Lecture 7: Comprehensive Notes on Bond Price Relationships, Convexity, and Valuation

Evaluation of Bond Price and Yield Relationships

• The primary relationship explored in bond metrics is the investment relationship between bond price and yields.

• This understanding arises from both a mathematical perspective and the application of the law of one price.

• An inverse relationship exists between bond price and yield; as yields increase, bond prices decrease, and vice versa.

Bond Price and Face Value Relationships Relative to Coupon Rates

• Understanding how bonds with different coupon rates trade at the same yield is critical for evaluating bond returns.

• The relationship between bond price and face value depends on the comparison between the coupon rate ($CR$) and the yield to maturity ($YTM$).

• Bonds Trading at Par:

  • Occurs when the coupon rate is equal to the yield to maturity ($CR = YTM$).

  • Example: If investors require a $10\%$ return and the bond offers a $10\%$ coupon, the bond provides exactly what is required.

  • The bond does not need to offer extra incentives like discounts; therefore, price equals face value.

• Bonds Trading at a Discount:

  • Occurs when the coupon rate is lower than the yield to maturity (CR < YTM).

  • If the coupon rate is $8\%$ and investors require a total return of $10\%$, the coupon alone is insufficient.

  • The bond must offer a discount to close the $2\%$ gap in return. This discount acts as a capital gain for the investor.

  • Example calculation:

  • Face Value: $1,000$

  • Price: $877.11$

  • Discount (Capital Gain): $1,000 - 877.11 = 122.89$

  • This capital gain, when added to the $8\%$ coupon, brings the total return to the required $10\%$.

  • Specifically, the present value of the missing $2\%$ return (e.g., $R20$ on a $1,000$ face value) over $10$ years at $10\%$ equals exactly $122.89$, the amount of the discount.

• Bonds Trading at a Premium:

  • Occurs when the coupon rate is higher than the yield to maturity (CR > YTM).

  • Example: The bond pays an $8\%$ coupon when investors only require a $6\%$ total return.

  • The bond price will be greater than the face value because it offers more than the market requires.

  • Example calculation:

  • Coupon: $R80$

  • Yield: $6\%$

  • Time: $10$ years

  • Price: $1,147.20$

  • The premium is the difference between price and face value: $1,147.20 - 1,000 = 147.20$.

  • The premium effectively acts as a built-in capital loss over the life of the bond. Paying $1,147.20$ today to receive $1,000$ at maturity ensures the total return is capped at $6\%$ despite the higher coupon payments.

  • This mechanism ensures the bond's total return remains in line with similar market instruments.

Positive Convexity in Bonds

• The relationship between bond prices and yields is not linear; it is convex. This feature is known as positive convexity.

• Positive convexity means that an equal change in yield up and down will result in a price change of different magnitudes:

  • Price appreciation (when yields drop) is greater than price depreciation (when yields rise) for the same basis point change.

• Numerical Summary of Convexity Example:

  • Starting Point: $CR = 8\%$, $YTM = 8\%$, Price = $1,000$.

  • Yield increases to $10\%$ ($+2\%$): Price drops to $877.11$, a decrease of $122.89$.

  • Yield decreases to $6\%$ ($-2\%$): Price rises to $1,147.20$, an increase of $147.20$.

  • Result: The price increase ($147.20$) is larger than the price decrease ($122.89$) for the same $2\%$ yield move.

• Graphical Characteristics:

  • The price-yield curve is steeper at low interest rate levels and flatter at high interest rate levels.

  • This asymmetry makes bonds with high convexity attractive to traders, as it offers more upside for falling rates than downside for rising rates.

Time Path of a Bond and Time Decay

• Time decay, or "pull to par," refers to how a bond price changes as it approaches maturity, assuming the yield to maturity remains constant.

• Par Bonds: The price remains constant at face value until maturity.

• Discount Bonds: The price increases toward face value over time as the discount is realized gradually as a capital gain.

• Premium Bonds: The price decreases toward face value over time as the premium is realized gradually as a capital loss.

Decomposition of Bond Price Changes

• There are two primary reasons a bond price changes:

  1. The change in the yield to maturity ($YTM$).

  2. The change in time (time decay/pull to par).

• It is essential to decompose these changes to evaluate a manager's skill. Price appreciation due to time decay on a discount bond is passive, whereas appreciation due to correctly anticipating yield changes is considered a skill.

• Case Study for Decomposition:

  • Given: A $20$-year bond (initially $12$ years in description), Face Value = $1,000$, Coupon = $9\%$ (paid twice a year, so $R45$ per period), Initial Yield = $12\%$ ($6\%$ per period).

  • Initial Price: $774.00$.

  • Scenario after 4 years: Time left = $16$ years ($32$ periods), New Yield = $8\%$ ($4\%$ per period).

  • Price after 4 years: $1,089.37$

  • Total Price Change: $1,089.37 - 774 = 315.37$.

• Decomposition Steps:

  • To find the price change due only to time, calculate the bond's value after $4$ years using the original yield ($12\%$).

  • Price after $4$ years at $12\%$ yield: $788.74$.

  • Change due to Time: $788.74 - 774 = 14.74$.

  • Change due to Yield Support: $315.37 - 14.74 = 300.63$.

Pricing Bonds Between Coupon Dates

• Pricing bonds at a specific coupon date occurs rarely in practice. Most bonds trade between coupon dates, introducing two complications:

  1. Fractional Periods: Time to maturity is not a whole number (e.g., $4$ years, $2$ months, $5$ days).

  2. Accrued Interest: The next coupon payment must be shared between the buyer and the seller.

• Accrued Interest and the "Give to Caesar" Principle:

  • Accrued interest is interest earned but not yet collected. It is shared based on the number of days each party held the bond.

  • Example: If a bond's annual coupon is $R80$ and the seller held it for $100$ days, they are entitled to $\frac{100}{365} \times 80$.

• Clean and Dirty Prices:

  • Dirty Price: The total price the buyer pays, which includes the clean price plus any accrued interest.

  • Clean Price: The price of the bond excluding accrued interest.

• Cum-Coupon vs. Ex-Coupon:

  • Cumulative Coupon (Cum Coop): The next coupon payment goes to the buyer. The buyer must then compensate the seller for the portion the seller earned during their holding period.

  • Ex-Coupon (Ex Coop): The next coupon payment goes to the seller. The seller must then compensate the buyer for the buyer's portion of the coupon period.

Questions & Discussion

• Question: Does it make sense to pay a premium for a bond, such as paying $1,147.20$ for a $1,000$ face value?

• Response: Yes, because the return matters more than the nominal price. Even if a capital loss is incurred, the buyer receives a higher coupon ($8\%$ vs $6\%$ market rate) that offsets the loss, resulting in the desired yield of $6\%$. Evaluation based on price alone is insufficient for bonds; the return (yield) is the defining metric.

• Question on Manager Performance: Who did a better job: Manager A, whose portfolio increased by $R300$ due to time decay, or Manager B, whose portfolio increased by $R300$ due to anticipating a yield change?

• Response: Manager B. Anticipating yield changes requires significant skill and market insight, whereas time decay is an automatic function of holding a discount bond. It is critical to decompose performance to distinguish between "real meat" (skill-based gains) and passive gains.