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Bonds

Interest - Discount Calculations

Accrued Interest

todo short description

Accrued Interest = P * R/M * A/D

Where:

A - Number of accrued days counted according to the day count basis.
D - Number of days in the period
M - number of periods per year
P - Par Value
R - annnula interest rate

Algorithm

def getAccrued(P, R, M, A, D):
        return P*(1.0 *R/M)*(1.0*A/D)

Example

What is the accrued interest of the following bond?
Par Value = 5 000 000
Rate = 13.6% (semi-annual)
Day count basis = 30/360
Settlement Date = 11/09/2001
First Interest Date = 09/23/2001

getAccrued(5000000, 13.6/100.0, 2, 46, 360)

>>>> 43444.4444444

Short Coupon

todo description

Accrued Interest = P * R/M*A/NL

Where:[[BR]]

A - Accrued days
M - Number of coupon periods per year
NL - Number of days representing the quasi-coupon period
P - Par Value
R - Annual coupon rate in decimal format

Algoirthm

def getAccruedShortCoupon(P, R, M, A, NL):
        return P*(1.0 *R/M)*(1.0*A/NL)

Example

A city issued $10 000 000 bonds on August 15. The bonds are dated June 1 and were setteled on September 15. The interest rate on the bonds is 7% paid semmi-annual. Assume 30/360 basis.

getAccruedShortCoupon(10000000, .07, 2, 104, 180)

>>>> 202222.222222

(taken from http://www.msrb.org/msrb1/glossary/view_def.asp?param=ACCRUEDINTEREST )

Long Coupon

*todo description
Accrued Interest = P * R/M*(sum{i=1,NC}[A_i/NL_i])

Where:

A_i - Number of accrued days for the i_th quasi-coupon period within odd period
M - Number of coupon periods per year
NC - Number of quasi-coupon periods that fit in add period. If the number contains a fractional part, raise it the next whole number
NL_i - Length of the quasi-coupon periods within odd perid (in days)
P - Par Value
R - Annual interest rate (in decimal format)

Algorithm

def getAccruedLongCoupon(P, R, M, A, NL, NC):
        sum = 0.0
        i = 1
        
        while i <= NC:
                sum+= 1.0*  A[i-1] / NL[i-1]
                i+=1
        
        return P*(1.0 *R/M)*sum

Example

todo get new example

Par Value: 10 000
Rate: 7.5 (semi-annual)
issue Date: 07/01/92
First Interest Date: 04/01/93
Settlement Date: 02/01/93

M = 2
A1 = 123 (10/01/92 - 02/01/93)
A1 = 92 (07/01/92 - 10/01/92)
Nc = 2 (10/01/92 - 04/01/93), (04/01/92 - 10/01/92)
NL1 = 182 (10/01/92 - 04/01/93) NL2 = 183 (04/01/92 - 10/01/92)

getAccruedLongCoupon(10000, .075, 2,[123, 92], [182,183], 2)

>>>>441.958656098

Zero Coupon Prices and the Money Multipliers

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Using Zeros to Value Cash Flows

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

No Arbitrage

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Bond Price Dynamics

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Bond Valuation

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Yield to Maturity or Internal Rate of Return

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Bond Rankings and Interest Rates

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Bond Price Movements and Duration

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Duration -> DurationDraft

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Modified Duration

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Bond convexity

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Odd First Coupons

Typical Uses

todo

Ingredients

todo

Price (given yield) with an odd short first coupon

P = [ RV/(1 + Y/M)^(N-1+DSC/E)] +

[(100 * R/M * DFC/E)/(1 + Y/M)^DSC/E] +

[E{k=2;N} (100 * R/M) / (1 + Y/M)^(K-1+DSC/E)] -

[100*R/M*A/E]

Where:

A - Number of days from begining of coupon period to settlement date (accured days)
DFC - Number of days from the begining of the odd first coupon period (dated date) to the first coupon date
DSC - Number of days from settlement date to first coupon date
E - Numberof days in quasi-coupon period in which the settlement date falls
K - A summation counter
M - Number of coupon periods per year (standard for the particular security involved)
N - Number of coupons payable between settlement date and redemption date (maturity date, call date, put date, etc.) (If this number contains a fractional part, raise it to the next whole number - ie 2.5 = 3)
P - Dollar price per $100 pae value
R - Annual interest rate (as a decimal)
RV - Redemption value of the security per $100 par value (RV = 100, except in those cases where call or put features must be considered)
Y - Annual yield ( as a decimal) on investment with security held to redemption
(Note: This formula does not apply to municipal securities)

Algorithm

def getOddShortFirstCoupon(A, DFC, DSC, E, M, N, R, RV, Y):

    coefficient = 1.0 + 1.0 * Y/M

    term1 = 1.0 * RV/(coefficient ** (N - 1 + (1.0 * DSC/E)))

    term2 = (100.0 * (1.0 * R/M) * (1.0 * DFC/E))/coefficient ** (1.0 * DSC/E)

    term3 = 0
    K = 2

    while K <= N:
        
        term3 += (100.0 * R/M) / coefficient ** (K - 1 + (1.0 * DSC/E))
        K += 1
        
    term4 = 100.0 * (1.0 * R/M) * (1.0 * A/E)

    return term1 + term2 + term3 - term4

Example

-> replace example

Calculate a dollar price given a yield for a Treasury Bond with an odd short first coupon

Settlement Date 11/11/92
Maturity Date 03/01/05
Issue / Dated Date 10/15/92
First Coupon Date 03/01/93
Day Count Basis Actual / Actual
A = 27 (10/15/92 - 11/11/92)
DFC = 137 (10/15/92 - 03/01/93)
DSC = 110 (11/11/92 - 03/01/93)
E = 181 (09/01/92 - 03/01/93) (based on the quasi-coupon date of 9/1/92)
M = 2 (semi-annual)
N = 25 (11/11/92 - 03/01/05)
R = 0.0785 (7.85%)
RV = 100.000
Y = 0.0625 (6.25%)
P = 113.597717
DOLLAR PRICE = 113.597717

Price (given yield) with an odd long first coupon

P = [ RV/(1 + Y/M)^{(N+Nqf+DSC/E)}] +

[(100 * R/M * [E{i=1;NCF} (DFC_i/NLF_i)])/(1 + Y/M)^Nqf+DSC/E] +

[E{k=1;N} ((100 * R/M) / (1 + Y/M)^{(K+Nqf+DSC/E)})] -

[100*R/M*[E{i=1;NCF} (A_i/NLF_i)]]

Where:

A_i - Number of occured days for the i^th quasi-coupon period within the odd period (accured days)
DFC_i - Number of days from dated date to the first quasi-coupon (i=1) or number of days in quasi-coupon (i=2,3,...,NCF)
NLF_i - Normal length in days of the full i^th quasi-coupon period within odd period
DSC - Number of days from settlement date to the next quasi-coupon date or next coupon date
E - Number of days in quasi-coupon period, or regular coupon period, in which the settlement date falls
K - A summation counter
M - Number of coupon periods per year
N - Number of coupons payable between ''the first real coupon date'' and redemption date (maturity date, call date, put date, etc.) (If this number contains a fractional part, raise it to the next whole number - ie 2.5 = 3)
NCF - Number of quasi-coupon periods that fit in odd period. (If this number contains a fractional part, raise it to the next whole number.)
Nqf - Number of ''whole'' quasi-coupon periods between settlement date and first coupon (this may be zero)
P - Dollar price per $100 pae value
R - Annual interest rate (as a decimal)
RV - Redemption value of the security per $100 par value (RV = 100, except in those cases where call or put features must be considered)
Y - Annual yield ( as a decimal) on investment with security held to redemption

(Note: This formula does not apply to municipal securities)

Algorithm

def getOddLongFirstCoupon(A, DFC, DSC, E, M, N, NCF,  NLF, Nqf, R, RV, Y):

    coefficient = 1.0 + 1.0 * Y/M
    K = 1
    term3 = 0.0
    sumAi = 0.0
    sumDFCi = 0.0
    
    while K <= N :
        term3 += 100.0 * R/M  / (coefficient **(Nqf + K + 1.0 * DSC/E))
        K += 1

    i = 1
    while i <= NCF :
        sumAi += 1.0 * A[i-1]/NLF[i-1]
        sumDFCi += 1.0 * DFC[i-1]/NLF[i-1]
        i += 1
        
    term1 = RV/(coefficient **(Nqf + N + 1.0 * DSC/E))
    term2 = 100.0 * (R/M) * sumDFCi / (coefficient **(Nqf + 1.0 * DSC/E))
    term4 = 100.0 * R/M * sumAi
    
    return term1 + term2 + term3 - term4

Example

-> replace example

Calculate a dollar price given a yield for a Treasury Bond with an odd short first coupon

Settlement Date 11/11/92
Maturity Date 03/01/05
Issue / Dated Date 10/15/92
First Coupon Date 03/01/93
Day Count Basis Actual / Actual
A1 = 78 (6/15/92 - 9/1/92)(based on the quasi-coupon date of 3/1/92 and 9/1/92)
A2 = 71 (9/1/92 - 11/11/92)(based on the quasi-coupon date of 3/1/92 and 9/1/92)
DFC1 = 78 (6/15/92 - 9/1/92)(based on the quasi-coupon date of 3/1/92 and 9/1/92)
DFC2 = 181 (9/1/92 - 3/1/93)(based on the quasi-coupon date of 3/1/92 and 9/1/92)
DSC = 110 (11/11/92 - 03/01/93)
E = 181 (09/01/92 - 03/01/93) (based on the quasi-coupon date of 3/1/92 and 9/1/92)
M = 2 (semi-annual)
N = 24 (3/1/93 - 03/01/05)
NCF - 2 (6/15/92 - 3/1/93)
NLF1 - 184 (3/1/92 - 9/1/92) (based on the quasi-coupon date of 3/1/92 and 9/1/92)
NLF2 - 181 (9/1/92 - 3/1/93) (based on the quasi-coupon date of 3/1/92 and 9/1/92)
Nqf - 0 (11/11/92 - 3/1/93)
R = 0.0935 (9.35%)
RV = 100.000
Y = 0.0775 (7.75%)

getOddLongFirstCoupon([78,71], [78,181], 110, 181, 2, 24, 2, [184,181], 0, 0.0935, 100.000, 0.0775)
>>>> 112.478106

References

Odd Last Coupons

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Odd First and Last Coupons

Odd Short First and Odd Long Last Coupons

Typical Uses

todo

Ingredients

P =
[RV/(1+Y/M)^{(N-1+DSC/E+Nql+DLQ/LQL)}] +
[(100*R/M*DFC/E)/(1+Y/M)^DSC/E] +
[sum{K=2,N}[(100*R/M)/(1+Y/M)^K-1+DSC/E]] +
[(100*R/M*sum{i=1,NCL}[DLC_i/NLL_i])/(1+Y/M)^{N-1+DSC/E+Nql+DLQ/LQL}] -
[100*R/M*A/E]

Where:

A - Number of occured days for the i^th quasi-coupon period within the odd period (accured days)
DFC - Number of days from the beginning of the odd first coupon (dated date) to the first coupon date
NLF_i - Normal length in days of the full i^th quasi-coupon period within odd period
DSC - Number of days from settlement date to the next quasi-coupon date or next coupon date
E - Number of days in quasi-coupon period, or regular coupon period, in which the settlement date falls
K - A summation counter
M - Number of coupon periods per year
N - Number of coupons payable between ''the first real coupon date'' and redemption date (maturity date, call date, put date, etc.) (If this number contains a fractional part, raise it to the next whole number - ie 2.5 = 3)
NCF - Number of quasi-coupon periods that fit in odd period. (If this number contains a fractional part, raise it to the next whole number.)
Nqf - Number of ''whole'' quasi-coupon periods between settlement date and first coupon (this may be zero)
P - Dollar price per $100 pae value
R - Annual interest rate (as a decimal)
RV - Redemption value of the security per $100 par value (RV = 100, except in those cases where call or put features must be considered)
Y - Annual yield ( as a decimal) on investment with security held to redemption

(Note: This formula does not apply to municipal securities)

==== Algorithm ===

def getOddShortFirstOddLongLast(A, DFC, DLC, DLQ, DSC, E, LQL, M, N, NCL, NLL, Nql, R, RV, Y):

    coefficient = 1.0 + 1.0 * Y/M
    K = 2

    term1 = RV/(coefficient **(N - 1 + 1.0*DSC/E + Nql + 1.0*DLQ/LQL))
    term2 = (100.0 * R/M * DFC/E) / coefficient ** (1.0*DSC/E)
    while K <= N :
        term3 += 100.0 * R/M  / (coefficient **(K - 1 + 1.0 * DSC/E))
        K += 1

    i = 0
    while i < NCL:
        term4 += 1.0 * DLC[i] / NLL[i]
        i += 1

    term4 = (100.0 * R/M * term4) / coefficient ** (N - 1 + 1.0*DSC/E + Nql + 1.0*DLQ/LQL)
    term5 = 100.0 * R/M * A/E

    return term1 + term2 + term3 + term4 - term5

Example

References

Stepped Coupons

Typical Uses

todo

Ingredients

todo

Algorithm

Example

References

Zero Coupon

Typical Uses

The zero coupon securities do not pay any periodic interest payment. This type of instruments are sold at discount rate (for example $50 for every $100 of the face value) and increase in value as they aproach the maturity. The securities mature to its face value. For example one may buy a zero coupon bond with par value of $1000 and 10 years to maturity for $503. In ten years the return on the investment will be $1000. This investment yields 7.0% of return paid at the maturity date.

Ingredients

Computations of price and yield of zero coupon securities are based on quasi-cupon periods. A quasi-coupon periods are the coupon periods which would exist if teh security was paying interest rate diffenet from zero.

Price

PV = [ RV / ( 1 + Y / M )^{Nq - 1 + DSC / E}]

Where:

PV - Present value per $100 of face value
RV - Redemption Value
Y - Anual yield (in a decimal form) on investment in security held to redemption
M - Number of quasi-coupon periods per year
E - Number of days in quasi-coupon period
Nq - Number of quasi-coupon periods between setelment date and redemption date. If the number contains a fractional part, it has to be rounded to the next whole number.
DSC - Number of days from settlement date to next quasi-coupon date as if the security paid periodic interest.

Algorithm

 int getPrice(double y, double rv, double nq, int e, int dsc, int m) {
  
        _nq = (int)Math.round(nq);
        
        double coefficient = (_nq -1 + dsc*1.0/e*1.0);
        
        return rv/Math.pow(1+y/m, coefficient); 

    }

def zeroCouponPrice(Y, RV, Nq, E, DSC, M):
  Nq = Math.ceil(Nq)
  coefficient = Nq - 1 + DSC / E
  PV = ( RV / ( 1 + Y / M )) ** coefficient
  return PV

Example

A municipal bond was setteled on 02/12/92. The maturity date of the bond is 07/01/05. What was the price of the bond if the yield is 10.55%?

Y = 10.55 (0.1055)
RV = 100
assume semi-annual quasi period:
  M = 2
  E = 180
  DSC = 139
  Nq = 27

getPrice(Y, RV, Nq, E, DSC, M)
>>>25.252446

Yield

Y = [ (RV / P)^{1/(Nq - 1 + DSC/E)} - 1]*M

Where:

Y - Anual yield (in a decimal form) on investment in security held to redemption
PV - Present value per $100 of face value
RV - Redemption Value
M - Number of quasi-coupon periods per year
E - Number of days in quasi-coupon period
Nq - Number of quasi-coupon periods between setelment date and redemption date. If the number contains a fractional part, it has to be rounded to the next whole number.
DSC - Number of days from settlement date to next quasi-coupon date as if the security paid periodic interest.

Algorithm

  int getYield(double p, double rv, double nq, int e, int dsc, int m) {
  
        _nq = (int)Math.round(nq);
        
        double coefficient = 1D/(_nq -1 + dsc*1.0/e*1.0);
        
        return (int)(Math.pow(rv/p, coefficient) - 1) * m * 100; 

    }

def zeroCouponYield(PV, RV, Nq, E, DSC, M) {
  Nq = Math.ceil(Nq)
  coefficient = 1/(Nq - 1 + DSC/E)
  Y = M *(RV /PV)** coefficient) -1)
  return Y

Example

A corporation issued a zero coupon bond with redemption value of $100. The bond is sold at $25.125 discount price. What is a yield on investment in such a bond with settlement date 08/26/2002 and maturity date 09/01/2015?

P = 25.125
RV = 100
assume semi-annual quasi period:
  M = 2
  E = 180
  DSC = 5
  Nq = 27

getYield(P, RV, Nq, E, DSC, M)
>>>10.900794

References

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