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# Chapter 3 - PowerPoint PPT Presentation

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

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?

• 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.

• 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.

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

• Current yield

• Yield-to-call

• Yield-to-put

• Yield-to-worst

• Cash flow 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.

• 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).

• 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

• 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?

• 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

• 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:

• 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:

• not simply weighted average of YTMs for all bonds in portfolio

• 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.

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

• 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.

• 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.

• 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.

• 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.

• 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.

• coupon payments

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

• reinvestment of coupon payments – interest on interest

• yields

• current

• YTM

• CF Yield

• coupon interest + interest on interest =

• interest on interest =

• example

• Total Dollar Return

• 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!

• 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.

• 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.

• 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.

• 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)