# Bond Yields, Returns, and Duration

Jim Wilcox

Bond Yields, Returns, Risks, and Duration

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Bonds and Loans

Yields and Returns

Price Volatility and Risk in Default-Free Bonds

Measuring Interest Rate Risk

Duration: Types, Calculation, Meaning, Uses

• Next Time: Chapter 11 re: Duration

Week # 2

January 28, 2014

1

Coming Soon!

What We Did

1.

2.

3.

4.

Week # 2

January 28, 2014

2

Yield to Maturity (YTM):

A Result, Not a Cause!

• YTM = percentage rate that equates (known) bond

price to PV of all promised (via bond) payments

• If the price of a coupon bond = its principal (or FV),

then YTM = the bond’s coupon rate (C/FV)

– If bond price exceeds its face value, YTM < coupon rate

Week # 2

January 28, 2014

3

Yields on U.S. Treasury Bonds, 2003-2012:

Short-Term Yields (but not Prices) Varied More

Week # 2

January 28, 2014

4

Bond Yields Differ from Returns,

in Concept and in the Data

• Returns (over some time span) = current yield (via coupon) plus percentage change in the bond’s price (over time span) • Longer-maturity-bond prices fall more increase in YTM

– A measure called Duration, D, will conveniently show us how much

Week # 2

January 28, 2014

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in case we want a blank slide…

Week # 2

January 28, 2014

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Relation of a Bond’s Price to Its Yield:

Negative and Non-Linear

Because?

Price

(of a bond)

Actual Price

Tangent Line at y*

(use to approximate price)

p*

y*

Week # 2

January 28, 2014

Yield

7

Bond Price Changes, Volatility, and Durations

• “Volatility”, V, as used in Fabozzi, text chapter 11: V = ((%∆P)/(∆y))/(1+y), when y = yield to maturity, P = bond price

• 3 measures of price sensitivity to yield changes

– DV01 = dollar value (of ∆P) of yield change, ∆y, = 1 basis point – Yield value, ∆y, of price change of 1 = ∆y/∆P

– D = Duration = (approximate) %∆P per ∆y

• Some duration measures

– MacD: Macaulay duration

– ModD: Modified Macaulay duration

– DollarD: Dollar duration (= MacD x P)

Week # 2

January 28, 2014

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Duration: Overview

• Duration accounts for size and timing of cash flows

– Duration is typically < time until bond matures

– Maturity is date of one payment, often final and largest payment

• PV-weighted average of time when payments are made

– Larger coupons bring cash sooner, thus reduce avg. time

– Coupons can range from zero to quite large (e.g., 12%)

• We use Duration as a measure of interest-rate sensitivity – Approximately, percentage change in market values:

(dP/P) = (-D) x (dy)

• We learn to use D to measure effects on all A, L, and E

– Can quantify how much interest rates raise or lower values • 1 bond, bond fund, high-net-worth client, banks, nonfinancial corp.

– Can boost or “immunize” effects of ∆y on A, L, and E values Week # 2

January 28, 2014

9

Calculating MacD with Semiannual Coupons

(as in Chapter 11 (Fabozzi))

• Macaulay duration with constant semiannual coupons:

1C

(1 + y )

1

Macaulay duration =

+

2C

(1 + y )

2

+...+

nC

(1 + y )

n

+

nM

(1 + y ) n

P

1, 2, 3, …n = periods until bond payments

P = price of the bond

C = semiannual coupon payment (in dollars)

y = one-half the yield to maturity (y)

n = total number of semiannual periods

(i.e., number of years x 2)

M = dollar payment at maturity date, or Face Value

Week # 2

January 28, 2014

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An Example of Calculating Duration

Calculation of Macaulay Duration and of Modified Duration

of a 5-Year Bond with Semiannual Coupons

Annual coupon rate: 9.00% Maturity = 5 years Initial yield: 9.00% Coupon

Period (t)

PV of $1 at 4.5%

PV of CF

Or Cash Flow

1

$ 4.50

0.956937

2

4.50

0.915729

4.120785

8.24156

3

4.50

0.876296

3.943335

11.83000

4

4.50

0.838561

3.773526

15.09410

5

4.50

0.802451

3.611030

18.05514

6

4.50

0.767895

3.455531

20.73318...

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