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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

- 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….

- 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? (2) A better way: the effective annual yield (EAY): Even though the BEY understates the yield earned by investors, it is the convention used on Wall Street.

- 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.

- EAY = (1 + periodic interest rate)m – 1
- EAY = (1.07)2 – 1 = 0.1449 (or 14.49%).

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

- 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.

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 Call price: Common practice is to calculate both YTC and YTM. Typically investors calculate 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.

- 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.

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

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

M* is the call price

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 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.

Cash Flow Yield For amortizing securities, calculate a 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.

- The rate that equates the PV of projected cash flows with the price.
- The difficulty is projecting the cash flows.

not simply weighted average of YTMs for all bonds in portfolio

Yield Spread Measures for Floaters Drawbacks of the discount margin method:

- 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

- 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

- 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 A measure of a bond’s yield should consider all three sources of a bond’s dollar return. However, YTM will be the actual (or promised) yield only if: If not, the actual yield may be more or less than the YTM.

- 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).

- The current yield deals only with the first source.
- The YTM deals with all three sources of return.

- The bond is held to maturity.
- The coupons are reinvested at the YTM.

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.

Example Total = $2,555.96: What if coupons can’t be reinvested at the YTM?

- 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 (= $3530)
- Interest-on interest = $1,275.36
- Capital gain = $230.60 (= $1,000 - $769,40)

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

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

Determinants of Reinvestment Risk Coupon Rate:

- 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.

- 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 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

- 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

- coupon interest + interest on interest =
- interest on interest =
- example
- Total Dollar Return

Total Return On A Bond 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:

- YTM only equals the promised yield when:
- A bond is held to maturity, and
- Coupons can be reinvested at the YTM.

- 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

- 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

- 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 Drawback of horizon analysis: However, the horizon analysis framework is amenable to scenario analysis:

- 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.

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

- 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 Example: How much did the yields change from month one to two?

- There are two ways to calculate and communicate yield changes:
- Absolute yield change (in basis points, or bps)
- Percentage yield change.

- Month 1: 4.45%
- Month 2: 5.11%

- 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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