3.3 Bonds and yield curves

Bonds and yield curves are essential concepts in actuarial mathematics. They provide insights into fixed-income securities and interest rate dynamics. Understanding these topics helps actuaries assess investment risks, value portfolios, and make informed financial decisions.

This section covers various bond types, pricing methods, and yield curve analysis. It also explores duration, convexity, and immunization strategies. These tools are crucial for managing interest rate risk and constructing effective bond portfolios in actuarial practice.

Types of bonds

• Bonds are debt securities issued by governments, corporations, and other entities to raise capital
• Bonds offer investors a fixed income stream in the form of regular interest payments and the return of principal at maturity
• Different types of bonds cater to various investor preferences and risk appetites

Government vs corporate bonds

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• Government bonds are issued by national governments and government agencies (U.S. Treasury bonds)
  • Considered to have lower default risk due to the government's ability to raise taxes and print money
  • Generally offer lower yields compared to corporate bonds
• Corporate bonds are issued by private and public corporations to finance operations, expansions, or acquisitions
  • Carry higher default risk than government bonds, as companies are more susceptible to financial distress
  • Offer higher yields to compensate investors for the increased risk

Coupon vs zero-coupon bonds

• Coupon bonds make regular interest payments to bondholders throughout the life of the bond
  • The coupon rate is the annual interest rate paid on the bond's face value
  • Coupon payments can be made annually, semi-annually, or quarterly
• Zero-coupon bonds do not make regular interest payments
  • Investors purchase these bonds at a discount to their face value
  • The difference between the purchase price and the face value represents the investor's return at maturity

Callable vs non-callable bonds

• Callable bonds give the issuer the right to redeem the bond before its maturity date
  • Issuers may choose to call bonds when interest rates fall, allowing them to refinance at a lower cost
  • Investors face reinvestment risk, as they may have to reinvest their funds at lower interest rates
• Non-callable bonds do not have a call provision
  • Investors are assured of receiving interest payments until the bond's maturity
  • Non-callable bonds offer more predictable cash flows and are preferred by investors seeking stable income

Bond pricing

• Bond pricing involves determining the fair value of a bond based on its expected cash flows and the prevailing market interest rates
• Understanding bond pricing is crucial for investors to make informed decisions and for actuaries to value bond portfolios accurately

Time value of money

• The time value of money is a fundamental concept in bond pricing
  • It states that a dollar received today is worth more than a dollar received in the future
  • Future cash flows must be discounted to their present value to determine a bond's fair price
• The discount rate used to calculate the present value of a bond's cash flows is based on the prevailing market interest rates and the bond's risk characteristics

Coupon rate vs yield to maturity

• The coupon rate is the annual interest rate paid on a bond's face value
  • It is fixed at issuance and remains constant throughout the bond's life
• Yield to maturity (YTM) is the total return an investor earns by holding a bond until maturity
  • YTM takes into account the bond's current price, coupon payments, and the time remaining until maturity
  • It assumes that all coupon payments are reinvested at the same rate

Clean vs dirty price

• The clean price of a bond is the quoted price without accrued interest
  • It represents the price an investor would pay for a bond if they purchased it immediately after a coupon payment
• The dirty price of a bond includes accrued interest
  • Accrued interest is the interest that has accumulated since the last coupon payment
  • The dirty price is the actual price an investor pays when purchasing a bond between coupon payment dates

Accrued interest calculations

• Accrued interest is calculated based on the bond's coupon rate, face value, and the number of days since the last coupon payment
• The day count convention used to calculate accrued interest varies depending on the bond market and the bond's terms
  • Common day count conventions include 30/360, actual/360, and actual/actual
• Accrued interest is added to the clean price to determine the dirty price, which is the total amount an investor pays for a bond

Yield curves

• A yield curve is a graphical representation of the relationship between bond yields and their maturities
• Yield curves provide valuable insights into market expectations, economic conditions, and the pricing of fixed-income securities

Definition and purpose

• A yield curve plots the yields of bonds with different maturities but similar credit quality
  • The x-axis represents the time to maturity, while the y-axis represents the corresponding yield
• Yield curves serve several purposes:
  • They help investors gauge the overall level of interest rates in the economy
  • They provide insights into market expectations about future interest rates and economic growth
  • They are used to price fixed-income securities and derivatives

Normal vs inverted yield curves

• A normal yield curve is upward sloping, with longer-term bonds offering higher yields than shorter-term bonds
  • This shape suggests that investors expect economic growth and inflation to remain stable or increase in the future
• An inverted yield curve is downward sloping, with shorter-term bonds offering higher yields than longer-term bonds
  • This shape suggests that investors expect economic growth and inflation to slow down or decline in the future
  • Inverted yield curves are often seen as a warning sign of a potential recession

Theories of yield curve shapes

• The expectations theory suggests that the shape of the yield curve reflects investors' expectations about future short-term interest rates
  • If investors expect short-term rates to rise, the yield curve will be upward sloping
  • If investors expect short-term rates to fall, the yield curve will be downward sloping
• The liquidity preference theory argues that investors prefer shorter-term bonds and demand a liquidity premium for holding longer-term bonds
  • This theory explains why longer-term bonds typically offer higher yields than shorter-term bonds
• The market segmentation theory suggests that different investors have distinct maturity preferences, creating separate markets for short-term and long-term bonds
  • This theory argues that the shape of the yield curve is determined by the supply and demand dynamics in each market segment

Constructing yield curves

• Yield curves are constructed using the yields of benchmark bonds with different maturities
  • Government bonds are often used as benchmarks due to their low default risk and high liquidity
• The yields of non-benchmark bonds can be estimated using interpolation or extrapolation techniques
  • Linear interpolation assumes a straight-line relationship between the yields of two benchmark bonds
  • More advanced techniques, such as cubic spline interpolation, can produce smoother yield curves
• Yield curve construction is an important task for actuaries, as it forms the basis for pricing and valuing fixed-income securities

Bond duration

• Bond duration is a measure of a bond's sensitivity to changes in interest rates
• It is an important concept for actuaries to understand, as it helps in managing interest rate risk and constructing bond portfolios

Definition and interpretation

• Duration measures the weighted average time until the bond's cash flows are received
  • It takes into account both the size and timing of the bond's coupon payments and principal repayment
• A bond's duration is expressed in years and can be interpreted as the percentage change in the bond's price for a 1% change in interest rates
  • For example, if a bond has a duration of 5 years, its price is expected to decrease by approximately 5% for a 1% increase in interest rates

Macaulay vs modified duration

• Macaulay duration is the weighted average time until a bond's cash flows are received, calculated using the bond's yield to maturity
  • It assumes that the bond's cash flows are reinvested at the same yield to maturity
• Modified duration is an adjustment to Macaulay duration that accounts for the effect of interest rate changes on the bond's yield to maturity
  • Modified duration is calculated by dividing Macaulay duration by (1 + yield to maturity)
  • Modified duration provides a more accurate estimate of a bond's price sensitivity to interest rate changes

Factors affecting bond duration

• A bond's duration is influenced by several factors:
  • Time to maturity: Bonds with longer maturities have higher durations, as their cash flows are spread over a longer period
  • Coupon rate: Bonds with lower coupon rates have higher durations, as a larger portion of their cash flows comes from the principal repayment at maturity
  • Yield to maturity: Bonds with lower yields to maturity have higher durations, as the present value of their cash flows is more sensitive to changes in interest rates
• Understanding these factors helps actuaries manage interest rate risk and make informed decisions when constructing bond portfolios

Duration and interest rate risk

• Duration is a key measure of a bond's interest rate risk
  • Bonds with higher durations are more sensitive to changes in interest rates and have greater price volatility
  • Bonds with lower durations are less sensitive to interest rate changes and have lower price volatility
• Actuaries use duration to assess the interest rate risk of individual bonds and bond portfolios
  • By matching the duration of assets (bonds) to the duration of liabilities (insurance policies or pension obligations), actuaries can help minimize the impact of interest rate changes on a company's financial position

Bond convexity

• Bond convexity is a measure of the curvature of the relationship between a bond's price and its yield
• It is an important concept for actuaries to understand, as it complements duration in assessing a bond's price sensitivity to interest rate changes

Definition and interpretation

• Convexity measures the rate of change of a bond's duration as interest rates change
  • It captures the non-linear relationship between a bond's price and its yield
• A bond with positive convexity will have a price that increases more when interest rates fall than it decreases when interest rates rise
  • This asymmetric price response is favorable for investors, as it provides a "cushion" against interest rate increases
• Bonds with higher convexity are more desirable, as they offer better protection against interest rate risk

Convexity vs duration

• Duration is a first-order approximation of a bond's price sensitivity to interest rate changes
  • It assumes a linear relationship between price and yield changes
• Convexity is a second-order approximation that captures the curvature of the price-yield relationship
  • It accounts for the fact that the relationship between price and yield is not perfectly linear
• Combining duration and convexity provides a more accurate estimate of a bond's price sensitivity to interest rate changes

Calculating bond convexity

• Bond convexity can be calculated using the following formula:
  • Convexity = (P₋ + P₊ - 2P₀) / (2P₀ × Δy²)
  • Where P₋ is the bond's price if yields decrease by Δy, P₊ is the price if yields increase by Δy, P₀ is the current price, and Δy is the change in yield
• The convexity calculation requires estimating the bond's price at three different yield levels
  • This can be done using the bond's cash flows and the corresponding discount rates
• Convexity is expressed as a positive number, with higher values indicating greater convexity

Convexity and bond price sensitivity

• Convexity helps to refine the estimate of a bond's price sensitivity provided by duration
• The total percentage change in a bond's price for a given change in yield can be approximated using both duration and convexity:
  • Percentage price change ≈ -Duration × Δy + 0.5 × Convexity × Δy²
  • Where Δy is the change in yield
• This approximation demonstrates that convexity becomes increasingly important for larger changes in yield
  • For small yield changes, the duration term dominates the price change estimate
  • For larger yield changes, the convexity term becomes more significant

Immunization strategies

• Immunization is an investment strategy that seeks to protect a bond portfolio against interest rate risk
• Actuaries use immunization techniques to manage the assets backing insurance policies or pension obligations

Definition and purpose

• Immunization involves structuring a bond portfolio so that its cash flows match the timing and amount of the liabilities it is intended to cover
  • The goal is to minimize the impact of interest rate changes on the portfolio's value relative to the value of the liabilities
• Immunization strategies are particularly important for insurance companies and pension funds
  • These entities have long-term liabilities that are sensitive to changes in interest rates
  • By immunizing their bond portfolios, they can reduce the risk of insufficient assets to meet their obligations

Duration matching vs cash flow matching

• Duration matching is an immunization technique that involves matching the duration of the bond portfolio to the duration of the liabilities
  • This approach aims to ensure that the portfolio's value changes in the same direction and magnitude as the liabilities when interest rates change
  • Duration matching is a relatively simple and widely used immunization strategy
• Cash flow matching is a more precise immunization technique that involves matching the cash flows of the bond portfolio to the expected cash outflows of the liabilities
  • This approach ensures that the portfolio generates sufficient cash flows to meet the liabilities as they come due
  • Cash flow matching requires a more granular analysis of the liability cash flows and may be more difficult to implement than duration matching

Rebalancing bond portfolios

• Immunization is not a one-time process; bond portfolios must be periodically rebalanced to maintain their immunized status
  • As time passes and interest rates change, the duration and cash flows of the portfolio may diverge from those of the liabilities
• Rebalancing involves adjusting the portfolio's holdings to realign its duration or cash flows with the liabilities
  • This may involve selling some bonds and purchasing others with different maturities or coupon rates
• The frequency of rebalancing depends on the specific immunization strategy and the volatility of interest rates
  • More frequent rebalancing may be necessary during periods of high interest rate volatility

Limitations of immunization

• While immunization strategies can help mitigate interest rate risk, they have some limitations:
  • Immunization assumes that interest rate changes affect all bonds equally, but in reality, different bonds may react differently to rate changes
  • Immunization does not protect against other risks, such as credit risk or liquidity risk
  • Perfect immunization is difficult to achieve in practice, as it requires a precise matching of cash flows or durations
• Despite these limitations, immunization remains an important tool for actuaries in managing the interest rate risk of bond portfolios

Credit risk and ratings

• Credit risk is the risk that a bond issuer will default on its obligations, failing to make interest payments or repay the principal
• Credit ratings are used to assess the creditworthiness of bond issuers and help investors make informed decisions

Credit risk assessment

• Credit risk assessment involves analyzing the financial health and ability of a bond issuer to meet its debt obligations
  • This includes evaluating factors such as the issuer's financial statements, cash flows, debt levels, and industry prospects
• Credit risk is an important consideration for actuaries when valuing bond portfolios and setting insurance premiums
  • Bonds with higher credit risk typically offer higher yields to compensate investors for the increased risk of default
• Actuaries use various models and techniques to quantify credit risk, such as default probability models and expected loss calculations

Bond rating agencies

• Bond rating agencies, such as Moody's, Standard & Poor's, and Fitch, provide credit ratings for bond issuers
  • These ratings reflect the agencies' opinions on the creditworthiness of the issuers and their ability to meet debt obligations
• Bond ratings are expressed as letter grades, with AAA (or Aaa) being the highest quality and C or D representing default or near-default
  • Ratings are divided into investment-grade (BBB-/Baa3 or higher) and high-yield or "junk" (BB+/Ba1 or lower) categories
• Bond ratings are important for investors, as they provide a standardized assessment of credit risk and help determine the appropriate yield for a bond

Investment-grade vs high-yield bonds

• Investment-grade bonds are those rated BBB-/Baa3 or higher by the major rating agencies
  • These bonds are considered to have a relatively low risk of default and are suitable for most institutional investors
  • Investment-grade bonds generally offer lower yields than high-yield bonds due to their lower risk profile
• High-yield bonds, also known as "junk" bonds, are those rated BB+/Ba1 or lower
  • These bonds are considered to have a higher risk of default and are often issued by companies with weaker financial profiles or in industries with greater uncertainty
  • High-yield bonds offer higher yields to compensate investors for the increased risk

Credit spreads and default risk

• Credit spreads are the difference in yield between a bond and a benchmark security, typically a government bond of similar maturity
  • Credit spreads reflect the additional yield investors demand for taking on the credit risk of a bond
• Bonds with higher credit risk (lower ratings) tend to have wider credit spreads, while bonds with lower credit risk (higher ratings) have narrower spreads
  • Changes in credit spreads can indicate changes in the market's perception of a bond issuer's creditworthiness
• Actuaries monitor credit spreads and use them to assess the default risk of bond portfolios
  • Widening credit spreads may signal increasing default risk and require adjustments to portfolio valuations or insurance premiums

Taxation of bonds

• The taxation of bonds is an important consideration for investors and can impact the after-tax returns of bond portfolios
• Actuaries need to understand the tax treatment of different types of bonds to accurately value portfolios and make investment decisions

Taxable vs tax-exempt bonds

• Taxable bonds are those whose interest payments are subject to federal, state, and/or local income taxes
  • Most corporate bonds and some government bonds (such as U.S. Treasury bonds) are taxable
  • The interest income from these bonds is included in an investor's taxable income and taxed at the applicable marginal rate
• Tax-exempt bonds