Chapter 3

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 • 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%

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 $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.

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 = 0.1449 (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: • 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 $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.

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 = • 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!

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 $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.

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)