# Bond Yields, Returns, and Duration

Topics: Bond, Bonds, Fixed income analysis Pages: 10 (545 words) Published: May 15, 2014
Financial Institutions and Markets
Jim Wilcox

Bond Yields, Returns, Risks, and Duration

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

5

in case we want a blank slide…

Week # 2
January 28, 2014

6

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

8

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

10

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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