Topic 4

Yield Measures
and The Yield Curves

FINA 4120 - Fixed Income 1

 Topic 4 will cover

• How do we think about the return on bond

investment?
– Various yield measures
– Return decomposition*

• More on yield curve

– Construction*
– Application
– Term Structure Theory*

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Various Yield Measures

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What determines the realized return (yield) on
a fixed-income investment?
• Three main components affect realized yield over the
investment horizon:
1. Periodic interest payments (coupon payments)
2. Income from reinvestment of periodic interest payments
3. Any capital gains or losses when the bond matures or is sold

– (2) reflects reinvestment risk

– (3) reflects interest rate risk

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 Traditional Yield Measures

• Current Yield
– Annual dollar coupon interest / Price
– Reinvestment and capital gain / loss are not considered
• YTM
– Recall its definition – Like an IRR
– Assume the bond is held to maturity
– Therefore, the coupon payments are assumed to be
reinvested at an interest rate equal to the YTM
– However, the actual reinvestment rate may be different

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 Premiums, Discounts and Yields

• Discount ( price < face )

– coupon rate < current yield < YTM
• Par ( price = face )
– coupon rate = current yield = YTM
• Premium ( price > face )
– coupon rate > current yield > YTM

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 Sample Question

Suppose a 10-year 9% coupon bond (with annual

coupon payment) is selling for $112 with a par value
of $100. What is the current yield for the bond?

a) 7.27%
b) 8.04%
c) 9.00%
d) 9.20%

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 Answer

b. Current yield = coupon / price = 9/112 = 8.04%

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 Practice Question

The following yields and prices were reported in the

financial press. Which set of them could be correct?

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 Answer

b.

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 Total (Dollar) Return

• Also called “horizon return” or “realized compound yield.”

• A calculation of future value, experimenting with

different interest rate changes.

• A popular method that avoids some of the shortcomings of

traditional yield calculations and that helps to identify
sources of risk.

• Low tech, easy to understand

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 Decomposing Dollar Returns

First find future value of coupon interest using future value of

an annuity formula:

é (1 + r ) n - 1ù
Cê ú
ë r û

C = semiannual coupon payment

n = # of six month periods to maturity or sale
r = effective reinvestment rate on a six month basis

This accounts for coupon interest plus interest on interest from

reinvesting.

Part of this, the total coupon interest, is nC.

Thus interest on interest is equal to:

é (1 + r ) n - 1ù
Cê ú - nC
ë r û

Finally, compute the capital gain, which equals:

Pn - P0

P0 = purchase price
Pn = sales price or face value

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Example: Decomposing Returns -- Par Bond Held to
Maturity

Invest $1,000 in a 7-year bond with 9% semiannual

coupon, selling at par.

We know that:
yield = 9% (bond equiv. basis)
earns 4.5% semiannually

If this yield were a sure thing, at maturity the

accumulated payments would be:

$1,000(1.045)14 = $1,852

Thus the total dollar return at maturity would be $852

($1,852 - $1,000).

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Let‘s decompose this:

1. Coupon interest plus interest on interest =

é (1045
. )14 - 1ù
$45ê ú = $852
ë .045 û

2. Interest on interest = $852 - 14($45) = $222

3. Capital gain = $0 (why?)

So interest on interest is $222/$852 = 26% of the total

return. This is the portion exposed to reinvestment
risk.

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Example: Decomposing Returns (Discount Bond)
Invest in a 20 year bond with a 7% semiannual
coupon, selling at $816 (per $1,000 face) to yield 9%.

We know that:
yield = 9% (bond equiv. basis)
earns 4.5% semiannually

If this yield were a sure thing, at maturity the

accumulated payments would be:

$816(1.045)40 = $4,746

Thus the total dollar return at maturity is

$4,746-816 = $3,930.

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Let’s decompose this:

1. Coupon interest plus interest on interest =

é (1045
. ) 40 - 1ù
$35ê ú = $3,746
ë .045 û

2. Interest on interest = $3,746 - 40($35) = $2,346

3. Capital gain = $1,000-$816 = $184

In sum:
Total coupon interest = $1,400
Interest on Interest = $2,346
Capital gain = $ 184

Total = $3,930

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Example: Sensitivity Analysis on Total Returns
Again invest in a 20 year bond with a 7% semiannual
coupon, selling at $816 (per $1,000 face) to yield 9%.

But now assume that the reinvestment rate will only

be 6% (i.e., 3% semiannually).

What is the effect on total returns?

1. Coupon interest plus interest on interest =

é (103
. ) 40 - 1ù
$35ê ú = $2,639
ë .03 û

2. Interest on interest = $2,639 - 40($35) = $1,239

3. Capital gain = $1,000-$816 = $184

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In sum:
Total coupon interest = $1,400
Interest on Interest = $1,239
Capital gain = $ 184

Total = $2,823

$2,823 + $816
Realized yield solves: $816 =
y
(1 + )40
2

y = 7.6%, significantly less than the 9% yield to maturity!

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 Example: Impact of Holding Horizon on Total
Dollar Return
• Suppose that you invested in the 10 year 8% US
Treasury Note sold at par
– Par bond implies that YTM = 8%

• Suppose that the next day after your purchase the

market yield falls to 6% and stays there

What will be the realized yield if you hold the bond (1) for 3
years; (2) for 7 years; (3) to maturity?

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• Holding period = 10 years (hold to maturity)
– Realized yield = 7.43 < 8%, why?
– IR ↓ => Reinvestment income ↓

• Holding period = 3 years

– Realized yield = 10.82% > 8%, why?
– IR ↓
• Reinvestment income ↓
• Selling price of the bond ↑ => Capital Gain
• Capital gain dominates => higher realized yield

• Holding period = 7 years

– Realized yield = 8.05% ≈ 8%, why?
– Capital gain offsets lower reinvestment income
– 7 turns out to be close to the bond duration

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 Sample Question
You purchased a 5-year, 9% coupon bond that pays interest
semi-annually. The price of this bond is $108.32. Assume the
interest rate changes to 6% (BEY) immediately after your
purchase and stays there indefinitely. You plan to hold the bond
for 4 years. Assume a flat yield curve.
a) What is the Yield to Maturity (BEY) of the bond when you
purchased it?
b) Compute the total dollar return and its three components (coupon
interest, interest on interest, capital gain or loss) for your
investment.
c) What is the realized yield (BEY) of your investment? Is it higher
or lower than the Yield to Maturity in (a)? Why?

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 Answer (using calculator)
a) 7%. In a financial calculator, set P/Y=C/P=2, enter N=10, PV=-
108.32, FV=100, PMT=4.5, then CPT I/Y.
b) Total coupon interest: 4.5*8=36
Interest on interest: 40.02 – 36 (In a financial calculator, set
P/Y=C/P=2, enter N=8, PV=0, I/Y=6, PMT=4.5, then CPT FV. FV
of the coupons you received is 40.02.)
Capital loss: 102.87 – 108.32= -5.45 (In a financial calculator, set
P/Y=C/P=2, enter N=2, FV=100, I/Y=6, PMT=4.5, then CPT PV.
The selling price of the bond is computed as 102.87)
Total dollar return is 40.02-5.45=34.57
c) Realized yield is (((34.57+108.32)/108.32)1/8-1)*2= 7.05%.
It is slightly higher than the YTM since the drop in interest rate led to a
smaller capital loss when the bond was sold. In addition, the duration
of the bond is close to 4, so the realized yield should be close to 7% if
you hold for 4 years.

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 Other Yield Measures

• Yield to Call
– The IRR assuming the bond will be called

Example: A 30yr bond with 8% coupon sells for $115, and is callable in
10 years at par

Cash Flow to Call Cash Flow to Maturity

Coupon payment $4 $4
Number of semi-annual periods 20 periods 60 periods
Final payment (principal) $100 $100
Price $115 $115
Yield to Call Yield to Maturity
5.98% 6.82%

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 Practice Question

Suppose that a 10-year bond with 9% coupon rate (semi-

annual coupon) can be called at the end of the 5th year at
$104.50. Its current price is $123.04. What is the yield-
to-call (quoted on BEY) of the bond?

a) 4.2%
b) 4.4%
c) 4.6%
d) 4.8%

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 Answer

c. In a financial calculator, set P/Y=C/P=2, enter N=10, PV=-

123.04, FV=104.50, PMT=4.5, then CPT I/Y.

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 Other Yield Measures

• Yield to Put
– The IRR assuming the bond will be putted

• Yield to Worse
– The IRR assuming the worst senario

• Cash Flow Yield

– Used for MBS and ABS
– The IRR on projected cash flows

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 Yield Decomposition

• The interest rate (or yield) offered on a bond can

be decomposed as:
– Risk Free Rate + Spread
• Risk free rate is the interest rate on a risk-free
instrument --- aka “level of interest rate”
– Recall who sets the level of interest rate
• The interest rate spread reflects perceived risks
associated with the issue
– Recall the risks associated with bond investments

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The Yield Curve

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 The Yield Curve
• The yield curve gives the yield (rate of return) on fixed income
securities as a function of their time to maturity
• It is also known as the "term structure of interest rates."

• We will study how the yield curve is used for:

– Pricing securities and fixed income derivatives (options, futures and
forwards)
– Looking for arbitrage opportunities
– Predicting market expectations of future spot rates

• The slope of the yield curve changes over time as economic

conditions and expectations of future economic conditions
change

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 The Yield Curve
• There are many types of yield curves...

• ...when people refer to "The yield curve", they mean the yield
curve for government securities, which is constructed using
Treasury bill and Treasury bond price data

• For pricing, we will focus on the (theoretical) spot rate curve

– Recall: spot rate = the yield on a zero-coupon security
– Treasury spot rate: Spot rate on Treasury security
– Treasury spot rate curve is used for valuing default-free cash flows
– LIBOR spot rate curve is used for valuing LIBOR-based debt
instruments and their derivatives

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 Constructing Spot Yield Curves
• Constructing a spot yield curve is easiest using the prices of pure
discount bonds such as Treasury bills and/or Treasury strips

• Example: Here is a set of prices and the implied yields (on a bond
equivalent basis): These yields are found using the logic that:
100
P= =>
Time To Maturity Price (P) Yield (1 + yt / 2) 2t
(in years) (per $100) (b.e.b.) éæ 100 ö (1 / 2t ) ù
1 $96 4.12% yt = 2êç ÷ - 1ú
2 $90 5.34% êëè P ø úû
3 $85 5.49%
4 $80 5.57% where t is the horizon in years.
yt is the spot yield for cash flows arriving in year t, expressed
on a bond equivalent basis.

Note that yt /2 can be described as an effective 6-month

rate.
To convert to an effective annual yield use (1 + yt /2)2 - 1.

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Constructing Spot Yield Curves for Coupon Bearing
Bonds by “Bootstrapping”
• Bootstrapping is a popular method for constructing a
spot yield curve when pure discount instruments are
unavailable or illiquid.

• Step 1: Gather price and coupon rate information.

• Step 2: Find effective spot yields sequentially, going

from the shortest to longest maturity, using the
formula:
C1 C2 CN
P= + + ... +
(1 + Y1) (1 + Y2 )2 (1 + YN ) N

– Ci is the cash flow in the ith period

– Yi is the effective spot yield for i periods
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 Bootstrapping
• EXAMPLE: Finding Semi-Annual Yields
The following information is available on Treasury bond
prices (32nds have already been converted to decimal form in the prices):

Maturity Coupon Rate Price

(months) (s.a. pmts) (per $100 face)
6 7 1/2 99.473
12 11 102.068
18 8 3/4 99.410
24 10 1/8 101.019

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 103.75
99.473 =
(1 + Y1 )
so Y1 = 4.3% ( = 8.6% on a b.e.b.)
5.5 105.5
102.068 = +
(1+.043) (1 + Y2 ) 2
so Y2 = 4.4% ( = 8.8% on a b.e.b.)
3 3 3
4 4 104
99.410 = 8 + 8 + 8
2
(1+.043) (1+.044) (1 + Y3 ) 3
so Y3 = 4.6% ( = 9.2% on a b.e. basis)

Continuing in this manner generates a yield curve of:

Y6 mo = 8.6% (b.e.b.)
Y1 yr = 8.8% (b.e.b.)
Y1.5 yr = 9.2% (b.e.b.)
Y2 yr = 9.6% (b.e.b.)

It is also possible to construct spot yield curves based

on other compounding periods (e.g., continuous,
weekly, monthly, yearly).

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A more general way to get bootstrapped rates
involves linear algebra and a computer: (optional)

For example, the information in the preceding

example can be summarized in matrix form as:

é103.75 0 0 0 ù é d1 ù é 99.473 ù
ê 55 . 1055. 0 0 ú êd ú ê102.068ú
ê úê 2 ú = ê ú
ê 4.375 4.375 104.375 0 ú êd3 ú ê 99.410 ú
ê5.0625 5.0625 5.0625 105.0625ú êd ú ê101019 ú
ë ûë 4 û ë . û

where the semiannual discount factor is:

1
di =
(1 + Yi )i

This equation can be represented as:

Md=P

M is the matrix of payoffs. P is the vector of prices.

The vector of discount factors is found by solving for
d:
d = M-1 P
FINA 4120 - Fixed Income 35
 Practice question
Below is a list of bond prices for various maturities. Assume
coupon bonds pay annual coupons and the par values are
$1,000.
Maturity (years) Coupon (%) Price ($)
1 0 943.40
2 0 873.52
3 8.5 1015.74

What are the spot rates corresponding to year 1, 2 and 3?

FINA 4120 - Fixed Income 36

 Answer

1000
= 943.40
1 + Y1
1000
= 873.52 Þ Y1 = 6%, Y2 = 7% and Y3 = 8%
(1 + Y2 ) 2

85 85 1085
+ + = 1015.74
1 + Y1 (1 + Y2 ) 2
(1 + Y3 ) 3

FINA 4120 - Fixed Income 37

Application 1: Using the Spot Yield Curve for
Pricing Other Bonds

Example:
These yields can be used to estimate the value of
other Treasury bonds, or any package of cash flows
with similar characteristics.

What is the value of a 1-year, 9% coupon bond

(semiannual pmts)?

4 .5 104.5
P= +
1.043 (1.044) 2

= 100.192
FINA 4120 - Fixed Income 38
 Application 2: Using the Spot Yield Curve to
Search for Arbitrage Opportunities

• An important use of the spot yield curve is to see whether

bonds are correctly priced relative to other bonds.

• If not, the discrepancy represents an arbitrage opportunity

(the ability to make a risk-free profit).

FINA 4120 - Fixed Income 39

Example: Looking for Arbitrage Profits

Assume that the yield curve is inverted. The effective

annual rates are given by:

Y1 = 9.9%
Y2 = 9.3%
Y3 = 9.1%

These are based on the prices of three zero coupon

bonds maturing in 1, 2 and 3 years respectively.

Also available is a 3 year 11% annual coupon bond

selling for $102 (per $100 face).

Is there an arbitrage opportunity? How would you

exploit it?

FINA 4120 - Fixed Income 40

The price of the coupon bond can be checked by discounting the promised
cash flows using the yields derived from the discount bonds:
11 11 111
p= + 2
+ 3
= 104.69
1099
. (1093
. ) (1091
. )
• The bond at $102 is clearly under-priced!

• How to profit?
• Buy the under-priced coupon bond and sell a set of discount bonds
whose payments mimic the cash flows of the bond you buy:
– sell $11 face of the 1 yr discount bond,
– sell $11 face of the 2 yr discount bond,
– sell $111 face of the 3 yr discount bond.
• These sales generate:
11 11 111
, ,and totaling...104.69
1.099 (1.093) 2 (1.091)3

FINA 4120 - Fixed Income 41

Therefore you collect $104.69 today.

At the same time, buy the under-priced bond for $102.

Profit today = $2.69 per $100 face value transaction.

All future cash inflows and outflows cancel, so this is

an arbitrage opportunity.

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 Application 3: Forward Rate

• A forward rate is the expected interest rate on an instrument

over some period in the future

• Forward rates that can be locked in by buying and selling

securities today are called implied forward rates
– These forward rates are imbedded in the spot yield curve
– Implied forward rates are informative about the market’s consensus
forecast of future interest rates
– They are also the key to pricing forward, future, and swap contracts

FINA 4120 - Fixed Income 43

 Example: The idea of an implied forward rate is easiest to see in a two period
example:

• Say we know that

– the one period effective spot yield, Y1 = 10%
– the two period effective spot yield Y2 = 11%

• Then the implied one period forward rate starting in one period, "f", solves:
– (1 + Y1)(1 + f) = (1 + Y2) 2
– (1.1)(1 + f) = (1.11)2
– f = 12%

• Interpretation:
– The one-period implied forward rate starting in one period is the rate an investor
must earn on a one period security purchased at the end of one period, so that the
return from buying and holding a two period security to maturity is equal to the
return from rolling one period securities.

FINA 4120 - Fixed Income 44

 Example: Locking in the implied forward rate by trading in the spot market

• Again, say we know that

– the one period effective spot yield, Y1 = 10%
– the two period effective spot yield Y2 = 11%

• Consider the following investment strategy:

– Buy today a two-period security with F=$100
– P = $100/(1.11)2 = $81.1622
– At the same time, sell a one-period security with a price of $81.1622
– F = $81.1622(1.10) = $89.2785
– Cash flows locked in:

0 1 2

$0 -$89.2785 +$100
FINA 4120 - Fixed Income 45
Example: Locking in the implied forward rate by trading in the spot market

• Forward return locked in is:

100 - 89.2785
= .12
89.2785

This is the forward rate in the yield curve, 12%!

0 1 2

$0 -$89.2785 +$100

FINA 4120 - Fixed Income 46

 Forward Rates in General

• Implied forward rates can be found for any future time period.
This requires some additional notation:
– mfn denotes the n period forward rate starting in m periods (from time 0)

• It is given by: 1/ n
é (1 + Ym+ n ) m+ n ù
(1 + m f n ) = ê m ú
ë (1 + Ym ) û

• Cautions:
– The only numbers you can plug into these formulas are effective rates per period
For instance, rates quoted on a b.e.b. must be divided by 2

– All spot and forward rates are assumed to be as of time 0

FINA 4120 - Fixed Income 47

Example: Find the implied 3-period forward rate
starting in 2 periods, given the following information
on effective spot yields:

Y1 = 2.0%
Y2 = 2.6%
Y3 = 3.0%
Y4 = 3.2%
Y5 = 3.4%

The implied forward rate solves:

(1+Y2)2(1+2f3)3 = (1+Y5)5
1/ 3 1/ 3
é (1 + Y5 )5 ù é (1.034)5 ù
(1 + 2 f 3 ) = ê 2ú
=ê 2ú
= 3.93%
ë (1 + Y2 ) û ë (1.026) û

FINA 4120 - Fixed Income 48

 Practice Question
• The 6-month Treasury bill spot rate is 4%, the 1-
year Treasury bill spot rate is 5%. The implied 6-
month forward rate 6 months from now is: (all
interest rates are BEYs)

a) 3.0%
b) 4.5%
c) 5.5%
d) 6.0%

FINA 4120 - Fixed Income 49

 Answer

• d. 0.5f0.5 = (1.025)2/1.02-1 = 3%, which is an

effective semi-annual interest rate, the
corresponding BEY is 6%.

FINA 4120 - Fixed Income 50

Constructing yield curves from forward rates
It is also possible to construct a yield curve from a
given a set of forward rates:

(1+Yn)n = (1+0f1)(1+1f1)(1+2f1)...(1+n-1f1)

This highlights the fact that long yields are geometric

averages of implied forward rates.

FINA 4120 - Fixed Income 51

 Example: Constructing yield curves from forward rates

Given the following set of one year forward rates, find the five year spot
yield curve and plot the results

0f1 = 5.2
1f1 = 5.6
2f1 = 5.8
3f1 = 5.4
4f1 = 5.0

How does the slope of the yield curve change when forward rates
increase? How does it change when they decrease? Why?

FINA 4120 - Fixed Income 52

 Answer
yield curve

5.6

5.5

5.4

spot rates 5.3 Series1

5.2

5.1

5
1 2 3 4 5
years

The curve slopes up when forward rates are increasing, and

slopes down when they are decreasing. This is because the
spot yields are a weighted average of the forward rates.

Using the formula relating spot and forward rates, Y1 = 5.20;

Y2 = 5.40 Y3 = 5.53 Y4 = 5.50 Y5 = 5.40
FINA 4120 - Fixed Income 53
 Shapes of the Yield Curve
• Typical Shapes:
– Normal (upward sloping)
• Most common
– Inverted (downward sloping)
• Occurs most often when interest rate is at historically high
– Flat
– Humpbacked
• upward sloping in the short term, but downward sloping in the long
term

FINA 4120 - Fixed Income 54

 Historical Patterns in Yield Curve

• Most common yield curve shape is upward-

sloping
• Declining yield curve occur when interest rates
are historically high
• Short-term interest rate is more volatile

Link to Yield Movie

FINA 4120 - Fixed Income 55

Humpbacked in Real World

FIN 40660 - Fixed Income 56

Humpbacked in Real World

FIN 40660 - Fixed Income 57

 Interpreting the Yield Curve

• Theories of the yield curve help to explain:

– The shape of the yield curve at a point in time
– How the yield curve moves over time
– What one can infer about the future from the yield curve

• Traditional Theories
– Market Segmentation / Preferred Habitat
– Unbiased Expectations Hypothesis
– Liquidity Preference

• Modern theories also take into account the effects of

volatility
FINA 4120 - Fixed Income 58
 The Market Segmentation Theory
• Some investors/borrowers like long maturities (e.g.,
life insurers and pension funds)

• Others like short maturities (e.g., banks)

• The forces of supply and demand operate

independently in these two essentially separate
markets.
L.T. L.T.
demand supply
yield

Y10
S.T.
supply
Y1

S.T. Loanable funds

demand

1 maturity 10

FINA 4120 - Fixed Income 59

 Preferred Habitat Theory

• A milder version of the market segmentation

theory

• It states that investors have preferred maturities in

which to invest, but if expected return differentials
become large they will switch habitats.

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 Is there support for segmented markets?

• The evidence is mixed

In 1971 Congress removed an interest rate ceiling on
• 30-year bond prices seemed to Treasury bonds.
increase when that maturity was Pre-1971 4.5% interest rate ceiling on bonds.
phased out Post-1971 ceiling lifted.
– Explained as a scarcity effect
Following the deregulation, the government rushed to
• But other historical episodes issue new long-term bonds.
suggest the effect is small (see
box) What does the theory predict about the relative
change in rates at different maturities?
• These effects are probably
relevant in some episodes, but What happened? Not much…
usually only over short periods
of time.

FINA 4120 - Fixed Income 61

 A key implication of the expectations hypothesis is the
interpretation of implied forward rates:

• The forward rates implied by the term structure are equal to the
market's expectation of future spot rates over the same period.

• The unbiased expectations theory relates current forward interest rates with expected
future spot rates with the simple equation:

• tfn = E(tYn)
– tfn is the forward rate for an n period loan beginning at time t, as of time 0
– tYn is the future spot rate (or yield) for an n period loan beginning at time t,
– E(Y) denotes the market's expectation of Y.

• It follows that long-term yields are geometric averages of current and expected
short-term yields.

FINA 4120 - Fixed Income 62

 The Expectations Hypothesis and Predicted Business
Cycle Conditions
• What does the Expectation Hypothesis say about the current yield
curve?

• Observation: The yield curve tends to slope up at the beginning of an

expansion, and is more likely to slope down at the end of an expansion.

• Demand Side Story

– The demand for business investment is high during expansions. High expected
demand for money implies high real interest rates.
– If the economy is expected to slow, expected future rates fall since investment
demand is expected to slacken.

• Supply Side Story

– People like to smooth their consumption.
– Therefore if they anticipate a recession they will want to save more, pushing
down rates.
FINA 4120 - Fixed Income 63
 Evidence on the Expectations Hypothesis and
Predicted Business Cycle Conditions
• Discussion based on “The Term Structure and World Economic Growth” by
Campbell R. Harvey

• Main finding: The spread between long and short rates is a remarkably
good predictor of GNP growth rates in many countries.

• Ran regression:
• ln(GNPt+5) - ln(GNPt) = a + b(TS)t + ut+5
– TS = spread between 90 day bill and bond with maturity at least five years.

• In the U.S. and Canada, this regression “explained” almost 50% of the growth
in GNP

FINA 4120 - Fixed Income 64

Dec 5 2018, Yield Curve Inverts for the First
Time in More than a Decade

FIN 40660 - Fixed Income 65

 Shortcomings of the expectations hypothesis

• Theoretically, it requires several strong assumptions that do not

hold in practice:
– Investors maximize expected returns, with no consideration of risk.
– Expectations are held with absolute certainty. (It can be shown that if there is
uncertainty about rates, then mathematically the theory must be false.)
– There are no transactions costs.

– Investors view securities with different maturities as perfect substitutes for one
another.

• More disturbingly, it appears to be seriously violated in

historical data
• Still, most experts agree that it is helpful in interpreting the
shape of the yield curve

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 The Liquidity Preference Theory
• The liquidity preference theory states that investors require a premium for
investing in longer-term debt. The required premium is called a "liquidity
premium" or “term premium.”

• This suggests modifying our interpretation of implied forward rates:

tfn = E(tYn) + tLn,

– tfn is the forward rate for an n period loan beginning at time t (as of time 0),
– tLn is the liquidity premium on an n period loan beginning at time t (as of time
0),
– E(tYn) is the expected future spot rate (or yield) for an n period loan beginning
at time t (as of time 0).

• Interpreting forward rates as the sum of the expected future spot rate and
a liquidity premium is called the “biased expectations theory.”

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Figure: Interpreting the yield curve in the presence of
a liquidity premium.

Expected Rates in the Presence of a Liquidity Premium

0.06

0.05

0.04

expected
rates

0.03
forward

0.02

0.01

0
1 2 3 4 5 6
period

How to read this graph:

The x-axis shows the number of periods spanned by a

particular forward rate. All these forward rates are for
periods starting at a fixed future time t. The rates are implied
by the current spot yield curve.
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 Evidence on the Liquidity Premium
• Forward rates tend to be higher than estimates of expected spot rates,
supporting the existence of a liquidity premium. (On average the yield
curve is upward sloping, even though on average interest rates don't
increase over time.)

• The measured premium is thought to increase with maturity over short

maturities, and level off for long maturities.

• Estimated premiums vary significantly over time

• Statistical analyses suggest that the size of the premium ranges from a few
basis points to 1%

• Intrinsically difficult to estimate since this requires predicting expectations

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 Practice Question

According to the pure expectations theory, what does a humped

yield curve suggest about the expectation of future interest rate?

a) The short-term interest rates are expected to keep on rising in the future
b) The short-term interest rates are expected to keep on falling in the future
c) The short-term interest rates are expected to rise for a time then begin to fall
d) The short-term interest rates are expected to fall for a time then begin to rise

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 Answer

C.

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 Practice question
With respect to the term structure of interest rates, the market
segmentation theory holds that:
a) An increase in supply for long-term bonds could lead to an inverted
yield curve.
b) Expectations about the future short-term interest rates are the major
determinants of the shape of the yield curve.
c) The yield curve reflects the maturity preferences of financial
institutions and investors.
d) The shape of the yield curve is independent of the relationship
between long-and short-term interest rates.

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 Answer

C.

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