Session 3b.pptx

FNCE 6043 Aug – Nov , 2023 Joe Zhang
SMU Classification: Restricted
Spot Rates Curve
1

 Spot rates and spot rate curve
 Constructing theoretical spot rate curve
 Bond pricing using spot rate curve
 Forward rates and relation between spot rates and forward rates
 Yield spreads
 Theories of the term structure of interest rates
2

 Previously, we use the market discount rate to price bonds. The same discount rate is used for
each cash flow.
 A more fundamental approach to calculate the bond price is to use a sequence of market
discount rates that correspond to the cash flow dates. These market discount rates are called
spot rates.
 Spot rates are yields-to-maturities on zero-coupon bonds maturing at the date of each cash flow.
 A general formula for calculating a bond price given the spot rates:
where ZX = spot rate, or the zero-coupon yield, or zero rate, for period x
3
𝑃𝑉 =
𝑃𝑀𝑇
1 + 𝑍1
+
𝑃𝑀𝑇
1 + 𝑍2
2
+ ⋯ +
𝑃𝑀𝑇 + 𝐹𝑉
1 + 𝑍𝑁
𝑁

 An example: suppose that the one-year spot rate is 2%, the two-year sport rate is
3%, and the three-year spot rate is 4%.
 What is the price of a three-year bond that makes a 5% annual coupon payment?
5
1.02 1 +
5
1.03 2 +
105
1.04 3 = 4.902 + 4.713 + 93.345
= 102.960
 What is the yield to maturity?
102.960 =
5
1+r 1 +
5
1+r 2 +
105
1+r 3 , 𝑟 = 0.03935
4

 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 interest rates
 The slope of the yield curve changes over time as economic conditions and
expectations of future economic conditions change
5

6
Yield
Maturity (a)
Positive
Inverted
Maturity
Yield
(b)
Humped
Maturity (d)
Flat
Maturity (c)
Yield
Yield

 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 two reasons:
 First, Treasury securities are free of default risk, and differences in credit worthiness do not
affect yields.
 Second, as the largest and most active bond market, the Treasury market offers the fewest
problems of illiquidity or infrequent trading.
7

8
 For pricing, we will focus on the (theoretical) spot rate curve
 Also called zero or “strip” curve
 A sequence of yields-to-maturity on zero-coupon (government) bonds
 It is necessary to derive the spot rate curve from theoretical considerations because the
most actively traded government and corporate bonds make coupon payments.

9
 A default-free theoretical spot rate curve can be constructed from the yield on
Treasury securities.
 The Treasury issues that are candidates for inclusion are
 on-the-run Treasury issues
 on-the-run Treasury issues and selected off-the-run Treasury issues
 all Treasury coupon securities, and bills
 Treasury coupon strips

10
 On-the-Run Treasury Issues
 The on-the-run Treasury issues are the most recently auctioned issue of a given maturity.
 including 3-month, 6-month, and 1-year Treasury bills; 2-year, 5-year, and 10-year notes;
and 30-year bond.
 Treasury bills are zero-coupon instruments; the notes and the bond are coupon securities.
 For each on-the-run coupon issue, the estimated yield necessary to make the issue trade at
par is used.
 The resulting on-the-run yield curve is called the par coupon curve.

11
 On-the-Run par coupon yield curve
 The goal is to construct a curve with 60 semiannual spot rates: 6-month rate to 30-year rate.
 Only eight maturity points available when only on-the-run issues are used. Missing maturity
points are extrapolated from the surrounding maturity points on the par yield curve.
 The following is calculated:
(yield at higher maturity – yield at lower maturity)/(number of semiannual periods between
the two maturity points)
 The yield for all intermediate semiannual maturity points is found by adding to the yield at the
lower maturity the amount computed here.

12
 The yields from par yield curve are:
r2-year=6%, r5-year=6.6%
 Calculate:
(6.6 – 6 )% / 6 = 0.10%
r2.5-year = 6.0% + 0.1% = 6.1 %
r3-year = 6.1% + 0.1% = 6.2 %
r3.5-year = 6.2% + 0.1% = 6.3 %
r4-year =6.3% + 0.1% = 6.4 %
r4.5-year = 6.4% + 0.1% = 6.5 %

13
Period Years Yield to Maturity/Coupon Rate (%)
1 0.5 5.25
2 1.0 5.50
3 1.5 5.75
4 2.0 6.00
5 2.5 6.25
6 3.0 6.50
7 3.5 6.75
8 4.0 6.80
9 4.5 7.00
10 5.0 7.10
… … …
All bonds except for the six-month and one-year issues are at par.

14
 We can convert the par yield curve into the theoretical spot rate curve using
bootstrapping.
 The basic principle is that the value of the Treasury coupon security should be
equal to the value of the package of zero-coupon Treasury securities that
duplicates the coupon bond’s cash flow.
 For example:
 Given these two spot rates, we can compute the spot rate for a theoretical 1.5-year zero-
coupon Treasury.
 Given the theoretical 1.5-year spot rate, we can obtain the theoretical 2-year spot rate and so
forth until we derive theoretical spot rates for the remaining 15 half-yearly rates.
 The spot rates using this process represent the term structure of interest rates

15
1 0.5 5.25
2 1.0 5.50
3 1.5 5.75
4 2.0 6.00

16
Period Years Spot rate (%)
1 0.5 5.25
2 1.0 5.50
3 1.5 5.76
4 2.0 6.02
Note that 5.25% and 5.50% are the 0.5-year and 1-year spot rates.
 How to construct the theoretical 1.5-year spot rate?
Step 1: Identify the cash flows for the 1.5-year treasury securities
Step 2: Pricing the security using the spot rate approach(Price=par)
Step 3: Solve for the 1.5-year sport rate.

17
1 0.5 5.25
2 1.0 5.50
3 1.5 5.75
4 2.0 6.00
z1 = 5.25%, z2 = 5.5%, z3 = ?

18
 There are two problems with using just the on-the-run issues.
(1) There is a large gap between some of the maturities points, which may result in misleading
yields for those maturity points when estimated using the linear interpolation method.
(2) The yields for the on-the-run issues themselves may be misleading because most offer the
favorable financing opportunities. The true yield is greater than the quoted (observed) yield.
 To mitigate this problem, some dealers and vendors use selected off-the-run
Treasury issues.
 Some argue that it is more appropriate to use all Treasury coupon securities and
bills to construct the theoretical spot rate curve.

 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):
19
Maturity
(months)
Coupon Rate
(s.a. pmts)
Price
(per $100 par)
6 7 1/2 99.473
12 11 102.068
18 8 3/4 99.410
24 10 1/8 101.019

20
Bootstrapping with All Available Treasure bonds
 Starting from the 6-month bond:
99.473 =
103.75
1 + 𝑍1
So 𝑍1 = 4.3%.
102.068 =
5.5
1 + 0.043
+
105.5
1 + 𝑍2
2
So 𝑍2 = 4.4%.
99.410 =
4
3
8
1 + 0.043
+
4
3
8
1 + 0.044 2
+
104
3
8
1 + 𝑍3
3
So 𝑍3 = 4.6%.
 Continuing in this manner generates a yield curve of:
 𝑟6 𝑚𝑜 = 8.6% 𝑟1 𝑦𝑟 = 8.8% 𝑟1.5 𝑦𝑟 = 9.2% 𝑟2 𝑦𝑟 = 9.6%

Alternatives for Fitting Yield Curves
 Generally there is not a set of spot rates that exactly fit Treasury bond prices.
 Interpolation is also required for maturities for which no bond is available.
 Methods include:
 Bootstrapping (generally off of close-to-par, or on-the-run bonds)
 Regressions (various specifications)
 Other fitting techniques (e.g., cubic splines, guess-timate)

 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 with semiannual payments?
𝑃 =
4.5
1.043
+
104.5
1.0442
= 100.192

 In the previous example, the cash flow of the 1-year coupon bond can be
replicated by the following two zero-coupon bonds with different par values
 Hence, the price of the 1-year coupon bond should equal the sum of the two zero
coupon bonds.
 Otherwise, can you profit from it? (Arbitrage)
Price @ time 0 CF @month 6 CF at 1 year
1-year coupon bond P = 100.192 Coupon: $4.5 Coupon $4.5 Par: $100
6-month zero P0(6mths) = 4.5/1.043 Par: 4.5
1-year zero P0(1yr) = 104.5/(1.044)2 Par: 104.5

 P=100.5> 100.192. How to arbitrage?
1-year coupon bond P = 100.5 > 100.192 Coupon: $4.5 Coupon $4.5 Par: $100
1-year zero P0(1yr) = 104.5/(1.044)2 Par: 104.5

 P=100.5> 100.192. How to arbitrage?
 In general, buy low and sell high
 Buy six-month zero (@$4.5 par) and 1-year zero (@104.5 par), and sell 1-year coupon bond (@$100 par)
 Initial profit: $100.5 – $100.192 = $0.308.
 Future P&L: $0 with certainty.
1-year coupon bond P = 100.5 > 100.192 Coupon: $4.5 Coupon $4.5 Par: $100
1-year zero P0(1yr) = 104.5/(1.044)2 Par: 104.5

Forward Rates
26

27
 From the yield curve we can extrapolate information on the market’s consensus of
future interest rates.
 Consider the following two investment alternatives for an investor who has a one-
year investment horizon:
Alternative 1: Buy a one-year instrument.
Alternative 2: Buy a six-month instrument and when it matures in six months, buy another
six-month instrument.

28
6 months 1 year
Today
 With alternative 1, the investor will realize the one-year spot rate and that rate is
known with certainty;
 with alternative 2, the investor will realize the 6-month spot rate, but the 6-
month rate 6 months from now is unknown.
$100(1 + 𝑧2)2
(1 + 𝑧2)2
1 + 𝑧1 1 + 𝑓 $100 1 + 𝑧1 1 + 𝑓

29
6 months 1 year
Today
 With alternative 1, the investor will realize the one-year spot rate and that rate is
known with certainty;
 with alternative 2, the investor will realize the 6-month spot rate, but the 6-
month rate 6 months from now is unknown.
$100(1 + 𝑧2)2
(1 + 𝑧2)2
1 + 𝑧1 1 + 𝑓 $100 1 + 𝑧1 1 + 𝑓
Q: if z1= 2.625%, z2= 2.75%, the expected 6-month rate 6 months from now is 2.8%,
which strategy would you take?

 Let z1 = 0.0525/2 = 0.02625, z2= 0.0550/2= 0.0275. If an investors are indifferent between
the two strategies, they must offer the same 1-year dollar return.
 Total dollar return for strategy 1:
 Total dollar return for strategy 2: where 𝑓 is the six-month rate six
months from now
 So, we must have
$100 1 + 𝑧1 1 + 𝑓 = $100(1 + z2)2 ⇒ 𝑓 = 0.028752
30
$100(1 + z2)2
$100 1 + 𝑧1 1 + 𝑓 ,

 In the previous example, The future interest rate inferred from the spot rate curve
is called forward rate
 These forward rates are imbedded in the spot yield curve.
 They are informative about the market’s consensus forecast of future interest rates.
 They are also the key to pricing forward, future, and swap contracts.
 In general, the forward rate (also referred to as implied forward rate, IFR) between
two spot rates are
31
1 + 𝑧𝐴
𝐴
× 1 + 𝐼𝐹𝑅𝐴,𝐵−𝐴
𝐵−𝐴
= (1 + 𝑧𝐵)𝐵

 How do you make use of this information in the previous example?
 If your assessment on the 6-month rate 6 months from now is higher than 𝑓,
you would:
 If your assessment on the 6-month rate 6 months from now is lower than 𝑓,
you would:
32

 Investor can hedge future interest rate (Locking in the implied forward rate) by
trading in the spot market .
 Say we know that
 the one-period spot yield, 𝑟1 = 10%
 the two-period spot yield, 𝑟2 = 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:
33
0 1 2
$0 -$89.2785 +$100

 Forward return locked in is:
100 − 89.2785
89.2785
= 0.12
 This is the forward rate in the yield curve, 12%!
34
0 1 2
$0 -$89.2785 +$100

35
 In general, the relationship between a t-period spot rate (𝑧𝑡), the current six-month
spot rate (𝑧1), and the six-month forward rates is
𝑧𝑡 = 1 + 𝑧1 1 + 𝑓1 1 + 𝑓2 … 1 + 𝑓𝑡−1
1/𝑡
− 1,
where 𝑓𝑡 is the six-month forward rate beginning t six-month periods from now.
 This highlights the fact that long yields are geometric average of forward rates.

 Given the following set of six-month forward rates, find the 2.5-year spot yield curve and
plot the results
𝑧1 = 5.2
𝑓1= 5.6
𝑓2= 5.8
𝑓3 = 5.4
𝑓4 = 5.0
 How does the slope of the yield curve change when forward rates increase? How does it
change when they decrease? Why?
36

yield curve
5
5.1
5.2
5.3
5.4
5.5
5.6
1 2 3 4 5
years
spot
rates
Series1
Answer
37
Example: Constructing yield curves from
forward rates (cont.)
 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, z2 = 5.40 z3 = 5.53 z4 = 5.50 z5 = 5.40

38
 Yields are often compared to a benchmark, usually the government yield curve, to
which are added various premiums
 The difference between the yield of a security and the benchmark yield is known
as the benchmark spread

39
 If the benchmark is a government bond, the yield spread is known as the G-spread
 I-spread or interpolated spread over swap curve
 The yield spread of a bond over the standard swap rate
 TED spread: The difference between LIBOR and the yield on a T-bill of matching
maturity.
 Reflecting counterparty risk and the risk of banking system.
 Use SOFR to replace LIBOR

40
 Z-spread or Zero-volatility spread
 The yield is a constant amount above the benchmark yield
 It is calculated as follows:
where r1, r2,…rN, are spot rates derived from the government yield curve. Z is the Z-spread
per period and is the same for all time period.
PV = PMT
+ PMT
+ … + PMT+FV
(1+r1+Z) (1+r2+Z)2 (1+rN+Z)N

Theories of Term Structure of Interest Rates
41

42
Yield
Maturity (a)
Positive
Inverted
Maturity
Yield
(b)
Humped
Maturity (d)
Flat
Maturity (c)
Yield
Yield

 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
 Unbiased Expectations Hypothesis
 Liquidity Preference
 Preferred Habitat
 Market Segmentation

 The forward rates implied by the term structure are equal to the market's expectation of future
spot rates over the same period.
 The pure expectation theory relates current forward interest rates with expected future spot rates
with the simple equation. For two-period horizon, we have
 𝑓1 = 𝐸 𝑟2
 where 𝑟2 is the one-period zero-coupon rate at time 2. (𝑟 is called short rate)
 More generally: t𝑓n = E(tzn)
 t𝑓n is the forward rate for an n period loan beginning at time t, as of time 0
 tzn is the future spot rate (or yield) for an n period loan beginning at time t,
 E(z ) denotes the market's expectation of z.
 It follows that long-term yields are geometric averages of current and expected short-term yields.
44

 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.
45

 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.
46

 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.
 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
47

 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”.
 Let 𝑧1=𝑧2=0.05, and the expected one-year zero-coupon rate one year from now
(𝑟2) is also 0.05. Consider the following example of two investment opportunities:
 Two consecutive one-year zero coupon bonds;
 One 2-year zero coupon bond for 2-years
 If investor cares only about future expected value, then we must have:
1000(1 + 𝑧2)2
= 1000 1 + 𝑧1 1 + 𝐸 𝑟2
= 1000 ∗ 1.05 ∗ 1.05 = $1102.5
where 𝑟2 is the one-year zero-coupon rate for year 2.

 Now consider a short-term investor with one-year investment horizon, she can
either:
i. invest in one-year zero coupon: one-period return R=5%
ii. invest in two-year zero coupon: E(R) = 5%
 But future rate is only an expectation. If realized 𝑟2 > 5%, what happens to
realized return?
 For a risk averse investor, which one would she choose? What must happen to the
𝐸 𝑟2 in order for the investor to be indifferent between the two options?

 This suggests modifying our interpretation of implied forward rates:
t𝑓n = E(tzn) + 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(tzn) 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”.
 Implication: An upward sloping yield curve could be the results of either higher
future interest rate, liquidity premium, or both.
FIN 70650 - Fixed Income

51
 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 the size of the premium ranges from a few basis points to 1%

52

 This theory adopts the view that the term structure reflects the expectation of the future
interest rates and a risk premium. But it rejects the assertion that the risk premium must
rise uniformly.
 Investors and borrowers have preferred maturity segments (habitats).
 Some may shift out of their preferred maturity segments when supply and demand
conditions in different maturity markets do not match.
 They will only do so if there’re better rates to compensate them (risk premium)
 The shape of yield curves can slope up/down, flat, humped, etc, depending on the relative
supply/demand.
53

 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.
 For the market segmentation theory, the shape of the yield curve is determined by the
supply of and demand for securities within each maturity sector.
54

maturity
yield
month t
month t+1
Consistent with Market Segmentation Theory?
parallel shift in yield curve
55

 CFA I V4 Fixed Income: Learning Module 3 (spot rates and forward rates)
 CFA II V4 Fixed Income: Learning Module 1 (obtaining spot-rate curve from par
curve by bootstrapping; swap rate curve; traditional theories of the term structure)
 FF Chapter 5
56

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