Structured Finance Course

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


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%


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


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


12

http://www.benetzkorn.com/wp-content/uploads/2011/08/YC-PLot.jpg


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


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


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


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


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


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


Price 0.991 0.983 0.952 0.89 0.828 0.763

Rate


Price 0.991 0.983 0.952 0.89 0.828 0.763

Rate 3,68% 3,49% 5% 6% 6.5% 7%


• Solution:

Deriving forward rates - Exercise

31


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


• Solution:

Deriving forward rates - Exercise

32


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%


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


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



4.048%?

Valuation of Coupon Bonds - Exercise 2

38



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


• Assume yield increase from 5% to 5.5%, what would be the price

change in %?

• Solution


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


• Assume yield increase from 5% to 5.5%, what would be the price

change in %?

• Solution


40

∆P/P ≅ −3.4 ∗ 0.5% +7.5

2∗ 0,5% 2 = -1.691%

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41

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42

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43

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44

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