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:

2 3(1 ) (1 ) (1 ) (1 ) (1 )N N

C C C C MP

y y y y y

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%

Years from Now

Cash Flows

1 $100

2 $100

3 $100

4 $1,100

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:

Periods from Now

Cash Flows

1 $50

2 $50

3 $50

4 $50

5 $50

6 $50

7 $50

8 $1,050

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?

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 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:

Annual dollar coupon interest

PriceCY

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:

$700.0910

$769.49CY

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 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:

2 3(1 ) (1 ) (1 ) (1 ) (1 )N N

C C C C MP

y y y y y

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

1. Coupon return: Return from coupon payments (current yield).2. Capital gain return: Capital gain/loss when bond matures, is sold or is

called.3. Reinvestment return: Interest income generated from the reinvestment of

coupons (also called interest-on-interest).

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:

2 3

*

(1 ) (1 ) (1 ) (1 ) (1 )N N

C C C C MP

y y y y y

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?

Annual Interest Semiannual Rate Present Value of 6 PV of $103 PV of CFsRate (%) y (%) Payments of $3.5 6 Periods from Now

5 2.5 $19.28 $88.82 108.105.2 2.6 19.21 88.30 107.515.4 2.7 19.15 87.78 106.935.6 2.8 19.09 87.27 106.36

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.

2 3

*

(1 ) (1 ) (1 ) (1 ) (1 )N N

C C C C MP

y y y y y

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:

1 2 1 2min( , , ,..., , , ,..., )j kYTW YTM YTC YTC YTC YTP YTP YTP

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:

31 22 3(1 ) (1 ) (1 ) (1 )

NN

Projected CF Projected CFProjected CF Projected CFP

y y y y

Yield for a Bond Portfolio

not simply weighted average of YTMs for all bonds in portfolio

Bond Coupon Rate (%) Maturity (years) Par Value Price YTM (%)A 7 5 $10,000,000 $9,209,000 9B 10.5 7 20,000,000 20,000,000 10.5C 6 3 30,000,000 28,050,000 8.5

Period CF Received Bond A Bond B Bond C Portfolio 1 $350,000 $1,050,000 $900,000 $2,300,0002 350,000 1,050,000 900,000 2,300,0003 350,000 1,050,000 900,000 2,300,0004 350,000 1,050,000 900,000 2,300,0005 350,000 1,050,000 900,000 2,300,0006 350,000 1,050,000 30,900,000 32,300,0007 350,000 1,050,000 — 1,400,0008 350,000 1,050,000 — 1,400,0009 350,000 1,050,000 — 1,400,000

10 10,350,000 1,050,000 — 11,400,00011 — 1,050,000 — 1,050,00012 — 1,050,000 — 1,050,00013 — 1,050,000 — 1,050,00014 — 21,050,000 — 21,050,000

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 Margin1. Determine the cash flows assuming the reference rate

does not change over the life of the security.

2. Select a margin (spread).

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

4. 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.

Discount Margin - exampleExhibit 3-1. Calculation of the Discount Margin for a Floating-Rate Security Floating-rate security: Maturity: six years Coupon rate: reference rate + 80 basis points Reset every six months

PV of CF at Assumed Annual Margin (bp)Reference Cash

Period Rate Flow 80 84 88 96 1001 10% 5.4 5.1233 5.1224 5.1214 5.1195 5.11852 10 5.4 4.8609 4.859 4.8572 4.8535 4.85163 10 5.4 4.6118 4.6092 4.6066 4.6013 4.59874 10 5.4 4.3755 4.3722 4.3689 4.3623 4.3595 10 5.4 4.1514 4.1474 4.1435 4.1356 4.13176 10 5.4 3.9387 3.9342 3.9297 3.9208 3.91637 10 5.4 3.7369 3.7319 3.727 3.7171 3.71228 10 5.4 3.5454 3.5401 3.5347 3.524 3.51869 10 5.4 3.3638 3.358 3.3523 3.3409 3.3352

10 10 5.4 3.1914 3.1854 3.1794 3.1673 3.161311 10 5.4 3.0279 3.0216 3.0153 3.0028 2.996512 10 105.4 56.0729 55.9454 55.8182 55.5647 55.4385

Present Value = 100 99.8269 99.6541 99.3098 99.1381

Important Comments on Yield

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

1. Coupon Income: Income from coupon payments.2. Capital Gain Income: Capital gain (or loss) when bond matures, is sold or is

called.3. 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:1. The bond is held to maturity.2. The coupons are reinvested at the YTM.

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

Determining Reinvestment Income

Coupon interest + interest-on-interest is calculated as:(1 ) 1ny

Cy

Coupon interest is calculated as nC. Therefore, interest-on-interest is calculated as:

(1 ) 1nyC nC

y

Interest-on-interest can be substantial.

Example

301.05 1$35 30($35)

0.05

$1,275.36

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)

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

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 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

1 1n

r C

r

1 1n

r C nC

r

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:

Bond Coupon (%)

Maturity (Yrs)

YTM (%)

A 5 3 9.0

B 6 20 8.6

C 11 15 9.2

D 8 5 8.0

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:1/

total future dollars1

purchase price of the bond

h

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 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 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)