Bond Pricing and Yield

Bond Pricing and Yield

  • Bond Pricing Formula:

    • The price of a bond reflects the present value of its expected future cash flows, which include the coupon payments and the par value at maturity.
    • Formula:
      P=extCouponPaymentimes1−(1+extYTM)−TextYTM+extParValue(1+extYTM)TP = ext{Coupon Payment} imes \frac{1 - (1 + ext{YTM})^{-T}}{ ext{YTM}} + \frac{ ext{Par Value}}{(1 + ext{YTM})^T}
    • Here:
    • P = price of the bond
    • T = total number of periods until maturity

  • Capital Gains Rate:

    • This measures the rate of change in bond prices over time, influenced by the bond's approaching maturity and shifts in yield requirements.
    • Formula:
      extCapitalGainsRate=extPriceinFuture−extPriceTodayextPriceTodayext{Capital Gains Rate} = \frac{ ext{Price in Future} - ext{Price Today}}{ ext{Price Today}}

  • Current Yield:

    • Focuses exclusively on the income generated by the bond relative to its current price, excluding capital gains.
    • Formula:
      extCurrentYield=extAnnualCouponPaymentextCurrentMarketPriceext{Current Yield} = \frac{ ext{Annual Coupon Payment}}{ ext{Current Market Price}}

Key Concepts

  • Interest Rates and Bond Prices:

    • The relationship is inverse. When interest rates rise, bond prices decrease and vice versa.

  • Premium vs. Discount Bonds:

    • Premium Bond: Market price > par value (coupon rate > required return).
    • Discount Bond: Market price < par value (coupon rate < required return).

  • Capital Gains / Losses:

    • If a bond is selling at a premium and it approaches par value, its price decreases, leading to a capital loss.

  • Federal Reserve Impact:

    • Purchases of securities by the Fed raise bond prices and reduce interest rates due to increased demand.

Problem Highlights

  1. Problem 1: Find the price of a bond with:

    • Coupon Rate = 10%, Par Value = $1,000, YTM = 12.5%, and 10 years to maturity.
    • Correct Price: $861.59

  2. Problem 2: Calculate the capital gains rate if the bond sells for $861.59 today and expected to change as it matures.

    • Correct Rate: 0.894%

  3. Problem 4: Calculate the yield to maturity (YTM) for a bond with a price of $838.79 and 7% coupon rate.

    • Correct YTM: 11%

  4. Problem 6: Confirm bond price moves inversely to interest rates.

    • Answer: True

  5. Problem 14: Examine how Fed purchases of securities affect interest rates.

    • Answer: Prices of securities rise, and interest rates decline.

Detailed Bond Problems

Problem 1: Bond Price Calculation

  • Given Data:

    • Coupon Rate = 10%
    • Par Value = $1,000
    • Remaining Time = 10 years
    • Required Return (YTM) = 12.5%

  • Steps:

    1. Calculate Annual Coupon Payment:
      extCouponPayment=extCouponRateimesextParValue=0.10imes1000=100ext{Coupon Payment} = ext{Coupon Rate} imes ext{Par Value} = 0.10 imes 1000 = 100
    2. Discount future cash flows (coupons) using:
      PextCoupons=extCouponPaymentimes1−(1+extYTM)−TextYTMP_{ ext{Coupons}} = ext{Coupon Payment} imes \frac{1 - (1 + ext{YTM})^{-T}}{ ext{YTM}}
    3. Discount Par Value:
      PextPar=extParValue(1+extYTM)TP_{ ext{Par}} = \frac{ ext{Par Value}}{(1 + ext{YTM})^{T}}
    4. Total Bond Price:
      P=P<em>extCoupons+P</em>extPar=861.59P = P<em>{ ext{Coupons}} + P</em>{ ext{Par}} = 861.59
Problem 2: Capital Gains Rate Calculation

  • Given Data:

    • Price Today = $861.59
    • Price Next Year = Calculated with 9 years remaining.

  • Steps:

    1. Recalculate Bond Price for T = 9.
    2. Calculate Capital Gains Rate:
      extCapitalGainsRate=extPriceinFuture−extPriceTodayextPriceTodayimes100ext{Capital Gains Rate} = \frac{ ext{Price in Future} - ext{Price Today}}{ ext{Price Today}} imes 100
Problem 4: Yield to Maturity Calculation

  • Given Data:

    • Current Price = $838.79
    • Coupon Rate = 7%
    • Par Value = $1,000
    • Time to Maturity = 15 years.

  • Steps:

    1. Use YTM formula:
      P=extCouponPaymentimes1−(1+r)−Tr+extParValue(1+rT)P = ext{Coupon Payment} imes \frac{1 - (1 + r)^{-T}}{r} + \frac{ ext{Par Value}}{(1 + r}^{T})
    2. Utilize trial and error or financial calculators to solve for r, yielding YTM of 11%.

Terminology

  1. Yield to Maturity (YTM):

    • The overall rate of return anticipated on a bond if held until it matures.

  2. Current Yield:

    • The ratio of a bond's annual coupon payment to its current market price.
    • Formula:
      extCurrentYield=extCouponPaymentextMarketPriceext{Current Yield} = \frac{ ext{Coupon Payment}}{ ext{Market Price}}

  3. Premium vs. Discount Bonds:

    • Premium bonds sell for more than par value; discount bonds sell for less.

Additional Notes

  • Master the formulas for YTM, Current Yield, and Capital Gain Yield.
  • Understand the implications of bond pricing with respect to interest rates and yields.