationship that exist between the price and yield
• Convexity Adjustment: $$% \Delta \text{Price} \approx -\text{Duration} \times \Delta \text{Yield} + \frac{1}{2} \times \text{Convexity} \times (\Delta \text{Yield})^2$$

Properties of Convexity

• Always positive for option-free bonds
• Bonds with lower coupons and longer maturities have higher convexity
• Explains why price increases from yield decreases are often be larger than price decreases from equivalent yield increases

Details on Yield Curve Shapes

• Normal (Upward-Sloping)
  • Lower short-term rates than long-term rates
  • Reflect expectations:
    • Future economic growth
    • Rising inflation
    • Higher future short-term interest rates
  • The most common shape historically that typically steepens early in economic recoveries
• Flat
  • Similar yields found across all maturities may indicate economic uncertainty
  • The yield curve often occurs during transition periods and may precede inversions that occur after a normal curve
• Inverted
  • Higher short-term rates than long-term rates
  • It's an indicator of economic recession that often happens approximately 6-18 months before recessions
  • Indicates Expectation
    • Economic Slowdown
    • Future interest rate decreases
    • Lower future inflation
• Humped
  • Medium-term yields are higher than both short and long-term yields
  • It's a relatively rare occurrence, and may indicate mixed economic signals or transition

Theories that Explain the Yield Curve

• Expectations Theory
  • Market expectations of future short-term rates determines long-term rates
  • Investors view short and long-term bonds as interchangeable
  • Formula: $(1+r_L)^n = (1+r_S)(1+f_1)(1+f_2)...(1+f_{n-1})$
    • Where $r_L$ is the n-period long rate, $r_S$ is the current short rate, and $f_i$ are the expected future short rates
• Liquidity Preference Theory:
  • Due to lower risk, investors prefer shorter maturities
  • Term premiums are driven by demand for longer maturities
  • Yield Curves are normally upward-sloping and can be expressed as: Long-term rate = Average of expected short-term rates + Liquidity premium
• Market Segmentation Theory
  • Different market participants operate in different maturity segments and independently determines supplies and demand
• Preferred Habitat Theory:
  • Investors have preferred maturity ranges but shift if yield differentials are sufficient and combines elements of expectations and segmentation theories
  • This theory explains both the general shape and occasional anomalies

Applications for Yield Curve Analysis

• Economic Forecasting:
  • Since 1955, Inverted curves indicated all U.S. recessions with few wrong signals
• Central Bank Policy:
  • Steepness is an indicator of the expected monetary policy direction
• Investment Strategy:
  • Guides bond portfolio positioning based on term premium expectations
• Yield Curve Spread Strategies
  • Bullet: Concentrate in a single maturity segment
  • Barbell: Invest in short and long maturities, avoiding intermediates
  • Ladder: Equal distribution across maturities

Real vs Nominal

• Nominal interest rate refers to the stated rate absent of inflation adjustment
• Real interest rate is inflation-adjusted returns that reflects actual purchasing power changes
• Ex-Ante real rate are rates based on expected inflation used for forward-looking
• Ex-Post real rates are rates based on actually realized inflation used for backward-looking

Relationship Between Rates

• Approximation Formula: $$r_{\text{real}} \approx r_{\text{nominal}} - \text{inflation rate}$$
• Exact Formula: $$1 + r_{\text{nominal}} = (1 + r_{\text{real}})(1 + \text{inflation rate})$$
• The previous formula can be rearranged into :$$r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + \text{inflation rate}} - 1$$

Example of Nominal vs Real

• Given a rate of 8% and inflation of 3%, the real interest in this example is calculated through approximation which = 5%
• Given a rate of 8% and inflation of 3%, the real interest in this example is calculated through the exact formula is

4. 85%

Impact of Inflation on Bonds

• Fixed-Rate Bonds: Value decreases with unexpected inflation
• Floating-Rate Bonds: Offers more protection because rates adjust periodically
• Inflation-Protected Securities: Is principal adjusted against an inflation indexed index
• Equities: Mixed impact, based on pricing power and cost structure
• Real Assets (real estate, commodities): Often serve as inflation hedges

Break-Even Inflation Rate

• Break-Even Inflation = Nominal Yield - TIPS Yield
• Represents inflation rate where investors don't care about holding a nominal or an inflation-protected

Fisher Effect

• The nominal interest rate in an economy equates to the sum of the real interest rate plus the expected inflation proposed by economist Irving Fisher
• Comprehensive formula: $$1 + r_{\text{nominal}} = (1 + r_{\text{real}})(1 + \text{inflation rate})$$
• Which expands to: $$r_{\text{nominal}} = r_{\text{real}} + \text{inflation rate} + (r_{\text{real}} \times \text{inflation rate})$$ $$r_{\text{nominal}} \approx r_{\text{real}} + \text{inflation rate}$$

International Fisher Effect

• Countries entailing elevated nominal interest rates should undergo currency depreciation relative to countries with lower rates
• The expected change in exchange rates should mirror the interest rate differential between different countries
• Formula: $$\frac{E_1 - E_0}{E_0} \approx (r_{\text{domestic}} - r_{\text{foreign}})$$
  • $E_0$ = Current exchange rate (domestic/foreign)
  • $E_1$ = Expected future exchange rate
  • $r_{\text{domestic}}$ = Domestic interest rate
  • $r_{\text{foreign}}$ = Foreign interest rate

Fisher Effect Applications

• Monetary Policy Analysis to help influence real rates and understand inflation expectations
• Investment Decision-Making to compare investments across different inflation environments
• International Investment when informing carry trade strategies