Bond Valuation, Dirty Price, Clean Price and Accrued Interest

A bond is a fixed-income instrument as the income it pays bond holders at payment intervals are fixed coupon payments. To briefly illustrate the mechanics of bond valuation, and the concepts of dirty price, clean price and accrued interest, a simple bond that has the following characteristics would be used:

Bond Face Value: $100
Bond tenor/Maturity :  5 years
Coupon rate: 6%
Payment frequency : Semi-annual
Yield to maturity of bond: 5%

The bond price will simply be the present value of the cash flows from the bond. Since the bond is a semi-annual bond, the coupon payment every 6 months (0.5 years) with be given by:

Coupon rate * Face Value / payment frequency = 0.06 * 100 / 2 = $3

Thus, there are 10 payments of $3, and one final payment of $100 (the face value). The  bond seller makes coupon payments of $3 every 6 months to the bond holders (including the maturity period) and a final payment of $100 when it matures.

The bond valuation can be done in two parts: the first part is the valuation of the coupon payments, which is a level annuity of $3 with 10 periods and a discount rate of 2.5% (half the yield as it semi-annual payments). The second part is the present value of the face value. (see blog section for annuities, perpetuities etc,).

Applying this two part valuation process gives:

Bond price  = (3/0.025)*[1 - 1/ (1+0,025)^10]  + 100 / (1 + 0.025)^10 = $104.38

Since this bond pays more than the rate at which the payments are discounted (i.e. coupon  rate greater than yield), the bond price is higher than face value (sells at a premium to face value) and the bond is a premium bond.

The valuation above is done at the time the bond is initially sold. The net present value of cash flows from the bond as a function of time determine the dirty price of the bond. An example will suffice to explain this. For convenience, let us imagine the bond makes semi-annual payments every six months, and the first payment is July 1. Let us price the bond 2 1/4 years into the bond's life.

The bond initially had 11 payments, 10 coupon payments and 1 principal payment. 2 1/4 years into the life of the bond, the bond now has 2 3/4 years left and six payments left, 5 coupon payments and 1 principal payment.

The present value of the outstanding cash flows is now $104.03 (See above). This is the dirty bond price.
To calculate the clean bond price, we need to calculate the accrued interest. The accrued interest is zero if the valuation of the bond is done on a coupon payment date, otherwise it is not. The accrued interest is that portion of the next coupon payment that will be earned by the bond seller that the seller does not receive as the bond is sold between coupon dates.

The accrued interest is calculated as:

(no of days since last coupon payment to settlement date * coupon payment)/(no of days between coupon payments).

For this bond, for convenience, we have calculated the number of days in years. Thus the number of days since the last coupon payment is 0.25 years and number of days between coupon payment is 0.5 years, thus the accrued interest is 0.25*$3/0.5 = $1.50 (See above table). Conventions exist for calculating number of days such as actual/actual or actual/360 depending on the type of bond. Actual/actual refers to counting actual number of days and actual/360 assumes that there are 360 days in any one year (rather than 365 or 366).

Given that the accrued interest calculated is $1.5, the clean bond price is the dirty bond price less the accrued interest or $104.03 - $1.50 = $102.53.  A discussion of bond duration, modified duration and convexity of the bond, important bond concepts for evaluating the interest rate sensitivity of the bond will be discussed in a separate blog entry.


Ref: Fabozzi, F. (2001),  The Handbook of Fixed Income Securities, Sixth Edition, McGraw Hill, New York

Annuities and perpetuities with continuous compounding

With continuous compounding, present value formulas and future value formulas are different from those with discrete compounding,  PV = FV / (1+r)^t    or  FV = PV * (1+r)^t  .

The present value (PV) and future value (FV) of a cash flow with continuous  compounding is given by:

PV = FV*e^(-rt)         or FV = PV*e^(rt)

where r is the continuously compounded interest or discount rate which is different from the discrete compounded or discount rate.

As the present value formulas are different, so are the formulas for the net present values of annuities, growing annuities, perpetuity and growing perpetuity with continuously compounded rates. However, we will make use of the geometric progression analysis available within this blog at the link (click link) . The summary results are the sum of the first n terms of a geometric progression and the sum of the infinite terms of a geometric progression.

Sn = [A/(1-R)]*(1- R^n)  
(formula for sum of the first n terms)

S∞ =A / (1-R)   
(formula for sum of infinite terms of the GP)
where A is the first term and R is the constant ratio or constant multiple.

We start with the cash flows and the associated present values, assuming the cash flows grow by a factor g (also a continuously compounded rate). The cash flows follow the following sequence:

C , C*e^g , C*e^2g, ..., C*e^(n-1)g, ...    (note: nth term has an index of (n-1)g

The present value of each cash flow, bearing in mind the formula PV = FV*e^(-rt) follow the following sequence:

Ce^(-r), C*e^(g-2r) , C*e^(2g-3r), ..., C*e^[(n-1)g-nr], ... 

On closer inspection, these terms are a geometric progression with first term, A = Ce^(-r) and a constant ratio R = e^(g-r). These two terms can be used to find the formulas for the annuity, growing annuity, perpetuity and growing perpetuity. Let's start with the growing annuity.

Growing Annuity and Annuity with continuous compounding

using Sn = [A/(1-R)]*(1- R^n)    (formula for sum of the first n terms)
then PV (GA) = [Ce^(-r)/(1-e^(g-r))]*[1- e^(g-r)n]

multiplying the numerator and denominator by e^(r-g) gives
PV (GA) = [{Ce^(-r)}*e^(r-g)/[{1-e^(g-r)}*e^(r-g)]]*[1- e^(g-r)n]  
or 
PV (GA) =[Ce^(-g)/(e^(r-g)-1)]*[1- e^(-(r-g)n)] 

the present value of an annuity with no growth (g=o) directly results as:

PV (A) =[C/(e^r-1)]*[1- e^(-rn)] 

Growing Perpetuity and Perpetuity with continuous compounding

Using S∞ =A / (1-R)    (formula for sum of infinite terms of the GP)
then PV (GP) = [Ce^(-r)/(1-e^(g-r))]

multiplying the numerator and denominator by e^(r-g) gives
PV (GP) = [{Ce^(-r)}*e^(r-g)]/[{1-e^(g-r)}*e^(r-g)]
or 
PV (GP) =[Ce^(-g)/(e^(r-g)-1)]

the present value of a perpetuity with no growth (g=o) directly results as:

PV (P) =C/({e^r}-1)


Summary results

Continuous compounding formulas

PV (A) =[C/(e^r-1)]*[1- e^(-rn)]    - Present value of an annuity
PV (GA) =[Ce^(-g)/(e^(r-g)-1)]*[1- e^(-(r-g)n)]  - Present value of a growing annuity
PV (P) =C/({e^r}-1) - Present value of a perpetuity
PV (GP) =[Ce^(-g)/(e^(r-g)-1)] - Present value of a growing perpetuity


Discrete compounding formulas

PV(A) = [C/r]* [1-{1/(1+r)^n}]    - Present value of an annuity
PV(GA) = [C/(r-g)]* [1-{(1+g)/(1+r)}^n]   - Present value of a growing annuity
PV(P) =  C/r    - Present value of a perpetuity
PV(GP) =  C/(r -g) - Present value of a growing perpetuity

The derivation of annuity, perpetuity, growing annuity and growing perpetuity formulas

In deriving the annuity, perpetuity, growing annuity and growing perpetuity formulas, it is necessary to understand that each of this series of payments represent a geometric progression  (GP) or geometric series. Thus, we would start by exploring the geometric series, then apply the result from that to our series of cash flows.

A geometric series can be described as series of numbers that differ by a constant ratio or constant multiple.
The following numbers sequence fits a geometric series:

A , AR , AR^2, ..., AR^(n-1), AR^n, ... AR^∞      (an infinite series).

The first term of the series is A and the constant ratio or constant multiple is R. Each successive term is r multiplied by the previous term. The sum of the first n terms of the GP, Sn is given by:

Sn = A + AR + AR^2+ ...+ AR^(n-1)  (notice the last term has a power of R of n-1)

To solve for Sn, we will use a trick. First, multiply Sn by R to give Sn*R

Sn*R= AR + Ar^2+ ...+ Ar^n     (notice the last term has a power of R of n)

we subtract Sn*R from Sn to give:

Sn - Sn*R=[A + AR + AR^2+ ...+ AR^(n-1)]-[AR + AR^2+ ...+AR^(n-1) +AR^n ] = A-AR^n
Sn(1-R) = A(1- R^n)
therefore Sn = [A(1- R^n)] / (1-R)   or  Sn = [A/(1-R)]*(1- R^n)  
(formula for sum of the first n terms)

A similar analysis can be used to show the sum to infinity of the series, provided the series converges (i.e. terms get smaller, which will be the case if the constant ratio is less than zero).

S∞ = A +AR + AR^2 +...+AR^(n-1)+AR^n +...
S∞*R= AR + AR^2 +...+AR^n+AR^(n +1)+...
S∞-S∞*R = [A +AR + AR^2 +...+AR^(n-1)+AR^n +... ]-[AR + AR^2 +...+AR^n+AR^(n +1)+... ]
S∞-S∞*R= A
S∞(1-r ) = A     or   S∞ =A / (1-R)   
(formula for sum of infinite terms of the GP)

These two formulas form the basis for deriving the annuity, growing annuity, perpetuity and growing perpetuity formulas.

Let is now apply these formulas to finance. r in this application is a discount rate, and not a constant ratio, R as used above..

An annuity is a series of payments of the same amount, C paid at regular intervals that end at period n. Thus the following payments are an annuity,

C,C,C,...,C   paid at periods 1,2,3,...,n (could be years 1,2,3,...,n if payments are annual).

The present values of these cash flows is also series payments of the form:

C/(1+r), C/(1+r)^2, C/(1+r)^3,...,C/(1+r)^n

On closer inspection, this is a GP with first term A=C/(1+r) and constant ratio R=1/(1+r)

including these terms in the equation for the first n terms of a GP gives

PV(A) = {[C/(1+r)] / [1 - {1/(1+r)}]}* [1-{1/(1+r)}^n]                
from      Sn = [A/(1-R)]*(1- R^n)

with some algebraic manipulation the above expression becomes:

PV(A) = {[C/(1+r)] / [(1 +r - 1)/(1+r)]} * [1-{1/(1+r)}^n]
PV(A) = {[C/(1+r)] / [ r/(1+r)]} * [1-{1/(1+r)}^n]
PV(A) = [C/r]* [1-{1/(1+r)^n}]            
the desired result for the present value of an annuity

It may be worthwhile to work through the derivation once to verify the algebra.

For the present value of a growing annuity, the present values of the cash flows is given by:

C/(1+r), C(1+g)/(1+r)^2, C(1+g)^2/(1+r)^3,...,C(1+g)^(n-1)/(1+r)^n

On closer inspection, this is a GP with first term A=C/(1+r) and constant ratio R=(1+g)/(1+r)

PV(GA) = {[C/(1+r)] / [1 - {(1+g)/(1+r)}]}* [1-{(1+g)/(1+r)}^n]              
from      Sn = [A/(1-R)]*(1- R^n)

with some algebraic manipulation the above expression becomes:

PV(GA) = {[C/(1+r)] / [(1 +r -1-g)/(1+r)]} * [1-{(1+g)/(1+r)}^n]
PV(GA) = {[C/(1+r)] / [( r-g)/(1+r)]} * [1-{(1+g)/(1+r)}^n]
PV(GA) = [C/(r-g)]* [1-{(1+g)/(1+r)}^n]            
the desired result for the present value of a growing annuity with g<r
where g is the growth rate of payments C between each period.

The formulas for perpetuities are easier. For a simple perpetuity with no growth, the cash flow sequence is:

C/(1+r), C/(1+r)^2, C/(1+r)^3,...,C/(1+r)^n , ...

On closer inspection, this is a GP with first term A=C/(1+r) and constant ratio R=1/(1+r)

The sum to infinity of the series is given by:

PV(P) = [C/(1+r)] / [ 1 -{1/(1+r)}]
from S∞ =A / (1-R) 

with some algebraic manipulation the above expression becomes:

PV(P) = [C/(1+r)] / [ (1+r -1)/(1+r)] = [C/(1+r)] / [ r/ (1+r)] = C/r
PV(P) =  C/r     which is the desired result.

For a growing perpetuity, the cash flow sequence is:

C/(1+r), C(1+g)/(1+r)^2, C(1+g)^2/(1+r)^3,...,C(1+g)^(n-1)/(1+r)^n, ...

On closer inspection, this is a GP with first term A=C/(1+r) and constant ratio R=(1+g)/(1+r)

PV(GP) = [C/(1+r)] / [ 1 -{(1+g)/(1+r)}]

with some algebraic manipulation the above expression becomes:

PV(GP) = [C/(1+r)] / [ (1+r -1-g)/(1+r)] = [C/(1+r)] / [(r-g)/ (1+r)] = C/(r-g)
PV(GP) =  C/(r -g)    which is the desired result.

Summary:
Sn = [A/(1-R)]*(1- R^n)     - Sum of the first n terms of a geometric progression
S∞ =A/(1-R)   - Sum of the infinite terms of a geometric progression

PV(A) = [C/r]* [1-{1/(1+r)^n}]    - Present value of an annuity
PV(GA) = [C/(r-g)]* [1-{(1+g)/(1+r)}^n]   - Present value of a growing annuity
PV(P) =  C/r    - Present value of a perpetuity
PV(GP) =  C/(r -g) - Present value of a growing perpetuity

Time Value of Money (TVM)

Time Value of Money (TVM) is the most important concept in finance. It is the underlying concept for majority of finance related calculations, such as project analysis (capital budgeting or investment appraisal), equity valuation, bond valuation and indeed any asset valuation.

Time value of money can be summarized in the statement, "£1 received today is more valuable than £1 received in the future." The reasons for this include the ability to invest £1 today to receive more than £1 in the future, and inflation, which is generally positive reduces the value of money in the future.

When doing most financial analysis, cash flows at different periods is involved. Some cash flow is typical involved now (year 0) and some cash flows in future periods. Any of these cash flows could be positive (cash inflows) or negative (cash outflows). Any of these cash flows can be converted to an equivalent value today, which is termed the present value of the future cash flow (PV).  If the sum of the present values of a series of future cash flows is sought, the approach is to find the present value of each cash flow, prior to adding them together to find the Net Present Value (NPV) of the series of cash flows. The concept of Net Present Value (NPV) is central to the study and understanding of finance. This is because the value of any asset can be evaluated by finding the NPV of the net cash flows generated by the asset.

In finding the present value of any cash flow, what is required is the amount of the cash flow or the future value of the cash flow (e.g. £1,000),  the timing of the cash flow (e.g. 3 years), and an appropriate discount rate to be applied to the cash flow (e.g. 10%). The formula for finding the present value, PV of a future value (or future cash flow), FV is given by:

PV = FV / (1+r)^t

where PV is the present value, FV is the future value, r is the discount rate in decimals and t is the time of the cash flow in years.
As an example, the PV of £1,000 to be received in 3 years time at a 10% discount rate is:

PV = £1000 / (1+0.1)^3 = £751.31    (to 2 decimal places).

Thus, at 10% discount rate, £1000 received in 3 years is equivalent to £751.31. We can look at this discount rate in another way. Let us consider the 10% as another investment opportunity available to us. Thus, if we invested £751.31 today, it would grow at 10% compounded annually to £1000. The 10% discount rate is now effectively a compound rate, or the opportunity cost of capital for the investment. When doing time value of money analysis of this type, the discount rate takes on a few names, such as opportunity cost of capital, capitalization rate, expected return on capital and hurdle rate.

The time value of money concept can be applied to more than just a single cash flow as mentioned earlier, such as when calculating a Net Present Value. A set of stylised payments are central to understanding financial products that involve multiple payments or cash flows such as mortgages, loans, bonds and stocks. These are annuities, growing annuities, perpetuities and growing perpetuities.  The formulas for the Net present value of these types of cash flow arrangements are given below:

NPV of an Annuity:   PV(A) = (C/r)*[1- {1/(1+r)^n}]
NPV of a growing Annuity: PV(GA) = [C/(r-g)]*[1- {(1+g)^n/(1+r)^n}]
NPV of a perpetuity: PV(P) =  C/r
NPV of a growing perpetuity: PV(GP) = C/(r-g)

These values assume that constant payments of C are made from period 1 (or year 1) or that the payments grow at a constant rate g after the first payment of A in period 1 (or year 1). As a reminder, Net Present Values refer to values now (period 0 or year 0). The formulas assume discrete compounding at an interest or discount rate of r. If the period of compounding is different from a year, the number of periods, n, needs to be adjusted as well as the interest rate, r in these formulas. The derivations of these formulas and equivalent formulas for continuous compounding are addressed in another post.

Mortgage calculations

Mortgages are loans made to home buyers (borrower) by financial institutions such as banks (lender) with the purchased home as collateral for the loan. The borrower typically makes fixed or variable payments on the loan depending on the interest rate charged on the loan and the amount borrowed (loan or mortgage principal). The loan typically has a fixed term (tenor).

Calculations involving mortgages are based on present value analysis. The amount borrowed is equal to the present value of the future mortgage payments made by the borrower. For a fixed interest loan, the fixed monthly payments made by the borrower results in the mortgage analysis becoming one that is analogous to the present value of an annuity. An example will be used to illustrate the point.

Let us assume the following loan/mortgage terms:
Loan amount - $200,000
Term - 20 years (or 240 months)
Mortgage interest rate - 6% per annum (or 0.5% per month from 6%/12 months)

The amount you will be paying each month can be derived from the present value of an annuity formula given by:

A = (C/r)*[1 - {1/(1+r)^n}]         (Present value of an annuity formula)

where A is the present value of annuity (Loan amount or 200,000), C is the monthly payment to be calculated, r is the effective monthly interest rate (0.5% or 0.005 in decimal), and n is the number of payment periods (in this case, months which is 240).  A little algebra will suffice in showing that C can be calculated as:

C = (A*r) / [1 - {1/(1+r)^n}]  = (200,000*0.005) / [1 - {1/(1+0.005)^240}]
C = 1000 / 0.6979 = $1432.86

This analysis assumes that you have a fully amortizing loan where the amount borrowed is paid off at the end of the 240 months or 20 years. The first payment is also assumed to be made one month after the loan is taken. If the loan were an interest only loan, monthly payments would be $1000 (0.5% per month multiplied by $200,000), but the principal of $200,000 would still be outstanding at the end of the term.

The present value of an annuity formula can be rearranged to find C, r , n or used as it is to find A once the other 3 variables are known. It is somewhat more intricate to find r and n as the use of logarithms may be required to find n and iteration to find r as r appears twice in the formula.