Date & Time Handling and the Yield Curves Mathematics & Excel Modeling

Joerg Hoerster Dr. Jan Rudl London, 2010

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Day count convention determines how interest accrues over time MOTIVATION FOR DAY COUNT CONVENTIONS

2010

2022

Sep. 14

Sep. 27 Time

Begin of an investment

interest payment date

How much interest should be transferred to the payment date

decided by day account convention!

What is the difference in years? Source: Maravon

1

There are three typical day count convention methods to calculate date differences DAY COUNT CONVENTION METHODS

Typical methods

30/360

Source: Maravon

Actual

Business day

2

30/360 method assumes 30 days per month and 360 days per year GENERAL DESCRIPTION

(d1 · m1 · y1 , d2 · m2 · y2 ) =?

= y2

Source: Maravon

y1 + (m2

m1 )/12 + (d2

d1 )/360

3

There are three typical variants of the 30/360 method METHOD VARIANTS

Rules

30E/360 (European)

30I/360 (Italian)

30U/360 (US)

•  If d1=31, change d1 to 30

•  The same as 30E/360 and additionally,

•  If d1=31, change d1 to 30

•  If d2=31, change d2 to 30

Source: Maravon

•  if m1=2 and d1=28 or d1=29, change d1 to 30, and the same for m2 and d2

•  If d2=31 and d1≥30, change d2 to 30

4

Actual method uses the actual number of days, but the number of days per year can differ GENERAL DESCRIPTION

(d1 · m1 · y1 , d2 · m2 · y2 ) =?

Days between(d1 · m1 · y1 , d2 · m2 · y2 ) = Days per year

Source: Maravon

5

There are three basic variants for actual method, which differ in number of days per year

Days per year

ACT/360

ACT/365

ACT/ACT

360

365

365 or 366

Depending on whether y1 or y2 is a leap year or Feb 29 is covered by (d1⋅m1⋅y1, d2⋅m2⋅y2) Source: Maravon

6

Business day method uses the actual number of business days, where weekends and holidays are excluded GENERAL DESCRIPTION

(d1 · m1 · y1 , d2 · m2 · y2 ) =?

=

Business days between(d1 · m1 · y1 , d2 · m2 · y2 ) Business days per year

Basic rules: ➞  business days per year = 250 or 252 (mostly) ➞  weekends differ from country to country, e.g. United Arab Emirates, South Korea etc. ➞  holidays differ extremely from country to country

Source: Maravon

7

Business day method requires day rolling conventions as the special handling of payment dates, which are non-business days DAY ROLLING CONVENTIONS

FOLLOWING

PREVIOUS

payment date

payment date

→ next business day

→ previous business day

•  If next business day is in the next month → PREVIOUS •  Otherwise → FOLLOWING

Day rolling conventions for payment dates as non-business days

•  If the previous business day is in the previous month → FOLLOWING •  Otherwise → PREVIOUS

MODIFIED FOLLOWING Source: Maravon

MODIFIED PREVIOUS 8

Understanding Day Count xls Conventions

9

Interests can be calculated in various ways

Variants of interest calculation

Simple interest

Source: Maravon

Compound interest

Compound interest with several compounding periods per year

Continuous interest

10

In the following we use four mathematical symbols for interest calculation

Source: Maravon

Symbol

Description

i

Interest rate

P

Principal/invested amount of money at time 0

n

Number of years

Vn

Value of P after n years

11

Using simple interest, the annual interests stay the same EXAMPLE

Value Vn is growing linearly with i=10% P=100

Vn = P · (1 + n · i)

Calculation of simple interests n annual interest Vn

Source: Maravon

1

2

3

…

10

10

10

10

…

10

110

120

130

…

200

12

Using compounding interest, the interests are also compounded EXAMPLE

Value Vn is growing expotentially

Vn = P · (1 + i)n

i=10% P=100

Calculation of compound interests n annual interest Vn

Source: Maravon

1

2

3

…

10

10

11=110 ⋅10%

12.1=121⋅10%

…

…

110

121

133.1

… 100 ⋅(1+10%)10=259.37

13

There can also be several annual compounding periods EXAMPLE

i m·n Vn = P · (1 + ) m

i=10% P=100 m=2

For fraction of years, day count conventions are used

Compounding frequency per year

Calculation of compound interests with annual compounding periods n annual interest Vn Source: Maravon

0.5

…

10

10% 10% 5 5.25=105 ⋅10% 5.25=110.25 ⋅10% … 2 2

10

105

1

110.76

1.5

115.76

… 100 ⋅(1+

10% 10*2 ) =265.33 2 14

If the number of compounding periods become more and more, we end up with the continuous compounding

EXAMPLE

m→ ∞ Compounding m times a year

" i% Vn = P ⋅ $ 1+ ' # m&

Compounding m → ∞ times a year If the number of compounding periods is ∞, how valuable is an investment of 1 Euro after 50 years

m⋅n

Vn = P ⋅ en⋅i

e: Euler's number: 2.71828…

Source: Maravon

15

The effective interest rate allows to compare different compounding methods MOTIVATION OF EFFECTIVE INTEREST RATE

How much should the annual compounding interest rate be, in order to achieve the same interest

Semi-annual compounding

n

Continuous compounding

n

Vn

Vn

1

2

3 …

10

110.25 121.55 134.01 … 265.33

1

2

3 …

10

110.52 122.14 134.99 … 271.83

Effective interest rate

Source: Maravon

16

Effective interest rate calculation is adapted for multi-annual and continuous compounding CALCULATION OF EFFECTIVE INTEREST RATE

Multi-annual Compounding

i: nominal interest rate ie: effective interest rate

i m·n ) = P · (1 + ie )n m i ! 1 + ie = (1 + )m m i m ! ie = (1 + ) 1 m

Vn = P · (1 +

Source: Maravon

Continuous Compounding

Vn = P · en·i = P · (1 + ie )n ! en·i = (1 + ie )n ! ei = (1 + ie ) ! ie = e i 1

17

Types of interest rates involved in the fixed-income jargon

Taxonomy of Rates

Coupon Rate and current yield

Source: Maravon

Yield to Maturity

Spot Zero-Coupon Rate

Forward Rates

Bond Par Yield

18

Coupon rate and current yield

The coupon rate is the stated interest rate on a security, referred to as an annual percentage of face value. It’s usually paid •  Semi-annually (e.g. in the US), or •  Annually (e.g. in France and Germany)

Let c denote the coupon rate, N the nominal value and P the current price of a bond. The current yield Yc is obtained from

Yc =

Example •  How much is the current yield of a bond with face value $1000, an annual coupon rate of 7% and a current price of $900?

Yc =

7% × 1000 = 7.78% 900

•  Note the coupon rate of 7% does not change in any event.

c×N P

Source: Maravon

19

Yield to Maturity

The yield to maturity (YTM) is the single rate that sets the present value of the cash flows equal to the bond price. Semi-annual payment of coupon 2T

P=∑

t =1

Where •  P: •  T: •  CFt: •  Y2: •  Y:

Source: Maravon

CFt Y ⎞ ⎛ ⎜1 + 2 ⎟ 2 ⎠ ⎝

Annual payment of coupon T

t

Po = ∑

Price of a bond Maturity Cash flow at the date t YTM on a semi-annual basis YTM on an annual basis

t =1

CFt (1+ Y)t

Example Consider a $1000 face value 3-year bond with 10% annual coupon, wich sells for $1010. The YTM Y of this bond is solved from

1010 =

100 100 1000 + 100 + + 1 + Y (1 + Y )2 (1 + Y )3

Y = 9.601 %

Remark Note the one-to-one correspondence between the price and the YTM of a bond. Therefore, bonds are often quoted in YTM.

20

Spot Zero-Coupon (or Discount) Rate

Example •  Let B(0,t) denote the market price at time 0 of a bond paying off $1 at date t. •  Let R(0,t) denote the spot zero-coupon rate that is implicitly defined by

B(0, t ) =

1 [1 + R(0, t )] t

•  In practice, if the spot zero-coupon yield curve t → R (0,t) and the future cash flows are known, the spot prices for all fixed-income securities can be derived.

Source: Maravon

Consider a 2-year zero-coupon bond that trades at $92. How much is the 2-year zero coupon rate R(0,2)?

92 =

100 [1 + R(0,2)]2

R(0,2) =

100 − 1 = 4.26% 92

21

Forward Rates

Definition

Characteristics

Let R(0,t) denote the spot zero-coupon rate. An implied forward rate F(0,x,y-x) (forward zero-coupon rate) between years x and y is defined as y

" [1 + R(0, y)] % F(0, x, y − x) = $ # [1 + R(0, x)]x '&

1 y− x

−1

that is the forward rate as seen from date t=0, starting at date t=x, and with residual maturity y-x.

Source: Maravon

F(0,x,y-x) is a rate

•  that can be guaranteed on a transaction occurring in the future (compare the example later on) •  that can be viewed as a break-even point that equalizes the rate of return on bonds across the entire maturity spectrum

22

Examples: Forward rates as a rate that can be guaranteed now on a transaction occurring in the future We simultaneously borrow and lend $1 repayable at the end of 2 years and 1 year, respectively. The cash flows generated by this transaction are as follows: Today Borrow

In 1 Year

1

In 2 Years - [1+R(0,2)]2

Lend

-1

[1+R(0,1)]1

Total

0

[1+R(0,1)]

- [1+R(0,2)]2

Borrowing in 1 years repayable in 2 years at the amount of

F(0,1,1) is the rate that can be guaranteed now for a loan starting in 1 year and repayable after 2 years with [1 + R(0, 2)]2 F (0, 1, 1) = 1 [1 + R(0, 1)] Source: Maravon

23

Instantaneous forward rate: a particular forward rate

Definition Recall the forward rate F(t,s,T-s) as seen from date t between years s and T. The instantaneous forward rate f(t,s) is defined as

f (t, s) = lim F (t, s,T − s) T −s→0

f(t,s) is the forward rate seen at date t, starting at date s and maturing an infinitely small instant later on.

Source: Maravon

Characteristics •  f(t,s) is a continuously compounded rate •  f(t,t)=R(t) is the short-term interest rate at date t that is the rate with a 1day maturity in the market. •  An instantaneous forward yield curve can be derived by making the date s vary between 1 day und 30 years. •  In practice, the market treats the instantaneous forward rate as a forward rate with a maturity of between 1 day and 3 months.

24

Bond par yield

YTM curve suffers from the coupon effect

•  Two bonds with the same maturity but different coupon rates do not necessarily have the same YTM.

•  A par bond is a bond with a coupon rate identical to its YTM. The bond price is therefore equal to its principal •  Let c(n) denote the par yield so that a n-year maturity fixed bond paying annually a coupon rate of c(n) with a $100 face value quotes par Bond par yield curve is the solution to overcome this coupon effects

100 × c(n) 100 × c(n) 100 + 100 × c(n) + + ... + = 100 1 2 [1 + R(0,1)] [1 + R(0, 2)] [1 + R(0, n)]n 1 [1 + R(0, n)]n c(n) = n 1 ∑ [1 + R(0,i)]i i=1 1−

•  Par yield curve n → c(n) is obtained with known zero-coupon rates R(0,1), R(0,2), …, R(0,n).

Source: Maravon

25

Definition and types of interest rates term structure •  Yield to maturity curve at date t with maturity Θ

•  The term structure of interest rates, called the yield curve, is the graph mapping interest rates corresponding to their respective maturity. •  The nature of interest rates determine the nature of the term structure.

Market curves that are derived from market data

Θ ↦ y(Θ) •  Swap rate curve at date t with maturity Θ Θ ↦ SR(Θ) •  Zero-coupon yield curve at date t with maturity Θ Θ ↦ R(t,Θ)

Implied curves that are constructed implicitly using market data

•  Par yield curve with maturity Θ Θ ↦ c(Θ) •  Forward rate curve at date t starting at date s with residual maturity Θ Θ ↦ F(t,s,Θ) •  Instantaneous forward rate at date t, starting at date s with infinitesimal maturity s ↦ f(t,s)

Source: Maravon

26

Properties of the term structure Not negative

•  Real interest rates may become negative •  Nominal rates cannot be negative •  Interest rates can neither be negative nor be assumed to have normal distribution.

Mean-reverting behavior

•  dr(t)= a[b-r(t)]dt+σdW(t) •  When rates reach high levels, they subsequently tend to decline rather than rise still further.

Not perfectly correlated changes

•  Correlation coefficients between interest rate movements are not equal to 1, but converge to 1 for short maturities.

Higher volatility of short-term rates than of long-term rates

•  The term structure of volatility is a function that is –  Decreasing for longer maturities and –  Increasing for maturities until 1 year, which we call the humped form.

More than 95% of yield curves movements is explained by three main factors

•  Principal Components Analysis (PCA): interest rate variation is mostly determined by –  The parallel movement component –  The slope oscillation component and –  curvature component

Source: Maravon

27

An overview of deriving the zero-coupon yield curve

Practical importance

Types of a zero-coupon yield curve to be derived

Source: Maravon

Deriving the zero-coupon yield curve enables investors at date t •  to know the discount factor curve and consequently •  to price at date t any fixed-income securities delivering known cash flows in the future, and •  to obtain implicit curves such as the forward rate curve beginning at T>t, the instantaneous forward rate curve and the par yield curve

•  Non default Treasury zero-coupon yield curve •  Interbank zero-coupon yield curve •  Credit spread zero-coupon yield curve

28

Deriving the non-default Treasury zero-coupon yield curve: a general approach •  Selection of a coherent basket of bonds •  Generation of a database providing information of each bond Input

Methodologies in the “black box”

Methods deriving the zero-coupon yield curve

Theoretical

Practical •  The bootstrapping method •  Alternative indirect methods

•  Direct methods

Output •  A zero-coupon yield curve

Source: Maravon

t à R(0,t)

29

Direct method is a standard methodology used to exact zero-coupon implied prices from the coupon bond market prices •  An n-dimensional vector of coupon bond prices at time t: Collect prices of n coupon (or zero-coupon) bonds delivering corresponding cash flows (coupons and principal)

(

Pt = Pt1 ,...,Ptj ,...,Ptn

)

T

•  An n x n-matrix of the cash flows:

( )

F = Fti(j)

i = 1, …, n j = 1, …, n

under the assumption that all bonds have the same cash flow dates ti Determine the zero-coupon bond prices at time t under the assumption of non-arbitrage. Furthermore, assume the matrix F is invertible

•  An n-dimensional vector of the zero-coupon bond prices (discount factors) at time t: T

Bt = ( B(t,t1),..., B(t,tn))

n

–  Due to:

j

Pt = ∑ Fti(j)B(t,ti) ⇔ Pt = F × Bt i=1

–  We have: Exacting the annual compounded zero-coupon rate R(t,ti-t) at time t with maturity ti-t and the continuously compounded equivalent Rc(t, ti-t)

Source: Maravon

•  R(t, ti-t) =

e

•  Rc(t, ti-t) = −

B t = F −1 × Pt −

1 ln [ B(t,ti)] ti− t

−1

1 ln [ B(t,ti)] ti − t

30

Example: deriving the zero-coupon curve until the 4-year maturity

Basket of four bonds from the market Annual coupon Bond 1 Bond 2

5.0 5.5

Exercises

Maturity 1 2

Price P01 = 101 P02 = 101.5

Bond 3

5.0

3

P03 = 99

Bond 4

6.0

4

P04 = 100

Source: Maravon

•  Write the matrix equations for the four bond prices using Pt = F × Bt •  Calculate B(0,1), B(0,2), B(0,3), B(0,4) •  Find the zero-coupon rates R(0,1), R(0,2), R(0,3), R(0,4)

31

Key to the preceding exercises

! 101 $ ! 105 # 101.5 & # 5.5 105.5 & =# 1 # 99 5 105 # & # 5 #" 100 &% #" 6 6 6 106

$ ! & # & ×# & # &% # "

B(0,1) B(0,2) B(0,3) B(0,4)

$ & & & & %

−1

2

3

Source: Maravon

⎛ B(0,1) ⎞ ⎡⎛105 ⎞⎤ ⎛ 101 ⎞ ⎛ 0.9619 ⎞ ⎜ ⎟ ⎢⎜ ⎟⎥ ⎜ ⎟ ⎜ ⎟ B(0,2) 5.5 105.5 101.5 0.91194 ⎜ ⎟ ⎢⎜ ⎟⎥ ⎜ ⎟ ⎜ ⎟ = × = ⎜ B(0,3) ⎟ ⎢⎜ 5 ⎟⎥ ⎜ 99 ⎟ ⎜ 0.85363 ⎟ 5 105 ⎜ ⎟ ⎜ ⎟⎥ ⎜ ⎟ ⎜ ⎟ ⎜ B(0,4) ⎟ ⎢⎜ 6 ⎟ ⎜ 100 ⎟ ⎜ 0.78901 ⎟ 6 6 106 ⎝ ⎠ ⎢⎣⎝ ⎠⎥⎦ ⎝ ⎠ ⎝ ⎠

⎛ -11ln[B(0,1) ] ⎞ − 1⎟ ⎛ 3.96% ⎞ ⎛ R(0,1) ⎞ ⎜ e ⎜ ⎟ ⎜ - 1ln[B(0,2) ] ⎟ ⎜ ⎟ 2 R(0,2) 4.717% ⎜ ⎟ ⎜ e ⎟ − 1⎟ = ⎜ = 1 ⎜ R(0,3) ⎟ ⎜ - ln[B(0,3) ] ⎟ ⎜ 5.417% ⎟ − 1⎟ ⎜ ⎜ ⎟ ⎜ e 3 ⎟ ⎜ R(0,4) ⎟ ⎜ 6.103% ⎟ 1 ln [ B(0,4) ] ⎟ ⎝ ⎝ ⎠ ⎜ 4 ⎠ e − 1 ⎝ ⎠

32

The Bootstrapping Method: a general description

Motivation form the practice

•  The direct method is under very limited circumstance, where finding many distinct linearly independent bonds with the same coupon dates is quasi-impossible in practice •  For real-situation the bootstrapping method is preferred as a common approach

Definition

•  Bootstrapping is the term for generating a zero-coupon yield curve from existing market data such as bond prices

Method Description

•  Bootstrapping can be viewed as a repetitive double-step procedure that we illustrate later on using an example, where interpolation is a useful component for bootstrapping

Source: Maravon

33

Interpolation is the key component for bootstrapping method Linear Interpolation

Cubic Interpolation •  We need four zero-coupon rates R(0,v), R(0,x), R(0,y), R(0,z) to implement the cubic interpolation with v < x < y < z. The to be interpolated rate R(0,w) with v