1. Chapter 3
Measuring Yield

2. 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.

3. Example What is the annual yield? Answer: 12%
Suppose a financial instrument is priced at $ and it has the following known annual cash flows:
Years from Now
Cash Flows 1
$100 2 3 4
$1,100
What is the annual yield?
Answer: 12%

4. 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 $ with the following semiannual cash flows:
Periods from Now
Cash Flows 1
$50 2 3 4 5 6 7 8
$1,050
What is the yield of this instrument?
After trial-and-error process we get 7%:
However, this is a semiannual yield.

5. 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 = (or 14.49%).
Even though the BEY understates the yield earned by investors, it is the convention used on Wall Street.

6. 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

7. Current Yield Current Yield: Example: The current yield ignores:
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.

8. 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).

9. 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

10. 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?

11. 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

12. 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:

13. 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:

14. Yield for a Bond Portfolio
not simply weighted average of YTMs for all bonds in portfolio

15. 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.

16. 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.

17. Discount Margin - example

18. 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.

19. 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.

20. Example Suppose we have:
A 15-year 7% coupon bond. The par value is $1,000 and the price is $ 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 = $ (= $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.

21. 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.

22. 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.

23. 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

24. Dollar Return coupon interest + interest on interest =
example
Total Dollar Return

25. 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!
Bond
Coupon (%)
Maturity (Yrs)
YTM (%) A 5 3
9.0 B 6 20
8.6 C 11 15
9.2 D 8
8.0

26. 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.

27. Example on Total Return
An investor has a 3-year horizon and is considering a 20-year 8% coupon bond for $
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.

28. 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.

29. 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)