BUS424 (Ch 3) 1

Yield, Total Return, and Reinvestment Risk

1. Internal rate of return

2. Yields to maturity

3. Other Yields

Current yield

Cash flow yield

Yield for a portfolio

4. Decompose total dollar returns

5. Reinvestment risk

6. Compute total return for a bond

BUS424 (Ch 3) 2

The yield on an investment can be found as:

Internal Rate of Return

nn

y

CF

y

CF

y

CF

y

CFP

)1(...

)1()1(1 33

221

The yield calculated from this relationship is the internal rate of return. the return to make the above equation hold.

What is the actual meaning of y:

(1)

(2)

Example: see page 36.

BUS424 (Ch 3) 3

Yield to Maturity of A Bond

The yield on an investment can be found as:

ny

MC

y

C

y

C

y

CP

)1(...

)1()1(1 32

What is meant by y:

BUS424 (Ch 3) 4

The coupons of corporate bonds are typically semiannually paid, doubling the periodic interest rate or discount rate (y) gives the annual yield to maturity.

But this is simple annual interest rate.

What does YTM really mean?

BUS424 (Ch 3) 5

Annualizing YieldsAlso known as effective annual yield (EAY)

EAY=(1+periodic interest rate)m-1

Examples:(1) The periodic interest rate is 4%, interest semi-annually paid, what is EAY?(2) The periodic interest rate is 2%, interest quarterly paid, what is EAY?(3) If APR is 8%, interest semi-annually paid, what is EAY?(4) If APR is 8%, interest quarterly paid, what is EAY?

BUS424 (Ch 3) 6

Current Yield

Current yield = annual dollar coupon interest/price

Only consider coupon payment

BUS424 (Ch 3) 7

Yield to Call

•Call price: the price at which the bond may be called

•Yield to call: –Fixed call price–Call schedule

ny

MC

y

C

y

C

y

CP

)1(

*...

)1()1(1 32

BUS424 (Ch 3) 8

Assumptions for Yield to Call

1. Cash flows can be reinvested at the yield to call until the assumed call date

2. The investor will hold the bond to the assumed call date

3. The issuer will call the bond on that date

Also note: callable bonds could be traded above the call price

BUS424 (Ch 3) 9

Bond price and call

This is a deviation from the yield to call, while it helps to understand the pricing of callable bonds

Example on page 375: Consider a callable bond with a 10-year 13% coupon rate that is callable in 1 year at a call price of 104. Suppose the yields on 10-year and 1-year bonds are 6% and 5%. Investors expect the bond would be called in 1 year. What is the price of the bond? Assume the bond pays coupons semiannually.

BUS424 (Ch 3) 10

Cash Flow YieldFor amortizing securities

Cash flow in each period includes

- coupon interest

- scheduled principal repayment

- unscheduled prepayments

Calculated as internal rate of return for a cash flow

BUS424 (Ch 3) 11

Yield for a Portfolio

There are multi-bonds, their yields to maturity are known or are calculable.

What is the yield to maturity of the portfolio?

Example on page 43.

BUS424 (Ch 3) 12

Yield Spread for Floating Rate Securities

Floating rate = reference rate + quoted margin

Calculate the discount margin by assuming the reference is constant over the life of a bond.

Page 44-45

BUS424 (Ch 3) 13

What do we know here?

Maturity: 6 years

Coupon rate: reference rate + 80bp

Initial reference rate: 10%

Price of the bond: 99.3098

Question: what is the yield spread of the bond?

yield spread = yield – reference rate

BUS424 (Ch 3) 14

Total Returns

1. If the bond is sold before maturity

2. If the bond is sold at maturity

BUS424 (Ch 3) 15

1. If Bonds are Sold Before Maturity

3-year investment horizon

purchasing a 20-year 8% coupon bond for $828.40. YTM is 10%

Reinvestment rate = 6%

YTM at the end of investment horizon is 7%

Total return for this bond?

BUS424 (Ch 3) 16

Step 1

Compute the total coupon payment + interest on interest

BUS424 (Ch 3) 17

Steps 2 and 3

Determine the resale value of the bond

Adding the amounts in steps (1) and (2)

BUS424 (Ch 3) 18

Steps 4 and 5

Obtain semi-annual total return

Calculate annual return

BUS424 (Ch 3) 19

Reinvestment Risk

Future reinvestment rates will be less than the YTM at the time the bond is purchased.

—Coupon payments—Length of maturity of the bond is shorter

than the holding period

Is there any reinvestment risk associated with zero-coupon bonds?

BUS424 (Ch 3) 20

A Simple Example – Bond Held until Maturity

15-year 7% bond, semi-annually paid, par $1000, price $769.40. YTM = 10%. The bond is held until maturity, reinvestment return = YTM=10%. Answer the following questions.

(1) What is the total dollar return of the bond

(2) Evaluate the return from each source

(3) What is the reinvestment return?

(4) What is the reinvestment risk?

BUS424 (Ch 3) 21

Decomposing Dollar Returns

• Assumption: reinvestment return=YTM• Three sources

–Periodic coupon interest payments made by the issuer-- nC

–Capital gain -- M-P

–Interest on interest – (reinvestment income, subject to reinvestment risk)

-- C[((1+r)n-1)/r]-nC

BUS424 (Ch 3) 22

Additions

Multiple reinvestment rate: see the example on page 53.

What is the implication of total return in practice? – horizon return/horizon analysis

How to reconcile the analysis we have done so far with the simple formula:

ret=(Ps-Pb+C)/Pb?

BUS424 (Ch 3) 23

Exercise