Copyright © 2009 Pearson Prentice Hall. All rights reserved. Chapter 5 Time-Varying Rates of Return and the Yield Curve

Copyright © 2009 Pearson Prentice Hall. All rights reserved.

Chapter 5Time-Varying Rates of Return and the Yield Curve

Copyright © 2009 Pearson Prentice Hall. All rights reserved.5-2

Chapter 5 Outline

5.1 Working With Time-Varying Rates of Return5.2 Inflation5.3 Time-Varying Interest Rates: U.S. Treasuries and the Yield Curve5.4 Why is the (Nominal) Yield Curve Usually Upward Sloping? 5.5 Corporate Insights about Time-Varying Costs of Capital from the Yield

CurveAppendix: The Finer Points of Bonds5.6 Extracting Forward Interest Rates5.7 Shorting and Locking in Forward Interest Rates5.8 Bond Duration5.9 Duration Similarity5.10 Duration Hedging5.11 Continuous Compounding5.12 Institutional Knowledge: Compounding, Price Quotes, and STRIPS


Time-Varying Rates of Return and the Yield CurveIntroduction

• We have assumed a constant interest rate environment up to now.

• In the real world, we see rates change for different length of investment. Usually, we see short-term rates lower than longer-term rates.

• Should we consider the impact of different rates? Yes.

• Can we do so and have our capital budgeting rules work? Yes.

• Rates do change, sometimes dramatically in a short time:

March 2008Date 1 mo 3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr

03/03/08 1.99 1.70 1.80 1.74 1.61 1.84 2.48 2.96 3.54 4.37 4.42

03/17/08 1.16 1.11 1.31 1.32 1.35 1.52 2.23 2.71 3.34 4.24 4.29

Source: U.S. Treasury


Time-Varying Rates of Return and the Yield CurveWorking with Time-Varying Rates of Return

• Compounding Different Rates of Return

• If rates are 20% in Year 1 and 30% in Year 2, what is the two-year holding rate of return?

To solve, you string together the yearly rates:

Two year rate = First Year • Second Year

• If rates are 30% in Year 2, 40% in Year 3, and 50% in Year 4, what is the holding period return for the 3 years from end of Year 1 to end of Year 4?

Three year rate = First Year • Second Year • Third Year

(1 r0,2 ) = (1 r0,1) • (1 r0,2 )

(1 r0,2 ) = (1 20%) • (1 30%) = 1. 56

r0,2 = 56%

(1 r1,4 ) = (1 r1,2 ) • (1 r2,3) • (1 r3,4 )

(1 r1,4 ) = (1 30%) • (1 40%) • (1 50%) = 2. 73

r1,4 = 173%


Time-Varying Rates of Return and the Yield CurveSpot and Forward Rates

• Spot rates are quoted on securities starting today and maturing later.

• r0,20 is a spot rate that exists from today until 20 years from today.

• Forward rates begin at a future time.

• r10,30 is a forward rate that exists between the end of Year 10 and Year 30.


Time-Varying Rates of Return and the Yield CurveAnnualized Rates of Return

• Most rates are quoted as annualized rates to make comparisons easier.

• If the 3-year holding period return is 173%, what is the average annualized return?

• To solve, we find the string of rates that equal a 173% return in 3 years:

Three year rate = First Year • Second Year • Third Year

(1 r0,3) = (1 r3) • (1 r

3) • (1 r

3)

1 +173% = (1 r3) • (1 r

3) • (1 r

3)

2.73 = (1 r3)3

2.731/3 = 1 + r3

1.3976 = 1 + r3

39.76% = r3


Time-Varying Rates of Return and the Yield CurveFinding the Annualized Rates from a Holding Period Return


Time-Varying Rates of Return and the Yield CurveFinding Annualized Rates from Payoffs

• What is the annual rate of return if you invest $100 today in a security that pays $240 in 30 years?

• To solve, we take the following steps:1. Find the 30-year return of 140% from the payoffs.2. Solve for the annualized return of 2.96%.

r0,30 C30 C0

C0

r0,30 $240 $100

$100r0,30 140%

Solve for the annual rate from the 30 year return:

(1 r30

)30 (1 r0,30 ) 2.40

(1 r30

) (1 r0,30 )1/30 1 r0,3030 2.401/30

(1 r30

) 1.0296

r30

2.96%


Time-Varying Rates of Return and the Yield Curve Annualized Rates from Positive and Negative Returns

• What if you doubled your money in one year and lost half the next year?

• Assume you had $100 and it grew to $200 in a year and then lost $100.• What is your annualized rate of return?

Two year rate = First Year • Second Year

• What is your average rate of return?(100% - 50%) / 2 = 25%

• The annualized return of 0% makes sense.

• Be aware that the securities industry likes to promote the average return!

(1 r0,2 ) = (1 r0,1) • (1 r1,2 )

(1 r0,2 ) = (1100%) • (1 50%) 1.00

r0,2 = 0%


Time-Varying Rates of Return and the Yield CurvePresent Values with Time-Varying Interest Rates

• When calculating PVs for NPV or other valuations, what changes with time-varying rates?

• If rates are changing, we cannot simply use the same annual rate over and over for every time segment. We’ll have to link together rates for each period to find holding period returns for 1 year, 2 years, 3 years, etc…

NPV PV (C0 ) PV (C1) PV (C2 ) PV (C3) .....

NPV C0 C1

1 r0,1

C2

1 r0,2

C3

1 r0,3

.....

NPV C0 C1

1 r1

C2

1 r2

C3

1 r3

.....

NPV C0 C1

1 r0,1

C2

(1 r0,1) (1 r1,2 )

C3

(1 r0,1) (1 r1,2 ) (1 r2,3) .....


Time-Varying Rates of Return and the Yield CurveExample: PV with Time-Varying Rates

• A project costs $12 and pays $10 in one year and $8 in five years.• What is the PV of the cash flows and the NPV of the investment?

Assume the one-year interest rate is 5% and the annualized five-year rate is 6%. • To solve, use these steps:

1. Find PV of the one-year cash flow using the one-year rate.2. Find PV of 5th-year cash flow using the five-year annualized rate and discounting back five years.3. Find PV – Cost = NPV.

PV ($10, one year away at 5%) $10

1.051 $9.52

PV ($8, five years away at 6%) $8

1.065 $5.98

PV (Total cash flows) $15.50

Find NPV

NPV PV Cost $15.50 $12 $3.50

NPV C0 C1

1 r0,1

C5

1 r0,5

NPV 12 $10

1.051 $8

1.065 $3.50


Time-Varying Rates of Return and the Yield CurveNPV of a Five-Year Project


Time-Varying Rates of Return and the Yield CurveInflation

• Inflation is the process by which goods cost more in the future than today.

• Inflation matters when you make nominal rate choices; your payoff is the same no matter what inflation does.

• Inflation doesn’t matter when you inflation-index your investments since the payoffs will go up if there is inflation.

• Most contracts in the United States are in nominal terms, so investors and firms choose to correct their estimates for the impact of inflation.

• If firms adjust the cost of capital for inflation, valuation is improved.


Time-Varying Rates of Return and the Yield CurveMeasuring the Inflation Rate

• Economists measure changes in the prices of representative baskets of goods to calculate inflation changes.

• In the U.S., the Bureau of Labor Statistics keeps inflation data at:www.bls.gov

• There’s a basket of consumer prices -- the Consumer Price Index, CPI.

• There’s a basket of business prices -- the Producer Price Index, PPI.

• There’s a broad index for the economy -- GDP Deflator.

• There are more, but these are the ones most quoted in the news.


Time-Varying Rates of Return and the Yield CurveReal and Nominal Rates

• A stated, or quoted rate, has two parts that comprise the nominal rate:

• Real rate, which is the return adjusted for inflation• Inflation rate, which is the change in price levels in the economy (r π)

• Nominal return is greater than 0 and includes real and inflation rates• Real return is the remainder if inflation is removed from the nominal rate

• Real rates can be negative if inflation rises above the nominal rate.

• If you know the inflation and real rates, you can find the nominal rate.

(1 + rnominal) = (1 + rreal) • (1 + π)

• If you know the nominal and the inflation rate, you can find the real rate.

(1 + rreal) = (1 + rnominal) / (1 + π)


Time-Varying Rates of Return and the Yield CurveCalculating Real Rates of Return

• Nominal = Real x Inflation holding period returns

• In December 2004, the 10-year U.S. Treasury bond had a YTM of 4.10% and the inflation rate was 3.30%. To find the real rate:

• The real return was 0.77%. This is less than 410bp minus 330bp due to compounding.

(1 rnominal ) (1 rreal ) (1 )

(1 rreal ) (1 rnominal )

(1 )

(1 rreal ) (1 4.1%)

(1 3.3%)

(1.041)

(1.033)

(1 rreal ) 1.0077 rreal 0.77%

(1 rnominal ) (1 rreal ) (1 )


Time-Varying Rates of Return and the Yield CurveInflation in NPVs: A Simple Rule to Find Correct NPVs

• Discount real cash flows with real rates….• …or discount nominal cash flows with nominal rates…

• ….both methods give the same answers…• ….but DO NOT

mix the two.

• The easiest method for valuing projects is to leave everything in nominal terms.

• What is the present value of a $100,000 nominal payment in one year if the inflation rate is 8% and the real rate is 3%?

PV $nominal

(1 rnominal )

$nominal

(1 rreal ) (1 )

PV $100,000

(1 3%) (1 8%)

PV $89,895.72


Time-Varying Rates of Return and the Yield CurveTime-Varying Rates: U.S. Treasuries and the Yield Curve

• The U.S. Treasury bond market is the world’s most important financial market.

• Treasuries are considered to be default-free bonds. The government can print the money it needs to repay debt. There is no repayment uncertainty.

• U.S. Treasury Market• Treasury Bills mature in less than 1 year.• Treasury Notes mature between 1 year and 10 years.• Treasury Bonds mature in more than 10 years.

• The Treasury market is large and liquid, with tremendous trading volume.• Annual trading volume in 2004 was $130 trillion, about 18X GDP.

• The interest rates differ on Treasuries depending on their maturities. • A graph of rates plotted against their maturities is called a yield curve.


Time-Varying Rates of Return and the Yield CurveYield Curve Shapes: Upward Sloping is most common.

FIGURE 5.1 History: Some Yield Curves


Time-Varying Rates of Return and the Yield CurveThe Yield Curve on December 31, 2004

FIGURE 5.2 The Treasury Yield Curve on December 31, 2004


Time-Varying Rates of Return and the Yield Curve The Yield Curve -- Holding Rates of Return

• Holding Rate of Return – the total return from investing over a period of time

• On December 31, 2004, the yield curve had the following annualized rates:Two-Year TBond Three-Year TBond Twenty-Year TBond

2.58% 2.83% 4.85%

• If we invested $500,000, how much would we have at the end of two years?To solve, find the 2-year future value:

• If we invested $500,000, how much would we have at the end of twenty years?To solve, find the 20-year future value:

C2 C0(1 r2) (1 r

2)

C2 $500,000 (1 2.58%) (1 2.58%)

C2 $500,000 (1.0523) $526,150

C20 C0 (1 r20

) (1 r20

).....

C20 $500,000 (1 4.85%)20

C20 $500,000 (2.5785) $1.29 million


Time-Varying Rates of Return and the Yield CurveThe Yield Curve -- Forward Rates of Return

• Forward Rates of Return -- we can find forward rates using the yield curve.• Recall the forward rate is the rate that exists between two future periods.

• On December 31, 2004, the yield curve had the following annualized rates:One Year TB Two Year TB Three Year TB Twenty Year TB2.23% 2.58% 2.85% 4.85%

• If we know the holding period return for two periods, we can use the compounding formula to solve for the forward rate between them.

• For interest-bearing securities, we need to know the differences among annualized returns, holding returns, and forward rates and how they relate to each other.

(1 r0,2 ) (1 r0,1) (1 r1,2 )

(1.0523) (1.0223) (1 r1,2 ) (1 r1,2 ) 1.0523

1.0223(1 r1,2 ) 1.0293 r1,2 2.93%


Time-Varying Rates of Return and the Yield CurveHolding Returns, Annualized Returns, and Year-by-Year Returns

TABLE 5.1 Relation between Holding Returns, Annualized Returns, and Year-by-Year Returns on December 31, 2004, by Formula


Time-Varying Rates of Return and the Yield CurveBond Payoffs and Your Investment Horizon

• In perfect markets, there is no way to “game” the system by buying longer maturity bonds and then selling them early when you need the cash.

• In perfect markets, investors earn the appropriate holding period return whether they buy a series of short term bonds or buy a long term bond and sell it early.

• On December 31, 2004, the yield curve had the following annualized and forward rates:Annual One Year TB Two Year TB Three Year TB

2.23% 2.58% 2.85%

Forward Spot 1yr Forward 1,2 Forward 2,32.23% 2.93% 3.39%

• If the investor buys the three-year bond and sells it in a year, the one-year return is 2.23%.The purchase price is $91.92.

• At the end of one year, with 2 years left, the three-year bond can be sold for $93.97.

• The return of $2.05 on $91.92 is 2.23%, exactly what the one-year Treasury bond offers.

PV $100

(1 .0285)3 $91.92

PV

$100

(1 .0293) (1 .0339) $93.97


Time-Varying Rates of Return and the Yield CurveEffect of Interest Rate Changes on Bonds

• Treasury bonds do not have default (credit) risk, but are long-term bonds riskier than short-term bonds? Yes, the value can change if rates change.

• Consider the changes in values when investors change the required rate of return:

• The value of a 20 year, zero coupon Treasury bond at 4.85% is $387.82.The 20yr bond has price volatility of 1.9% for every 10bp rate change.

If rates move up 10bp to 4.95%, it sells for $380.50 (-1.89%).If rates move down 10bp to 4.75%, it sells for $395.29 (+1.93%).

• The value of a 1-year, zero coupon Treasury bond at 4.85% is $953.74.The 1yr bond has price volatility of 0.095% for every 10bp rate change.

If rates move up 10bp to 4.95%, it sells for $952.84 (-0.095%).If rates move down 10bp to 4.75%, it sells for $954.65 (+0.095%).

• For $1 million in bonds, the 20-year moves $19,100 and the 1-year moves $950.

• These paper losses are real. The changes in value cost someone money.


Time-Varying Rates of Return and the Yield CurveUpward Sloping Yield Curve Facts

• Why is the Yield Curve Upward Sloping?

• Two possibilities:• Upward slope implies the next period’s one year rates will be higher.• Investors demand more compensation for holding bonds longer.

• Does an Upward Sloping Yield Curve Predict Higher Future Inflation?• No, comparing TIPS rates to Treasury rates finds little evidence for this.

Maturity 5yr 7yr 10yr 20yrTIPS rate 0.97% 1.35% 1.73% 2.13%Treasury rate 3.35% 3.75% 4.10% 4.83%Implied Inflation 2.36% 2.36% 2.33% 2.66% (T-TIPS)

• Little of the increase in Treasury yields is due to rising inflation expectations.


Time-Varying Rates of Return and the Yield CurveMore Upward Sloping Yield Curve Facts

• Does an Upward Sloping Yield Curve Predict Higher Future Rates?• Studies find little evidence that these forecasts are correct.

• Does an Upward Sloping Yield Curve Mean Bargains on the Long End?

• Research finds investors earn more buying the longer maturities.• Investors require more compensation to own longer maturities due to

risk.

• Does an Upward Sloping Yield Curve Compensate Investors for Risk?• Yes, long-term bonds have more interest rate risk.

• Corporate Insights on Time-Varying Costs of Capital• Corporate projects have the same pattern as the yield curve.• Longer-term projects must offer greater returns or investors will not

invest.

Copyright © 2009 Pearson Prentice Hall. All rights reserved.

Chapter 5Appendix

The Finer Points of Bonds

Additional Chapter Art


TABLE 5.2 Locking in a Future Interest Rate via the Long-Short Forward Interest Rate Spread


FIGURE 5.3 Physics Analogy Illustrating Plain Duration

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Copyright © 2009 Pearson Prentice Hall. All rights reserved. Chapter 5 Time-Varying Rates of Return and the Yield Curve