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Chapter 3. Measuring Yield. Introduction. The yield on any investment is the rate that equates the PV of the investment’s cash flows to its price:. This yield is also called the internal rate of return . The yield is found through a trial-and-error process. Example .

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Chapter 3

Chapter 3

Measuring Yield


  • The yield on any investment is the rate that equates the PV of the investment’s cash flows to its price:

  • This yield is also called the internal rate of return.

  • The yield is found through a trial-and-error process.


  • Suppose a financial instrument is priced at $939.25 and it has the following known annual cash flows:

  • What is the annual yield?

  • Answer: 12%

Be aware
Be Aware….

  • The yield you get is commensurate with the spacing of the cash flows.

  • For example, suppose we have a four year instrument priced at $880.57 with the following semiannual cash flows:

  • What is the yield of this instrument?

  • After trial-and-error process we get 7%:

    • However, this is a semiannual yield.

How do we annualize yields
How Do We Annualize Yields?

  • We can annualize the 7% yield two ways:

  • (1) Multiply by 2: 7  2 = 14%

    • Called the bond equivalent yield (BEY).

    • The BEY is a simple interest rate (i.e., ignores compounding) and thus understates the true yield earned by investors.

  • (2) A better way: the effective annual yield (EAY):

    • EAY = (1 + periodic interest rate)m – 1

    • EAY = (1.07)2 – 1 = 0.1449 (or 14.49%).

  • Even though the BEY understates the yield earned by investors, it is the convention used on Wall Street.

  • Conventional yield measures
    Conventional Yield Measures

    • There are several bond yield measures used by portfolio managers:

      • Current yield

      • Yield-to-maturity (discussed already)

      • Yield-to-call

      • Yield-to-put

      • Yield-to-worst

      • Cash flow yield

    Current yield
    Current Yield

    • Current Yield:

    • Example:

      • What is the current yield for a 15-year 7% coupon annual pay bond with a par value of $1,000 selling for $769.49:

    • The current yield ignores:

      • The positive return from buying a discount bond and holding to maturity.

      • The negative return from buying a premium bond and holding to maturity.

  • The yield-to-maturity does not ignore these sources of return.

  • Yield to maturity
    Yield To Maturity

    • YTM is the yield that equates the PV of the bond’s future CFs to the bond’s price.

    • We briefly discussed it at the beginning of the chapter:

    • As we will see later YTM measures three sources of a bond’s return:

      • Coupon return: Return from coupon payments (current yield).

      • Capital gain return: Capital gain/loss when bond matures, is sold or is called.

      • Reinvestment return: Interest income generated from the reinvestment of coupons (also called interest-on-interest).

    Yield to call
    Yield to Call

    • With some bonds, the issuer may be entitled to call a bond prior to the stated maturity date.

      • This alters the maturity of the bond and the number of cash flows.

  • Call price:

    • For some issues the call price is the same as the par value. For others, the call price can be different from the par value and depend on a call schedule.

  • Common practice is to calculate both YTC and YTM.

    • YTC assumes issuer will call the bond at some assumed call date and call price.

  • Typically investors calculate

    • Yield to first call, yield to next call, yield to first par call, yield to refunding

  • Yield-to-call:

  • M* is the call price

    Yield to call example
    Yield to Call - example

    • 8 year 7% coupon bond with maturity value of $100 selling for $106.93

    • first call date is end of year 3

    • call price of $103

    • What’s the yield to call?

    Yield to put

    • Some bonds give the bondholders the right to sell the bond issue back at a specific price.

    • Just as there is a call schedule with a callable bond, there is a put schedule with a puttable bond.

    • YTP is calculated exactly like YTC except with the put price instead of the call price.

    M* is the put price

    Yield to worst

    • A practice in industry is to calculate the YTM, YTC, and YTP for every possible call date and put date.

      • The minimum of all of these yields is called yield-to-worst.

      • Gives investors a measure of the worst possible outcome from holding the bond.

  • Yield-to-Worst:

  • Cash flow yield
    Cash Flow Yield

    • For amortizing securities the cash flow each period consists of three components:

      • Coupon interest.

      • Scheduled principal repayment (according to an amortization schedule).

      • Prepayments – borrowers in the underlying securities can pay more principal than is specified in the amortization schedule. This excess amount is called prepayment.

  • For amortizing securities, calculate a cash flow yield:

    • The rate that equates the PV of projected cash flows with the price.

    • The difficulty is projecting the cash flows.

  • Cash flow yield:

  • Yield for a Bond Portfolio

    not simply weighted average of YTMs for all bonds in portfolio

    Yield spread measures for floaters
    Yield Spread Measures for Floaters

    • The coupon for floating rate securities changes periodically based on the coupon reset formula.

    • Since the future floating rate cannot be known we can’t determine a floater’s cash flows or YTM.

    • Instead, there are several measures used as spread or margin measures.

      • The most popular of these measures is the discount margin.

      • discount margin estimates the average margin over the reference rate

  • Drawbacks of the discount margin method:

    • It assumes the reference rate doesn’t change over time.

    • It ignores caps and floors that may be in place.

  • How to calculate discount margin
    How To Calculate Discount Margin

    • Determine the cash flows assuming the reference rate does not change over the life of the security.

    • Select a margin (spread).

    • Discount CFs in step 1 by reference rate + margin selected in step 2.

    • Compare PV of CFs in step 3 with the price. If the PV is equal to security’s price, then the discount margin is the margin assumed in step 2. If PV is not equal to price, try a different margin.

    Important comments on yield
    Important Comments on Yield

    • The dollar return of a bond potentially comes from three sources:

      • Coupon Income: Income from coupon payments.

      • Capital Gain Income: Capital gain (or loss) when bond matures, is sold or is called.

      • Reinvestment Income: Interest income generated from the reinvestment of coupons (also called interest-on-interest).

  • A measure of a bond’s yield should consider all three sources of a bond’s dollar return.

    • The current yield deals only with the first source.

    • The YTM deals with all three sources of return.

  • However, YTM will be the actual (or promised) yield only if:

    • The bond is held to maturity.

    • The coupons are reinvested at the YTM.

  • If not, the actual yield may be more or less than the YTM.

  • Determining reinvestment income
    Determining Reinvestment Income

    • Coupon interest + interest-on-interest is calculated as:

    • Coupon interest is calculated as nC.

    • Therefore, interest-on-interest is calculated as:

    • Interest-on-interest can be substantial.


    • Suppose we have:

      • A 15-year 7% coupon bond. The par value is $1,000 and the price is $769.40 with a YTM of 10%. What is the reinvestment interest?

    • How much of total return is the reinvestment return?

      • Total coupon interest = $1,050 (= $3530)

      • Interest-on interest = $1,275.36

      • Capital gain = $230.60 (= $1,000 - $769,40)

  • Total = $2,555.96:

    • Reinvestment return is 50% of the bond’s total return (it’s important!)

  • What if coupons can’t be reinvested at the YTM?

    • The risk that the reinvestment rate will be less than YTM is called reinvestment risk.

  • Determinants of reinvestment risk
    Determinants of Reinvestment Risk

    • Two characteristics of a bond determine the importance of the interest-on-interest component and thus its reinvestment risk:

    • Maturity:

      • For a given YTM and coupon rate, the longer the maturity of the bond the more dependent the bond’s total dollar return is on interest-on-interest return (i.e., more reinvestment risk).

      • For long-term bonds, interest-on-interest may be as much as 80% of a bond’s potential dollar return.

      • YTM may tell us little about the actual return of a long-term bond if the bond is held to maturity.

  • Coupon Rate:

    • For a given YTM and maturity, the higher the coupon rate of the bond the more dependent the bond’s total dollar return is on interest-on-interest return (i.e., more reinvestment risk).

    • Holding maturity and YTM constant, premium bonds have more reinvestment rate risk than discount bonds.

    • Note: Zero-coupon bonds have no reinvestment risk if held until maturity.

  • Cash flow yield1
    Cash Flow Yield

    • So far we have assumed reinvestment risk on non-amortizing bonds.

    • For amortizing securities, reinvestment risk is even greater. Why?

      • The investor must reinvest periodic principal repayments in addition to the periodic coupon payments.

      • Also, the cash flows are usually monthly, not semiannually so the cash is invested longer and more frequently.

    Sources of bond return
    Sources of Bond Return

    • coupon payments

    • capital gain/loss on sale of bond (or when called)

    • reinvestment of coupon payments – interest on interest

    • yields

      • current

      • YTM

      • CF Yield

    Dollar return
    Dollar Return

    • coupon interest + interest on interest =

    • interest on interest =

    • example

    • Total Dollar Return

    Total return on a bond
    Total Return On A Bond

    • YTM only equals the promised yield when:

      • A bond is held to maturity, and

      • Coupons can be reinvested at the YTM.

  • YTM can be problematic when finding the best bond to invest in.

  • Example: Suppose an investor with a 5-year horizon is considering the following bonds:

    • Which bond is best?

    • Difficult to tell:

      • Bond C has highest YTM, but it has 15-years until maturity (won’t know it’s value in 5 years) and a high coupon rate.

      • Bond A has a high YTM, but 3-year horizon….reinvestment risk!

      • YTM does not answer the question for us!

    Computing the total return for a bond
    Computing the Total Return for a Bond

    • Procedure:

      • 1. Compute the total coupon payments plus interest-on-interest assuming a given reinvestment rate (not YTM)

      • 2. Determine projected sale price at end of investment horizon (equal to the PV of the remaining CFs when the bond is sold, discounted at the projected YTM at that time).

      • 3. Add the above two amounts. This is the total future dollars received from the investment, given the assumptions and projections.

      • 4. Obtain the semiannual total return:

    • 5. Double the amount found above. This is the bond’s total return.

    Example on total return
    Example on Total Return

    • An investor has a 3-year horizon and is considering a 20-year 8% coupon bond for $828.40.

    • The YTM of the bond is 10% and the investor expects to be able to reinvest coupon payments at an annual interest rate of 6%.

    • At the end of the investment horizon (at which time the bond will have a 17 year maturity), the investor expects YTM to be 7%.

    • Find the total return on the bond.

    Comments on total return
    Comments on Total Return

    • When a portfolio manager evaluates bonds based on total return, it is referred to as horizon analysis.

    • When a total return is calculated over an investment horizon, it is referred to as a horizon return.

      • Horizon return and total return are used interchangeably.

  • Drawback of horizon analysis:

    • Requires the analyst’s assumptions regarding (1) reinvestment rates, (2) future yields, and (3) future investment horizon.

  • However, the horizon analysis framework is amenable to scenario analysis:

    • The portfolio manager can run many scenarios and see how sensitive the bond’s performance will be to each scenario for reinvestment rates and future market yields.

  • Communicating yield changes
    Communicating Yield Changes

    • There are two ways to calculate and communicate yield changes:

      • Absolute yield change (in basis points, or bps)

      • Percentage yield change.

  • Example:

    • Month 1: 4.45%

    • Month 2: 5.11%

  • How much did the yields change from month one to two?

    • Absolute = |5.11 – 4.45|  100 = 66 bps

    • Percentage = 100 x ln(5.11/4.45) = 13.83% (Note: the “ln” computes a continuously compounded annual return)

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