3_bond_yields

Bond Yields

Computing Yield

• Yield on any investment is the interest rate that equates the present value of cash flows to the investment price.

• Yield equation:

  • P = CF1 / (1 + y) + CF2 / (1 + y)^2 + ... + CFN / (1 + y)^N

    • Where:

      • P = price of the investment

      • CF = cash flow in period t (t = 1,2,3,...,N)

      • y = yield

      • N = number of periods

Determining Yield Periodicity

• The calculated yield reflects the period of cash flows:

  • Semiannual cash flows yield is semiannual.

  • Monthly cash flows yield is monthly.

• The simple annual interest rate multiplies the yield for the period by the number of periods in a year.

• The effective annual interest rate incorporates effects of compounding.

Effective Annual Yield

• Effective annual yield formula:

  • effective annual yield = (1 + periodic interest rate)^m − 1

    • Where:

      • m = payment frequency per year

• Example: For quarterly payments with a periodic interest rate of 0.08/4 = 0.02:

  • effective annual yield = (1 + 0.02)^4 − 1 = 8.24%

Periodic Interest Rate

• The periodic interest rate for a given annual interest rate can be calculated using:

  • periodic interest rate = (1 + effective annual yield)^(1/m) − 1

• Example: If the effective annual interest rate = 12%, for quarterly payments:

  • periodic interest rate = (1 + 0.12)^(1/4) − 1 = 2.87%

Investment Yield Example

• Investment Case:

  • Price: $62,321.30

  • Cash flow: $100,000 in 6 years

• Yield calculation:

  • P = CF / (1 + y)^n

  • y = (100,000 / 62,321.30)^(1/6) − 1 = 8.2%

• Simple and effective annual yields are to be determined.

Yield Measures

Current Yield

• Relates the annual coupon interest to market price:

  • formula:

    • current yield = annual dollar coupon interest / price

Example of Current Yield

• Current yield for a 15-year 7% coupon bond (par value $1,000, market price $769.42):

  • current yield = $70 / $769.42 = 9.1%

Yield to Maturity (YTM)

• YTM is the annualized interest rate making the PV of cash flows equal to price (assuming bond held to maturity).

• Semiannual bond YTM calculation:

  • P = C / (1 + y) + C / (1 + y)^2 + ... + C / (1 + y)^n + M / (1 + y)^n

  • For semiannual bonds, double the periodic y for YTM.

Zero-Coupon Yields

• For a zero-coupon bond:

  • y = (M / P)^(1/n) − 1

  • Where:

    • M = maturity value

    • P = bond's price

    • n = number of periods

Example of Zero-Coupon Yield

• For a 10-year zero-coupon bond, maturity value $1,000, price $439.18:

  • y = (1,000 / 439.18)^(1/20) − 1 = 4.2%

  • To get YTM, double this yield: 8.4%.

Yield to Call

• The call price is the price when a bond may be called.

• YTC assumes the issuer calls the bond at an assumed date and specific call value:

  • P = C / (1 + y) + .... + C / (1 + y)^n* + M* / (1 + y)^n*

    • For semiannual bonds, double the periodic yield (y).

Yield to Sinker and Yield to Put

• Yield to sinker: Based on a sinking fund provision requiring some bond retirement at scheduled dates (Same calculations as YTM).

• Yield to put: Allows bondholders to sell back to issuer at specified prices on put dates. Similar calculations to yield to call apply, but with put value and periods until put.

Yield Changes

Absolute Yield Change

• Absolute yield change measured in basis points:

  • absolute yield change = |initial yield − new yield| × 100

Percentage Yield Change

• Percentage change calculated as:

  • percentage change yield = 100 × ln(new yield / initial yield)

Example of Yield Changes

• Yield changes observed for three months:

  • Month 1: 4.45%, Month 2: 5.11%, Month 3: 4.82%

Potential Sources of Bond Dollar Return

1. Periodic coupon interest payments.

2. Capital gain/loss when sold/matured.

3. Interest income from reinvesting periodic cash flows.

Interest on Interest Component

• substantial percentage of bond potential return.

• Calculation:

  • coupon interest + interest on interest = C * [(1 + r)^n - 1] / r

    • Where:

      • C = coupon interest

      • r = semiannual reinvestment rate

      • n = number of periods.

Total Dollar Return Calculation

• Total dollar return can include:

  1. Total coupon interest (nC).

  2. Interest on interest calculated as:

     • interest on interest = (C * [(1 + r)^n - 1] / r) - nC

Example Calculation of Total Dollar Return

• Given values: coupon interest = $50, reinvestment rate = 4.5%, and periods = 40.

• Coupon interest + interest on interest = $50 * [(1.045^40 - 1) / 0.045] = $5,351.52.

Total Return Concept

• Yield to maturity is a promised yield if:

  1. Bond held to maturity.

  2. All coupon payments reinvested at YTM.

• Reinvestment risk can affect actual returns. Total return measures yield with an assumption of the reinvestment rate.

Total Return Calculation Steps

1. Compute total future dollars from investment considering a specific reinvestment rate.

2. Determine total return as the interest rate growing initial investment to total future dollars.

Example of Total Return Calculation

• Investment analyzed: 20-year bond purchased at $828.40, with an expected reinvestment rate of 6% and projected selling yields at 7%.

Coupon Payment Example

• Total coupon payments during 3 years = $40 every six months for 6 periods:

  • Total coupon interest plus interest-on-interest = $258.736

Projected Sale Price Calculation

• Present value of cash flows computed with given yield; price estimate obtained:

  • Selling price = $1,098.503

Final Total Return Calculation

• Total future dollars = total coupon interest + projected sale price:

  • $258.736 + $1,098.503 = $1,357.239

Semiannual Total Return Calculation

• Semiannual total return = (total future dollars / purchase price of bond)^(1/h) - 1

  • Example yield = 8.577%, doubling for annual total return = 17.15%.

Zero-Coupon Yields

For a zero-coupon bond, the yield can be calculated using the formula:

y = (M / P)^(1/n) − 1 Where:

• M = maturity value

• P = bond's price

• n = number of periods

Example of Zero-Coupon YieldFor a 10-year zero-coupon bond, maturity value $1,000, and price $439.18:

y = (1,000 / 439.18)^(1/20) − 1 = 4.2%.To obtain Yield to Maturity (YTM), double this yield: 8.4%.

Yield to Maturity (YTM)

YTM is the annualized interest rate making the present value of cash flows equal to the price of the bond (assuming the bond is held to maturity).

Semiannual bond YTM calculation: P = C / (1 + y) + C / (1 + y)^2 + ... + C / (1 + y)^n + M / (1 + y)^n For semiannual bonds, double the periodic yield (y) to find YTM.

Example of Yield to Maturity Calculation

Consider a 15-year 7% coupon bond with a par value of $1,000 selling for $769.42.

1. Coupon Payment (C):

   • Annual coupon payment = 7% of $1,000 = $70

   • Semiannual coupon payment = $70 / 2 = $35

2. Total number of periods (n):

   • 15 years = 15 * 2 = 30 periods

3. Price of the bond (P):

   • Market price = $769.42

Using YTM formula (approximately):

• This requires iterative methods or financial calculators, but the yield can be estimated using a YTM approximation. A rough formula can be:

YTM ≈ [C + (M - P) / n] / [(P + M) / 2] Where:

• C is the annual coupon payment.

• M is the maturity value (par value).

Plugging in values: YTM ≈ [70 + (1,000 - 769.42) / 15] / [(769.42 + 1,000) / 2] YTM ≈ [70 + (230.58 / 15)] / [884.21] YTM ≈ [70 + 15.04] / 884.21 YTM ≈ 85.04 / 884.21YTM ≈ 0.096 or 9.6% approximately.