page 1 january 2013 bond arithmetic adapted version with the permission of dr. gunther hahn, cfa...
TRANSCRIPT
Page 1January 2013
Bond Arithmetic
Adapted version with the permission of Dr. Gunther Hahn, CFA
Frankfurt, January 2013
Page 2January 2013
Overview
• Discounting and the time travelling machine
(compounding vs. discounting)
• Value of a Bond
(pricing formula)
• Special Bonds
(Zero Coupon, Consol, Floater)
• Price Quotation in the market
(Clean vs. Dirty Price, Day Count Conventions)
• Price Behaviour of bonds
(Discount vs. Premium Bond, Price vs. time, Price vs. yield)
Page 3January 2013
Overview II
• Yield Changes and Performance of Bonds
(Duration)
• A closer look at Duration
(Performance Approximation)
• McCauley Duration
(Average time, Price elasticity, Immunization)
Page 4January 2013
Literature
Bond Basic:Fabozzi, F. (1993): „Fixed Income Mathematics“, McGraw-Hill
Bonds and Yield Curves :Luenberger, D. (1998): „Investment Science“, Oxford, pp. 40 – 101
Bonds and xls examples:Benninga, S. (2008): „Financial Modelling“, 3rd edition, MIT press, pp. 669-717
Page 5January 2013
Discounting and the time travelling machine
• Assume you invest today 100€ at 10% interest.
Which amount can you expect after one year?
Amount + Interest
100 + 100 * 10% = 100 * (1 + 10%) = 110
• And after 2 years ?
Amount + Interest
100 * (1 + 10%) + 100 * (1 + 10%) * 10% = 100 * (1 + 10%)2 = 121
• And after n years ?
100 * (1 + 10%)n = Amount * (1 + interest)n
Page 6January 2013
• Now assume you receive 110€ in 1 year from today.
How much is this worth today, if the interest level is at 10% ?
Amount + Interest = 110
? + ? * 10% = ? * (1 + 10%) = 110
? = 110 / (1 + 10% ) = 100
• Assume you receive X € in n years. How much is this worth at y % interest?
Todays Value = X / (1 + y)n
Page 7January 2013
Value of a Bond
A Bond represents the right to receive future Cash Flows.
The Cash Flows consists out of Coupon and principal payment.
Today 1 st Coupon date
2 nd Coupon date
… Maturity
Pay for bond CouponPrincipal +
CouponCoupon
Page 8January 2013
Example: Assume you buy a 5% Bond for 80 € with a maturity of 4.3 years.
0 0.3 1.3
-80 1055
2.3 3.3 4.3
5 5 5
Page 9January 2013
Idea of Valuation: Each individual Cash Flow can be valued and aggregated to the total value !
0 0.3 1.3
-80 1055
2.3 3.3 4.3
5 5 5
5 / (1+ 10%)0.3 = 4,86
5 / (1+ 10%)1.3 = 4,42
5 / (1+ 10%)2.3 = 4,02
5 / (1+ 10%)3.3 = 3,65
105 / (1+ 10%)4.3 = 69,69
86,64
Page 10January 2013
Pricing Formula
The Value of the bond consists out of the sum of the individual values.
3.43.33.23.13.0 %101
105
%101
5
%101
5
%101
5
%101
5
Or in a more formal way.
T
tt
t
y
CFP
1
Notation
P Price (dirty) of Bond
T Time to maturity
t index
CFt Cash Flow at time t
y interest (yield) of bond
Page 11January 2013
Special Bonds
TyP
1
100
• British Consol
Bond that never matures. The Bond pays its coupon forever and needs to be bought back by the issuer in order to mature.
0 0.3 1.3
5
2.3 3.3 …
5 5 5 …
• Zero Coupon Bond
Bond that pays no Coupon. Only at maturity the principal is repaid.
0 0.3 1.3
0
2.3 Maturity
0 0 100
y
CouponP
Page 12January 2013
Special Bonds II• Floater
Bond that pays a floating rate (on a quartely basis) depending on the level of the interest rate. At the beginning of the period the rate is observed and at the end the rate is paid and the new rate is
observed.
0 0.25 0.5
X1=3 Month-Rate
Maturity
100 + Xn / 4X1 / 4 is paid
X2=3 Month-Rate
X2 / 4 is paid
X3=3 Month-Rate
...
...
resetnexttotimei
y
XP
1
4/100
On each of the reset days the value of the floater is 100.
The idea behind this logic is that the cash flow from a floater can be duplicated easily. On each of the reset days a fixed term deposit for 3 Month earning the 3 Month-Rate is opened. At the end of the period the 3 Month-Rate is earned and the 100 are recieved back.
resetnexttotimei
y
XP
1
4/100
resetnexttotimei
y
XP
1
4/100
resetnexttotimei
y
XP
1
4/100
On each of the reset days the value of the floater is 100.
The idea behind this logic is that the cash flow from a floater can be duplicated easily. On each of the reset days a fixed term deposit for 3 Month earning the 3 Month-Rate is opened. At the end of the period the 3 Month-Rate is earned and the 100 are recieved back.
Page 13January 2013
Price Quotation in the market
• So far the valuation was equal to the amount which needs to be paid. This amount is called the dirty price.
• The price which is quoted on Bloomberg or in the newspaper is the clean price of the bond, which accounts for the accrued interest.
Dirty Price = Clean Price + Accrued Interest
Next coupon
payment
CF
Last coupon
payment
CF
today
Accrued Interest = (1-t) * CF
t1 - t
Page 14January 2013
Example: Assume you buy a 8% Coupon Bond with 4.25 years to maturity. The clean Price is 90€. How much do you pay to receive the Bond?
Clean Price 90 €
Accrued Interest 6 € (1 – 0.25) * 8
Dirty Price 96 €
Page 15January 2013
Example Accrued Interest
accrued
interest
Page 16January 2013
Example Accrued Interest continued
Clean Price 1.000 € * 103.19% = 1031.90
Accrued Interest (230 Days) 1.000 € * 7.125% *(230+1)/365 = 45.09
Dirty Price (107.699%) 1076.99 €
Next coupon
payment
20.04.2012
Last coupon
payment
20.04.2011
today
6.12.2011
230 Days230 Days230 Days
Page 17January 2013
Day Count Conventions
The difference between two dates can be calculated according to different market standards.
• Actual / Actual : real Number of days are counted.
• Actual / 365 : real Number of days are counted; the number of days in a year is counted as 365 (even if it is a leap year).
• Actual / 360 : real Number of days are counted;the number of days in a year is counted as 360.
• 30 / 360 : every month is counted as 30 days and every year as 360 days;- If the period starts on the 31st then the start is moved on the 30th- If the period ends on the 31st then the end is moved on the 1st- If the period ends on the 31st and starts on the 31st then the end is moved on the 30th.
• 30 E / 360 : every month is counted as 30 days and every year as 360 days;- If the period starts on the 31st then the start is moved on the 30th- If the period ends on the 31st then the end is moved on the 30th.
Page 18January 2013
Example Day Count Conventions
Page 19January 2013
Example: Pricing of a Bond
7.12.2011
7,125 7,125 107,1257,1257,125
20.04.2012 20.04.2013 20.04.2014 20.04.2015 7.12.2016
-107,699
tt
t
y
CFAccruedP
1
Page 20January 2013
Discount vs. Premium Bonds
• Discount Bond
Bond which a coupon rate below the market interest rate. Consequently the Price of the bond is cheaper than 100.
Page 21January 2013
• Premium Bond
Bond which a coupon rate above the market interest rate. Consequently the Price of the bond is greater than 100.
Page 22January 2013
Yield Changes and Performance of Bonds
The following picture shows how the dirty price changes if we vary the market interest rate.
Dirty Price
Interest Rate
Page 23January 2013
In order to compute the price change approximately, we calculate the first derivative of the dirty price function. Using the derivative we can approximate the change in price.
T
t
tt
T
tt
t yCFy
CFP 1
1
• We start with the pricing function …
• And calculate the first derivative with respect to the interest rate y.
T
tttt
T
tt yy
CFtytCF
y
P
1
1
11 1
T
ttt
y
CFt
yPyP
P
11
11
• Changing to percentage change in Price gives:
Page 24January 2013
Using the modified Duration we can approximate percentage price change.
Dirty Price (P)
Interest Rate (y)
T
ttt
y
CFt
yPD
yP
P
11
11mod
Current Interest
yDPP
mod
Page 25January 2013
Example: Assume you have a bond with a modified duration of 6. The dirty price is 120€. Suddenly the yield decreases from 4% to 3.5%. Will you gain or loose? How much is the percentage change in price and absolute change?
• Since the yield decreases the price of the bond will increase. This way investors are compensated for a lower yield level.
• Percentage change in dirty price = - modified Duration * change in yield
Percentage change in dirty price = - 6 * -0,5% = 3%
• Absolute change in dirty price = 3% * 120€ = 3,6€
The Price will increase approximately from 120€ to 123,6€.
Page 26January 2013
A closer look at Duration
Using the modified Duration and yield curve we can approximate the Performance of a bond over a period of time.
yDtyePerformanc mod
Interest Rate / yield curve
Time to MaturityTodayToday - Δt
Δt
Δy
Page 27January 2013
Example: Assume you hold a bond for half a year. When you buy the bond, the Duration was 6 and the yield 3%. At the end of the period the yield increased to 3.5%. Which approximate Performance did you earn?
• The formula gives:
Performance = 3% * 0,5 - 6 * 0,5% = 1,5% - 3% = -1,5%
Page 28January 2013
McCauley Duration
Besides the modified Duration, the McCauley Duration is often used as well. For its computation we start with the modified Duration:
T
ttt
y
CFt
yPyP
PD
11
11mod
Now we multiply both sides with (1+y) to obtain the McCauley Duration:
yD
y
CFt
Py
yP
PD
T
tt
tMcCauley
1
1
1
1mod
The McCauley Duration represents the percentage price change over the percentage yield change. So the McCauley Duration is an elasticity
(% Change / % Change).
Page 29January 2013
McCauley Duration – Calculation Example
The following table is helpful to calculate the McCauley Duration:
yD
y
CFt
Py
yP
PD
T
tt
tMcCauley
1
1
1
1mod
Page 30January 2013
McCauley Duration – Interpretation
The McCauley Duration has various interpretations (average time, price elasticity etc.).
• Average Time to maturity (balances discounted Cash Flows)
t
T
tt
T
t
tt
McCauley wtPy
CF
tD1
Disc. CF
• Price elasticity (can be used to calculate percental price changes)
yyP
PDMcCauley
1
If yields rise from 5% to 6% the denominator is not 1%,
but 1%/1.05 = 0.95%. For this reason of complexity
modified duration is more often used.yyP
PDMcCauley
1