The document discusses problems with the conventional discounted cash flow approach to bond valuation and introduces the arbitrage-free spot rate bootstrapping approach. Specifically: - The conventional approach discounts all cash flows using a single yield, but each cash flow has a different risk profile and should be discounted using a corresponding spot rate. - Spot rate bootstrapping values a bond as a package of zero-coupon bonds, with each cash flow discounted using the spot rate for its corresponding maturity. - Reinvestment risk arises because the conventional approach assumes coupon payments can be reinvested at the yield to maturity, which may not be realistic if rates change over time. Longer-term and higher-coupon bonds have greater dependence

1. From Fabozzi: Ch. 5 and 6
AS: Ch. 3, Section 2.4 and Ch. 7 Section 2
Lecture 2.1:
Spot Rate Bootstrapping
– The Arbitrage-Free Approach of Bond
Valuation

2.  The conventional approach of BondValuation (the Discounted Cash Flow, or the DCF Approach):
 𝑃𝑟𝑖𝑐𝑒 =
𝐶𝐹1
(1+𝑦𝑖𝑒𝑙𝑑)1 +
𝐶𝐹2
(1+𝑦𝑖𝑒𝑙𝑑)2 + … . . +
𝐶𝐹𝑁
(1+𝑦𝑖𝑒𝑙𝑑)𝑁
 EX: For a 3-year 10% bond, assuming that the 3-year yield is 12%:
 𝑃𝑟𝑖𝑐𝑒 =
$10
(1+12%)1 +
$10
(1+12%)2 +
$110
(1+12%)3 = $95.1963
 What is the flaw of this approach?
I. Problems with the Conventional DCF
Approach of Bond Pricing

3.  The DCF Approach: The same interest rate is used to discount the bond’s
cash flows throughout the life of the bond.
 The flaw is that it views each of the cash flows from period 1 to N as of the
same risk and discounts all of them by one interest rate.
 It will provide a ‘close’ approximation, but not the most accurate.
 In fact the cash flow received in year 1 will have a different risk from cash
flow received in year 2. And therefore each cash flow would have to be
discounted with an interest rate that is corresponding to the risk of
respective cash flow.
I. Problems with the Conventional DCF
Approach of Bond Pricing

4. 2.4 Pricing Bonds with Spot Rates
0 1 2 T=N
CF1 CFN+FV
CF2
P = ?
0 1 2 T=N
CF1 CFN+FV
CF2
P = ?

5.  Each cash flow from a coupon bond can be treated as the cash flow for
a zero coupon bond with the same maturity. That is, cash flow
received at the end of the year 1 from a coupon bond is equivalent to
the cash flow of a one-year zero coupon bond, cash flow for year 2
from a coupon bond is equivalent to the cash flow of a two-year zero
bond, etc.
 And therefore a correct approach for bond pricing will discount the
cash flow in year 1 of a coupon bond with a 1-year zero (spot) rate,
the cash flow in year 2 with a 2-year zero (spot) rate, the cash flow in
year 3 with a 3-year zero (spot) rate, etc. This is what an arbitrage
free approach will do.
I. Problems with the Conventional Approach of
Bond Pricing

6.  Consider each cash flow as a zero-coupon bond:
I. Problems with the Conventional DCF Approach of Bond
Pricing
0 1 2 T=N
CF1 CFN+FV
CF2
P = ?
...

7.  The flaw of the conventional approach is caused by the uncertainties
from reinvesting those coupon incomes received at different points of
time. To see what the reinvestment risk is about, we start with
decomposing bond yields into different sources of income.
 Bond investors receive returns from:
 Coupon interest – periodic income, usually semi-annually payments.
 Capital gain or loss – the profit or loss of market value (selling / maturity price
minus purchase price).
 Reinvestment income is the interest income generated by reinvesting coupon
interest payments and any principal payments from the time of receipt to the
bond’s maturity.
 Yield calculations should consider all three of them.
II. Decomposing Bond Yields

8.  YTM assumes that the coupon payments from a bond can be reinvested
at an interest rate equal to the yield to maturity.
 The yield to maturity will only be realized if the interim cash flows can
be reinvested at the yield to maturity and the bond is held to maturity.
This assumption can be highly misleading because of reinvestment risk
 Reinvestment risk is the risk an investor faces that future interest rates
for reinvesting coupon incomes could be less than the yield to maturity
at the time a bond is purchased.
II. Decomposing Bond Yields

9.  Suppose an investor has a 15-year 8% semi-annual coupon bond purchased at par
($100). Since the bond is purchased at par, its YTM = 8%.
 At YTM = 8%, this translates into total future dollars: $100 x (1.04)30 = $324.34

II. Decomposing Bond Yields

10.  Suppose an investor has a 15-year 8% semi-annual coupon bond purchased at par ($100).
Since the bond is purchased at par, its YTM = 8%.
 At YTM = 8%, this translates into total future dollars: $100 x (1.04)30 = $324.34
 The future value of $324.34 is obtained from:
 Principal repaid = $100
 Coupon incomes received = $100*4%*30 = $120
 Capital gain = $100 - $100 = $0
 Surplus (reinvestment income) = $104.34
 The surplus is the income from reinvesting those coupons received from period 1 to 30.
 Without reinvestment income, the dollar return would be $120 of coupon income and $0
capital gain (because the bond is purchased at par)
II. Decomposing Bond Yields

11. II. Decomposing Bond Yields
0 1 2 T=30
CF1 CFN+FV
CF2
P = ?
...

12.  The dollar return shortfall is $224.34 - $120 = $104.34. This shortfall is made up
if the coupon payments are reinvested at a yield of 8% (the interest rate at the time
the bond was purchased)
 The reinvestment income as a percentage of total dollar income is:
 % of Reinvestment Income = $104.34/224.34 = 46.5%
 The investor will only realize the YTM of 8% if:
 The coupon payments can be reinvested at the YTM of 8% (most of the time it is not, and
therefore there is a reinvestment risk)
 The bond is held to maturity (if the bond is not held to maturity, the investor faces the risk of
selling for less than the purchase price which is known as interest rate risk)
 These are large and questionable assumptions, and therefore YTM is only a approximate
measure for a bond’s annual rate of return.
II. Decomposing Bond Yields

13.  For the above 15 years 8% semiannual coupon bond purchased at par to yield a YTM of 8%, if the
market interest rate drops to 6% right after the purchase, what will be the rate of return (YTM) if
held to maturity?
 Coupon incomes received will now be reinvested at 6%, rather than at 8% (semiannual coupon
income = $4, and semiannual annual discount rate = 6%/2 = 3%):
 Total Cash Flows at Maturity:
 Principal at maturity = 100
 Coupon income + Reinvestment income: 4 ∗ 1 + 3% 29 + 4 ∗ 1 + 3% 28+…+ 4 ∗ 1 + 3% 1 + 4 ∗
1 + 3% 0 = 190.3017
 Capital Gain = 0
 Total cash flow at maturity: 190.3017 + 100 + 0 = 290.3017
 YTM calculation: 100 =
290.3017
(1+𝑌𝑇𝑀)30 , 𝑌𝑇𝑀 = 3.6164%*2=7.232%
 Conclusion: If interest rates drop after the purchase of the bond, reinvestment income decreases
which lowers down the YTM.
II. Decomposing Bond Yields

14.  Reinvestment risk is the risk an investor faces that future reinvestment
rates will be less than the yield to maturity at the time a bond is
purchased.
 The longer the maturity and the higher the coupon rate, the more a
bond’s return is dependent on reinvestment income to realize the yield
to maturity at the time of purchase.
III. Reinvestment Risk

15.  The two factors affecting reinvestment risk are:
 1. For a given YTM and a given non-zero coupon rate, the longer the
maturity, the more the bond’s total dollar return depends on
reinvestment income to realize the YTM at the time of purchase. That
is, the greater the reinvestment risk.
 The implication is that the YTM for a long-term, high coupon bond
may have a large amount of the total dollar return as reinvestment
income (which is more risky).
III. Reinvestment Risk

16.  2. For a coupon bond, for a given YTM and maturity, the higher the coupon
rate, the more dependent the bond’s total dollar return will be on the
reinvestment of the coupon payments in order to produce the YTM at the
time of the purchase.
 The implication is that bonds selling at a premium (which has a larger
coupon rate than the market rate) will be more dependent on reinvestment
income than bonds selling at par. This is because the reinvestment income
has to make up the capital loss due to amortizing the price premium when
holding the bond to maturity.
 Conversely, a bond selling at a discount will be less dependent on
reinvestment income.
III. Reinvestment Risk

17.  Exhibit 1 shows the percentages of total dollar return from
reinvestment income for three bonds with different coupons (and
selling prices) and terms to maturity.
 It shows:
 Reinvestment risk is greater for longer maturity bonds
 Reinvestment risk is greater for higher coupon bonds (those selling at a
premium)
III. Reinvestment Risk

18. III. Reinvestment Risk
This table shows:
1. The longer the
maturity, the larger
the percentage of
return from
reinvestment income
2. The higher the
coupon rate, the
larger the
percentage of return
from reinvestment
income

19.  For the 2-year, 7% semiannual coupon bond:

 (1) At 8% BEY, the price is:
 𝑃𝑟𝑖𝑐𝑒 =
$3.5
(1+4%)
+
$3.5
(1+4%)2 +
$3.5
(1+4%)3 +
$103.5
(1+4%)4 = $98.19

 (2) If this $98.19 is invested at 8% for 2 years, it will grow to $98.19*(1+4%)4 = $114.87
 (3) Total dollar income = $114.87 - $98.19 = $16.68
 Coupon income = ($100*3.5%*4) = $14
 Capital Gain = $100 - $98.19 = $1.81
 Reinvestment Income = $16.68 - $14 - $1.81 = $0.8684
 (4) % of Reinvestment Income of Total Dollar Income = $0.8684/$16.68 = 5.21%
III. Reinvestment Risk

20. III. Reinvestment Risk
 For the 3-year, 7% semiannual coupon bond:

 (1) At 8% BEY, the price is
 𝑃𝑟𝑖𝑐𝑒 =
$3.5
(1+4%)
+
$3.5
(1+4%)2 +
$3.5
(1+4%)3 + ⋯ +
$103.5
1+4% 6 = $97.38
 (2) If this $97.38 is invested at 8% for 3 years, it will grow to $97.38*(1+4%)6 = $123.22
 (3) Total dollar income = $123.22 - $97.38 = $25.84
 Coupon income = ($100*3.5%*6) = $21
 Capital Gain = $100 - $97.38 = $2.62
 Reinvestment Income = $25.84 - $21 - $2.62 = $2.22
 (4) % of Reinvestment Income of Total Dollar Income = $2.22/$25.84 = 8.58%

21.  The arbitrage-free approach values a bond as a package of zero-
coupon bonds with each zero bond’s maturity the same as each of cash
flow payment dates.
IV. The Arbitrage-Free Approach of Bond
Pricing
0 1 2 T=N
CF1 CFN+FV
CF2
P = ?
...
CF1 (Cash flow for a 1-period zero bond)
CF2 (cash flow for a 2–period zero-bond)

22.  Thus a 10-year, 8%, semiannual payment Treasury should be viewed
as 20 zero-coupon bonds. Cash flow for each of periods will be
discounted with a corresponding spot rate: (where zs are spot rates)
 𝑃𝑟𝑖𝑐𝑒 =
𝐶𝐹1
(1+𝑧1)1 +
𝐶𝐹2
(1+𝑧2)2 + … +
𝐶𝐹𝑁
(1+𝑧𝑁)𝑁
 𝑃𝑟𝑖𝑐𝑒 =
4
(1+𝑧1)1 +
4
(1+𝑧2)2 + … +
104
(1+𝑧20)20
IV. The Arbitrage-Free Approach of Bond
Pricing

23.  For the 3-year 10% bond:
 If the traditional valuation approach is used, assuming that 3-year yield is 12%:
 𝑃𝑟𝑖𝑐𝑒 =
$10
(1+12%)
+
$10
(1+12%)2 +
$110
(1+12%)3 = $95.1963
 If the Arbitrage-Free approach is used, assuming Treasury spot rates of 10%, 12%, and
14% for 1, 2, and 3-year yields:
 𝑃𝑟𝑖𝑐𝑒 =
$10
(1+10%)
+
$10
(1+12%)2 +
$110
(1+14%)3 = $91.3097

IV. The Arbitrage-Free Approach of Bond
Pricing

24.  This is the proper way to value a securities because it doesn’t allow for
arbitrage profit by taking apart or “ stripping” a security and selling
off the stripped securities at a higher aggregate value than it would
cost to purchase the security in the market.
IV. The Arbitrage-Free Approach of Bond
Pricing

25.  Arbitrage-free pricing approach – Assumes that no arbitrage profits
are possible in the pricing of the bond.
 Each of the bond’s cash flow (coupons and principal) is priced
separately and is discounted at the corresponding zero-coupon
government bond rate.
 Since each bond’s cash flow is known with certainty, the bond price
today must be equal to the sum of each of its cash flows discounted at
the corresponding spot rate – or otherwise arbitrage is possible.
IV. The Arbitrage-Free Approach of Bond
Pricing

26.  Some notes on arbitrage:
 Arbitrage is the simultaneous buying and selling of an asset at two different prices in
two different markets.
 The arbitrageur buys low in one market and sells for a higher price in another.
 If arbitrage is possible, it will be immediately exploited by arbitrageurs.
 If a synthetic asset can be created to replicate anther asset, the two assets must be
priced identically or else arbitrage is possible.
 The fundamental principle of finance is the “law of one price.”
IV. The Arbitrage-Free Approach of Bond
Pricing

27.  By viewing a bond as a package of zero-coupon bonds (Exhibit 4), it is
possible to value the bond as a package of zero-coupon bonds.
 If they are priced differently, arbitrage profits would be possible.
 To implement the arbitrage-free approach, it is necessary to determine
the interest rate (spot rate) for each coupon for each maturity:
 The Treasury spot rate is used to discount a default-free cash flow with the
same maturity.
 Adding a credit spread for an issuer to the Treasury spot rate curve gives the
benchmark spot rate curve used to value that issuer’s security.
 The value of a bond based on spot rates is called the arbitrage-free value.
IV. The Arbitrage-Free Approach of Bond
Pricing

28.  The following example shows that using the DCF method to price a
bond provides an opportunity for arbitrage.
 EX: Assuming a 10-year, 8% semi-annual coupon bond, if the market
rate is 6%, using the DCF method, the price is $114.88, as computed
in Exhibit 4.
IV. The Arbitrage-Free Approach of Bond
Pricing

29. IV. The Arbitrage-Free Approach of Bond
Pricing
 Exhibit 4: Price of a 8%,10-year Treasury valued at 6% discount rate,
using the DCF Approach
Period Year Cash Flow ($) Discount Rate (%) Present Value ($)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
104
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.8835
3.7704
3.6606
3.5539
3.4504
3.3499
3.2524
3.1576
3.0657
2.9764
2.8897
2.8005
2.7238
2.6445
2.5674
2.4927
2.4201
2.3496
2.2811
57.5823
Total $114.8776

30.  Exhibit 5 takes this10-year, 8% semi-annual coupon bond and creates
20 zero-coupon bonds with different maturities.
 If given the spot rate (annual discount rate) for each maturity, it is
possible to compute the individual present values of the 20 bonds.
 The summation of the present values is equal to the arbitrage-fee
bond value.
IV. The Arbitrage-Free Approach of Bond
Pricing

31. IV. The Arbitrage-Free Approach of Bond
Pricing
 Exhibit 5: Determination of Arbitrage Free Value of a 8%, 10-year Treasury
Period Year Cash Flow ($) Spot Rate (%) Present Value ($)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
104
3.0
3.3
3.5053
3.9164
4.3764
4.752
4.9622
5.065
5.1701
5.2772
5.3864
5.4976
5.6108
5.6643
5.7193
5.7755
5.8331
5.9584
6.0863
6.2169
3.9409
3.8712
3.7968
3.7014
3.5843
3.4743
3.3694
3.2747
3.1971
3.0829
2.9861
2.8889
2.7916
2.7055
2.6205
2.5365
2.4536
2.3581
2.2631
56.383
Total $115.2621

32. IV. The Arbitrage-Free Approach of Bond
Pricing
 Price = $114.8776 if priced with the DCF method
 Price = $115.2621 if priced with spot rates
 The difference in prices using these two valuation methods provides an
arbitrage opportunity because it would be possible to buy the bond for
$114.8775 and “strip” it to create 20 zero-coupon bonds worth a combined
$115.2621
 The sum of present value of the arbitrage profits would be $0.3846
(=$115.2621-$114.8776), which could amount to enormous profits for the
arbitrageur.
 A bond issue could be with several hundred millions of dollars, this would be very profitable!

33.  Exhibit 8 shows the opportunities for arbitrage profit.
 Note: in order to create profits for the 4.8% bond, it would be necessary to
“reconstitute” stripped bonds.
 The process of stripping and reconstituting assures that the price of
a Treasury will not depart materially from its arbitrage-free value.
 The Treasury spot rates can be used to value any default-free
security.
IV. The Arbitrage-Free Approach of Bond
Pricing

34. IV. The Arbitrage-Free Approach of Bond Pricing
 Exhibit 8 Arbitrage Profit from stripping the 8%, 10-year Treasury
Period Year Cash Flow Sell for Buy for Arbitrage Profit
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
104
3.9409
3.8712
3.7968
3.7014
3.5843
3.4743
3.3694
3.2747
3.1971
3.0829
2.9861
2.8889
2.7916
2.7055
2.6205
2.5365
2.4536
2.3581
2.2631
56.383
3.8835
3.7704
3.6606
3.5539
3.4504
3.3499
3.2524
3.1576
3.0657
2.9764
2.8897
2.8005
2.7238
2.6445
2.5674
2.4927
2.4201
2.3496
2.2811
57.5823
0.0574
0.1008
0.1363
0.1475
0.1339
0.1244
0.1770
0.1770
0.1134
0.1065
0.0964
0.0834
0.0678
0.0611
0.0531
0.0439
0.0336
0.0086
-0.0181
-1,193
115.2621 114.8775 0.3846

35.  Spot Rates:
 The yield on a zero-coupon or stripped Treasury security is called the Treasury
spot rate.
 Even though the Department of the Treasury does not issue zero-coupon bonds
with maturities greater than one year, government dealers synthetically create
zero-coupon securities by separating the coupon and principal payments.
 These zero-coupon bonds have no reinvestment risk. Therefore, these become
purer measures of yield and maturity and are used to measure the Treasury yield
curve.
 The relationship between maturity and Treasury spot rates is known as the term
structure of interest rates, or represented by a spot yield curve.
V. Spot Rate Bootstrapping (from Treasury
Yields)

36.  There are not many zero-coupon bonds issued and traded in the market, and as a result, spot rates
are typically bootstrapped from the Treasury yield curve by a method called bootstrapping.
 The basic principle underlying the bootstrapping method is that the value of a Treasury coupon
security is equal to the value of a package of zero-coupon Treasury securities that duplicates the
coupon bond’s cash flows:

 Value of a Treasury coupon bond if valued with a single YTM rate:

 𝑃𝑟𝑖𝑐𝑒 =
𝐶𝐹1
(1+𝑌𝑇𝑀)1 +
𝐶𝐹2
(1+𝑌𝑇𝑀)1 + … +
𝐶𝐹𝑁
(1+𝑌𝑇𝑀)𝑁
 If valued with spot rates:
 𝑃𝑟𝑖𝑐𝑒 =
𝐶𝐹1
(1+𝑧1)1 +
𝐶𝐹2
(1+𝑧2)1 + … +
𝐶𝐹𝑁
(1+𝑧𝑁)𝑁
 In equilibrium these prices are the same, which can be used to derive the spot rates in a bootstrapping
process.
V. Spot Rate Bootstrapping (from Treasury Yields)

37.  Example:
 Assuming that the yields on coupon Treasury bonds are trading at par (given in the
table).
 YTM for the bonds is expressed as a bond equivalent yield (semi-annual YTM).
 Notice that here the price for each bond has been computed with its respective
YTM.
V. Spot Rate Bootstrapping (from Treasury
Yields)

38.  Steps for a bootstrapping:
 Notice first: Treasury bills with maturity up to one year are zero coupon bonds as they are
not paying any coupons.
 1. Begin with the 6-month spot rate (z1).
 2. Set the value of the 1-year bond equal to the present value of the cash flows with the 1-
year spot rate (z2) divided by 2 as the only unknown. Solve for the 1-year spot rate.
 3. Use the 6-month and 1-yar spot rates and equate the present value of the cash flows of
the 1.5 year bond equal to its price, with the 1.5 year spot rate (z3) as the only unknown.
Solve for the 1.5 year spot rate.
 4. The process continues until all the spot rates needed are bootstrapped.
V. Spot Rate Bootstrapping (from Treasury
Yields)

39.  Step 1: Begin with the 6-month spot rate
 The bond with six months left to maturity is a zero-coupon bond and its yield is the
6-m spot rate.
 Since the 6-m bond has a semi-annual discount rate of 5%/2 = 2.5% or 5% on a
bond equivalent yield basis, therefore the 6-m spot rate is 2.5%.
 Since this bond will only make one payment of $102.50 (=$100+$5/2) in six
months, the YTM is the spot rate for cash flows to be received six months from now.
 The bootstrapping process proceeds from this point using the fact that the 6-month
annualized spot rate is z1 = 5%.
V. Spot Rate Bootstrapping (from Treasury
Yields)

40.  Step 2: Set the value of the 1-year bond equal to the present value of the cash flows with the 1-year spot rate
divided by 2 as the only unknown. Solve for the 1-year spot rate.
 The 1-year bond will make two payments, $3 in six months and $103.0 in one year, and that the appropriate
spot rate to discount the coupon payment (which comes 6 months from now), is written as:
3.0
(1+2.5%)1 +
103
(1+𝑧2)2 = 100
where z2 is the annualized 1-year spot rate.
 Solve for z2/2 as:
 1-yr Spot rate (z2) = 3.0076% x 2 = 6.0152%
V. Spot Rate Bootstrapping (from Treasury
Yields)

41.  Step 3: Use the 6-month and 1-year spot rates and equate the present value of the cash
flows of the 1.5 year bond equal to its price, with the 1.5 year spot rate as the unknown
 Now that we have the 6-month and 1-year spot rates, this information can be used to
price the 18-month bond.
 Set the bond price equal to the value of the bond’s cash flows (with a coupon rate of
7%/year) as:
$3.5/(1.025)1 + $3.5/(1.030076)2 + $103.5/(1+z3/2)3 = $100
where z3 is the annualized 1.5-year spot rate.
 Solve for the 1.5-yr Spot rate (z3) = 3.5244% x 2 = 7.0488%
V. Spot Rate Bootstrapping (from Treasury
Yields)

42.  Because the theoretical spot rates are to be constructed from the Treasury yields,
the first step of Bootstrapping is to derive a Treasury yield curve for on-the-run
Treasury issues, for the reason that they don’t have credit risk and liquidity risk.
• Because there are a limited number of on-the-run Treasury securities (3-month, 6-
month, 2-year, 3-year, 5-year, and 10-year notes (the 30-year has recently been reissued)
traded in the market, (linear) interpolation is required to obtain the yield for
interim maturities;
 Hence, the yield for most maturities used to construct the Treasury yield curve are interpolated
yields rather than observed yields.
V. Spot Rate Bootstrapping: Interpolation of
Yield Curves

43.  To fill in the gap for each missing one year maturity, it is possible to
start with the lowest maturity and work up to the highest maturity
with the following formula:

(𝑦𝑖𝑒𝑙𝑑 𝑎𝑡 ℎ𝑖𝑔ℎ𝑒𝑟 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 −𝑦𝑖𝑒𝑙𝑑 𝑎𝑡 𝑙𝑜𝑤𝑒𝑟 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦)
𝑁𝑜.𝑜𝑓 𝑌𝑒𝑎𝑟𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑃𝑜𝑖𝑛𝑡𝑠
 The estimated on-the-run yield for all intermediate whole-year
maturities is found by adding the amount computed to the yield at the
lower maturity.
V. Spot Rate Bootstrapping: Interpolation of
Yield Curves

44. Example: 2-year 4.52%, 5-year 4.66%, 10-year 4.80%, 30-year 5.03%
 Using the above information, to bootstrap the 3- and 4-year Treasury rates, the following
interpolation of .0466% was computed as follows:
(4.66% −4.52%)
3 𝑦𝑒𝑎𝑟𝑠
= 0.0466%
Then the interpolated 3-year rate = 4.52% + .0466% = 4.567%
The interpolated 4-year rate = 4.567% + .0466% = 4.614%
 Therefore, when a yield curve is shown, many of the points are only approximations. Exhibits 4
and 5 show an interpolated “bootstrapped” Treasury yield curve.
 This method produces only a ‘crude approximation’
V. Spot Rate Bootstrapping: Interpolation of
Yield Curves

45.  Treasury Yield Curve:
V. Spot Rate Bootstrapping (from Treasury
Yields)

46. V. Bootstrapping Spot Rates
Notice that these YTM rates are derived from an linear
interpolation of on-the-run Treasury yields. The YTM for
each bond is the coupon rate
These spot rates are derived
from bootstrapping

47.  In Exhibit 4, the 6-month and 12-month Treasuries are zero coupon bonds, and
therefore their annualized zero rates are:
 z1 = 3.00% and z2 = 3.30%
 To bootstrap the 18-month zero rate with 3.5% as coupon rate:
 First: we take the cash flows from a 18m Treasury:
 Cash flow at 0.5 year = 3.5% * $100 * 0.5 = $1.75
 Cash flow at 1.0 year = 3.5% * $100 * 0.5 = $1.75
 Cash flow at 1.5 year = 3.5% * $100 * 0.5 + $100 = $101.75
 Second: As the 18m Treasury is priced at par (100), the cash flows discounted at
zero rates should also be equal to 100, otherwise there will be arbitrage
opportunities.

V. Spot Rate Bootstrapping (from Treasury
Yields)

48.  Since the value of a Treasury coupon security is equal to the value of the
package of zero-coupon Treasury securities that duplicates the coupon
bond’s cash flows :
 $1.75/(1+z1) + $1.75/(1+z2)2 + $101.75/(1+z3)3 = $100
 With Z1 = 1.5% (3%/2), Z2 = 1.65% (3.3%/2), the spot rate for the 1.5 year
Treasury can be calculated as:
 $1.75/(1+1.5%) + $1.75/(1+1.65%)2 + $101.75/(1+z3)3 = $100
 To get z3 = 1.7527% for 6 months (semiannually), doubling this yield, we
obtain the bond-equivalent yield of 3.5053%.
V. Spot Rate Bootstrapping (from Treasury
Yields)

49.  By the same token, given these three spot rates, we can bootstrapping
for a theoretical 2 year spot (zero-coupon) rate:
 Since all the coupon bonds are sold at par, the YTM for each bond is
the coupon rate. The cash flows for the 2 year Treasury (with a
coupon rate of 3.9%) are:

 Cash flow at 0.5 year = 3.9%*$100*0.5 = $1.95
 Cash flow at 1.0 year = 3.9%*$100*0.5 = $1.95
 Cash flow at 1.5 years = 3.9%*$100*0.5 = $1.95
 Cash flow at 2.0 years = 3.9%*$100*0.5 + $100 = $101.95
V. Spot Rate Bootstrapping (from Treasury
Yields)

50.  Since the value of a Treasury coupon security is equal to the value of the package
of zero-coupon Treasury securities that duplicates the coupon bond’s cash flows :
 $1.95/(1+z1) + $1.95/(1+z2)2 + $1.95/(1+z3)3
 + $101.95/(1+z4)4 = $100
 With Z1 = 1.5% (=3%/2), Z2 = 1.65% (=3.3%/2), and Z3 = 1.7527% (=3.5053%/2),
the spot rate for the 2.0 year Treasury can be calculated as:
 $1.95/(1+1.5%) + $1.95/(1+1.65%)2
 + $1.95/(1+1.17527%)3 + $101.95/(1+z4)4 = $100
 To get z4 = 1.9582% for 6 months, doubling this yield, we obtain the bond-
equivalent yield of 3.9164%.
V. Spot Rate Bootstrapping (from Treasury
Yields)

51.  Exhibit 6 shows the plot of the theoretical spot rates and the par value
Treasury yield curve.
 Notice that the theoretical spot rate curve lies above the par yield
curve. This will always be the case when the par yield curve is
upward sloping. When the par yield curve is downward sloping, the
theoretical spot rate will lie below the par yield curve.
V. Spot Rate Bootstrapping (from Treasury
Yields)

52.  Spot Yield Curve:
V. Spot Rate Bootstrapping (from Treasury
Yields)