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Structured Finance Course
Lesson 3 – Introduction to Bonds Math
Prof. Riccardo Bruno
Luiss Guido Carli
1. Zero Coupon Bonds and Forward rates
Valuation of Zero Coupon Bonds
3
• No coupons, single payment at maturity
• Bond trades at “discount” to face value
• These are also referred as “strips”1
• Valuation is just based on NPV
F is the future value (say € 1,000)
r is the spot rate to time tP0 =
F
1 + r T
1. From the acronym used in the US Treasury markets when STRIPS stands for Separate Trading of Registered Interest and Principal Only
Valuation of Zero Coupon Bonds
4
• If r varies over time, Rt is one-year spot rate of interest in year t. Then,
• Do we observe Ri today? No, these are future spot rates
• We only observe P0 for each maturity of the strips market
• Hence we can define from the strips market a r0,T which is the T-year
spot rate, derived inverting the following equation
• T-year spot rate today embeds (“averages”) all one-year rates between
today and time T
P0 =F
1 + R1 1 + R22…(1 + RT)
T
𝑃0 =F
1 + 𝑟0, 𝑇𝑇
Valuation of Zero Coupon Bonds: Example
5
• Strips for different maturities have the following prices
• For the 5-year Strips we have
Maturity (years) 1/4 1/2 1 2 5 10 30
Price 0.991 0.983 0.967 0.927 0.797 0.605 0.187
0.797 =1
1 + r0,55 ⟹ r0,5 =
1
0.79715
− 1 = 4.64%
Valuation of Zero Coupon Bonds
6
• We can more in general infer that
and build a curve of spot rates
• In other words the strips market imply a term structure of rates
• Term structure contains information about future rates
• Also, from various strips prices we can infer rate at which values of
bond with t maturity varies from t-1 to t
• With this we derive “forward rates” (in the above case “one-year forward
rates”)
{P0,1, P0,2, P0,3, … . P0,T} ⇒ {r0,1, r0,2, r0,3… . r0,T}
P0,t−1P0,t
= 1 + ft =(1 + r0,t)
t
(1 + r0,t−1)t−1
Forward rates
7
• In general, forward interest rates are today’s rates for transactions
between two future rates
• Example: you expect to earn € 10 k in 1 year from now and want to use
that amount to purchase goods in 2 year from now. You want to “lock”
the amount you can spend on goods in year 2 today
• The current interest rates are:
• Strategy:
t 1 2
r0,t 0.05 0.07
Borrow 10 / (1+0.05) = 9.524
Invest 9.524 @ 7% for 2 years → 10.904
9.524 accrues to 10 at year 1 end
10 are received at year 1 end → debt is paid
10.904 received in year 2 vs. 10 in year 1
→ 9.04% is the one-year forward rate from end t=1 to t=2
Valuation of Zero Coupon Bonds: Example
8
• Quote a rate for forward loan of 1 year in year 3
• Solution:
t 1 2 3 4
Pt 0.9524 0.8900 0.8278 0.7629
r0,t 0.05 0.06 0.065 0.07
f4 =1 + r0,4
4
1 + r0,33 − 1 =
1.07 4
1.065 3 − 1 = 8.51%
2. Coupon Bonds
Valuation of Coupon Bonds
10
• Coupon paying bonds include intermediate payments + final principal
• Can trade at discount or premium to face value
• Valuation is NPV based
• Example: 5 year, par value € 1,000, 3% coupon annually paid
t = 0 1 3 4 52
P0 3030 3030 30+1,000
Valuation of Coupon Bonds
11
• For a bond paying coupons C, principal F on maturity, with T year
maturity
• But future spot rates are not observable, hence we can simplify with a
single rate, which is how bonds are normally quoted in the market
• Quotes for a bond can be interchangeably given as price, yield, or
spread (see later)
• y is not generally solved by closed-form equation solution → need to
use iterations, numerical methods as need to solve a Tth degree
polynom
• yield curve is the graph that plots y versus bond maturity
P0 =C1
(1 + R1)+
C2(1 + R1)(1 + R2)
+ ⋯+CT + FT
1 + R1 1 + R2 …(1 + RT)
P0 =𝐶1
(1 + 𝑦)+
𝐶2(1 + 𝑦)2
+⋯+𝐶𝑇 + 𝐹𝑇(1 + 𝑦)𝑇
=
𝑡=1
𝑇𝐶𝐹𝑡
(1 + 𝑦)𝑡where CFt is year
t cash flow
Valuation of Coupon Bonds
13
• Given
• By setting z =1
1+ywe can calculate a shorter version of the bond price
formula
or
• This formulation can be useful in both calculation and analysis of the P,
y, T relationship
where, remember, Ct = Coupon at time t + F (i. e. the Principal)
P0 =
t=1
TCt
(1 + y)t
P0– z P0 = Cz − CzzT + F zT − zT+1
𝑃0 = 𝐶
1 −1
1 + 𝑦 𝑇
𝑦+
𝐹
1 + 𝑦 𝑇
Valuation of Coupon Bonds
14
• For bonds paying coupons several times in a year (generally
semiannually or quarterly) price is calculated as (in a semiannual
example)
• and in the generic case of n coupon payments in the year and a T years
to maturity bond
=
𝑡=1
𝑇𝐶/2
(1 +𝑦2)𝑡+
𝐹
1 +𝑦2
2𝑇 =𝐶
𝑦1 −
1
1 +𝑦2
2𝑇 +𝐹
1 +𝑦2
2𝑇
𝑃0 =𝐶
𝑦1 −
1
1 +𝑦𝑛
𝑛𝑇 +𝐹
1 +𝑦𝑛
𝑛𝑇
Valuation of Coupon Bonds: Example 1
15
• What is the price of a 5 year Eurobond with coupon of 5% and € 1,000
nominal, if the required yield is 6%?
• Using the long formula
• Using the short formula
= € 957.87634
P0 =50
6%1 −
1
1 + 6% 5+
1,000
1 + 6% 5= 833.33 × 0.25274 + 1,000 × .7472 = € 957.8763
Valuation of Coupon Bonds: Example 2
16
• What is the price of US Treasury bond (semi-annual coupons) with a
$100 nominal, 4% coupon, 10 year maturity and a required yield of
4.048%?
P0 =$4
0.04081 −
1
1 +12 0.0408
20 +$100
1 +12 (0.0408)
20
= 32.628 + 66.981 = 99.609$
Coupon bonds accrued yield
17
• “Clean” price of a bond is what normally gets quoted in the market
• It is, at any certain date, the PV of future cash flows excluding the
interest matured, “accrued”, on the bonds since the last coupon
payment
• “Dirty” price = Clean Price + Accrued Interest
where number of days in coupon period is set by typical market
conventions (e.g. 30/360, ACT/ACT)
NB the i calculation is subject to different conventions in different
markets. Conventions in general relate to the number of days assumed
in an interest period (ACT, 360, 365) as well as the calculation of the
effective days in a subperiod (each month may be split in the same
number of dd, eg 30, rather than using the ACT number of days, etc)
C xNumber of days from last coupon payment to settlement date
Number of days in coupon period
• Different approaches to calculate yields of a bond
• Simplest of all → Current yield = Coupon / Bond price
• Current yield does not consider return coming from capital gains and
from coupon re-investment
• YTM (what we analysed so far) is the return that makes
∑ DISC [CF] = BOND PRICE
• YTM is realised ex-post (i.e. equals the return of the bond) only if
– bond held at maturity
– re-investment of coupons occurs at same YTM
• YTC (yield to call): yield calculation assuming bond is redeemed on call
date at call price
• YTW (yield to worst): worst return potential on a callable bond assuming
no default (i.e. on callable bonds the yield is calculated on all possible
call dates and the worst is taken as YTW)
Other measures of yield of a bond
18
3. Interest rate sensitivity
• Bond pricing for given coupon and maturities decreases in yield
• Sensitivity of bond price to y measures risk
Measures of interest rate sensitivity
20
Price
y
100
c
• Duration
• (Macaulay Duration)
• the weighted average of the times until those fixed cash flows are
received, where each weight is the proportion of the total price
represented by each cash flow at time t
• Modified Duration
Definion of Duration
21
• Starting from
• We can derive
• or
Measures of interest rate sensitivity
22
𝑃 =
𝑡=1
𝑇𝐶𝑡
1 + 𝑦 𝑡+
𝐹
1 + 𝑦 𝑇
𝜕𝑃
𝜕𝑦= -
1
(1+𝑦)σ𝑡=1𝑇 𝑡𝐶
𝑡
1+𝑦 𝑡 +𝑇𝐹
1+𝑦 𝑇
𝜕𝑃
𝜕𝑦= -
1
(1+𝑦)
1𝐶1
1+𝑦+
2𝐶2
1+𝑦 2+⋯+𝑇(𝐶+𝐹)
1+𝑦 𝑇
• We also notice that
• Where substituting the Modified Duration we have the following
sensitivity
Measures of interest rate sensitivity
23
Measures of rates sensitivity of a bond
24
Modified!
• Example 2: a 5% annual coupon bond is trading at par with a modified
duration of 2.639 and convexity of 9.57
• Assume yield rise from 5% to 7%, price will fall by
• Note that first order approx would be an overestimation of the price fall
Measures of rates sensitivity of a bond
25
∆P ≅ −MD ∗ 2% +CV
2∗ 2% 2 = -5.0866%
4. Exercises
Acquisition of a bond
• On 31 October, an investor acquires a bond with a nominal value of
€1.000, paying semi-annually a 3 1/4 % coupon (June 30th and
December 31st ; 30/360 convention) at 98,61 (settlement assumed on
same date), please calculate its total cash-out at the acquisition and the
cash-in due to the coupon expected on 31 December
• Accrued interests=3,25%*1000*(120/360)=10,833
• Price to pay=1000*98,61/100 +10,833=996,933
• Cash in= 1000*3,25%/2=16,25
27
Strictly Private and Confidential
Valuation of Zero Coupon Bonds
28
• Valuation is just based on NPV
F is the future value (say € 1,000)
r is the spot rate to time TP0 =
F
1 + r T
Maturity (years) 5
Price 0.797
Rate 4.64%
0.797 =1
1 + r0,55 ⟹ r0,5 =
1
0.79715
− 1 = 4.64%
P =1
1 + r0,𝑇T ⟹ r0,𝑇 =
1
P1T
− 1
• Calculate rates for each single price
• Solution:
Valuation of Zero Coupon Bonds - Exercise
29
Maturity (years) 1/4 1/2 1 2 3 4
Price 0.991 0.983 0.952 0.89 0.828 0.763
Rate
• Calculate rates for each single price
• Solution:
Valuation of Zero Coupon Bonds - Exercise
30
Maturity (years) 1/4 1/2 1 2 3 4
Price 0.991 0.983 0.952 0.89 0.828 0.763
Rate
Maturity (years) 1/4 1/2 1 2 3 4
Price 0.991 0.983 0.952 0.89 0.828 0.763
Rate 3,68% 3,49% 5% 6% 6.5% 7%
• Quote a rate for forward loan of 1 year in year 3
• Solution:
Deriving forward rates - Exercise
31
Maturity (years) 1/4 1/2 1 2 3 4
Price 0.991 0.983 0.952 0.89 0.828 0.763
Rate 3,68% 3,49% 5% 6% 6.5% 7%
1 + ft =(1 + r0,t)
t
(1 + r0,t−1)t−1 ft =
(1 + r0,t)t
(1 + r0,t−1)t−1 − 1
• Quote a rate for forward loan of 1 year in year 3
• Solution:
Deriving forward rates - Exercise
32
Maturity (years) 1/4 1/2 1 2 3 4
Price 0.991 0.983 0.952 0.89 0.828 0.763
Rate 3,68% 3,49% 5% 6% 6.5% 7%
1 + ft =(1 + r0,t)
t
(1 + r0,t−1)t−1 ft =
(1 + r0,t)t
(1 + r0,t−1)t−1 − 1
f4 =1 + r0,4
4
1 + r0,33 − 1 =
1.07 4
1.065 3 − 1 = 8.51%
Valuation of Coupon Bonds
33
• Long formula for coupon bond valuation
• Short formula
P0 =𝐶1
(1 + 𝑦)+
𝐶2(1 + 𝑦)2
+⋯+𝐶𝑇 + 𝐹𝑇(1 + 𝑦)𝑇
=
𝑡=1
𝑇𝐶𝐹𝑡
(1 + 𝑦)𝑡where CFt is year
t cash flow
𝑃0 = 𝐶
1 −1
1 + 𝑦 𝑇
𝑦+
𝐹
1 + 𝑦 𝑇
Valuation of Coupon Bonds
34
• For bonds paying coupons several times in a year (generally
semiannually or quarterly) price is calculated as (in a semiannual
example)
• and in the generic case of n coupon payments in the year and a T years
to maturity bond
=
𝑡=1
𝑇𝐶/2
(1 +𝑦2)𝑡+
𝐹
1 +𝑦2
2𝑇 =𝐶
𝑦1 −
1
1 +𝑦2
2𝑇 +𝐹
1 +𝑦2
2𝑇
𝑃0 =𝐶
𝑦1 −
1
1 +𝑦𝑛
𝑛𝑇 +𝐹
1 +𝑦𝑛
𝑛𝑇
• What is the price of a 5 year bond with coupon of 6% and € 1,000
nominal, in case of an yield to maturity of 5%?
• Please use the long formula and the short one
• Solution
Valuation of Coupon Bonds - Exercise
35
• What is the price of a 5 year bond with coupon of 6% and € 1,000
nominal, in case of an yield to maturity of 5%?
• Please use the long formula and the short one
• Solution
Valuation of Coupon Bonds - Exercise
36
P0 =60
5%1 −
1
1 + 5% 5+
1,000
1 + 5% 5= € 1043.295
P0 =60
(1 + 5%)+
60
(1 + 5%)2+⋯+
60 + 1000
1 + 5% 5= €1043.295
Valuation of Coupon Bonds - Exercise 2
37
• What is the price of US Treasury bond (semi-annual coupons) with a
$100 nominal, 4% coupon, 10 year maturity and a required yield of
4.048%?
Valuation of Coupon Bonds - Exercise 2
38
• What is the price of US Treasury bond (semi-annual coupons) with a
$100 nominal, 4% coupon, 10 year maturity and a required yield of
4.048%?
P0 =$4
0.040481 −
1
1 +12 0.04048
20 +$100
1 +12 (0.04048)
20
= 32.627 + 66.981 = 99.608$
∆𝑃
𝑃= −MD ∆𝑦 +
1
2CV ∆𝑦 2 + approx error
• Where MD is the Modified Duration and CV is convexity
• Example: a 5% annual coupon bond is trading at 102 with a modified
duration of 3.4 and convexity of 7.5
• Assume yield increase from 5% to 5.5%, what would be the price
change in %?
• Solution
Measures of rates sensitivity of a bond
39
∆𝑃
𝑃= −MD ∆𝑦 +
1
2CV ∆𝑦 2 + approx error
• Where MD is the Modified Duration and CV is convexity
• Example: a 5% annual coupon bond is trading at 102 with a modified
duration of 3.4 and convexity of 7.5
• Assume yield increase from 5% to 5.5%, what would be the price
change in %?
• Solution
Measures of rates sensitivity of a bond
40
∆P/P ≅ −3.4 ∗ 0.5% +7.5
2∗ 0,5% 2 = -1.691%
Bloomberg screen shot calculation
41
Investment grade issuer redemption
42
HY issuer redemption
43
Italian goverment redemption (BTP only)
44
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