Bond Formulas

This page lists the formulas used in calculations involving money, credit, and bonds. If you want to learn about these topics in detail, read the referring page.

Present Values and Future Values of Money

From The Present Value and Future Value of Money.

Future Value of a Dollar (FVD)
FV=P(1+i)nFV = Future Value of a Dollar
P = Principal
i = interest rate per year
n = number of years
The Present Value of a Dollar (PVD)
PVD=FVD
─────
(1+i)n		PVD = Present Value of a Dollar
FVD = Future Value of a Dollar
i = interest rate per time period
n = number of time periods
The Interest Rate of a Discount (IRD)
i=(FV
─────
PV		)1
―
n		 - 1 i = Interest Rate of Discount per time period
n = number of time periods
FV = Future Value
PV = Present Value
orFormula for the equivalent interest rate of a discounted bond, expressed as an equation. 

From The Present Value and Future Value of an Annuity.

Future Value of an
Ordinary Annuity (FVOA)
FVOA=A *(1 + i)n - 1
──────────
i
Future Value of an Annuity Due (FVAD)
FVAD=A *(1 + i)n - 1
─────────
i		+ A(1+i)n-A
The Present Value of an Annuity (PVA-∑ notation)
PVA=n

∑

k=1		A
─────
(1+i)k		PVA = Present Value of Annuity Amount
A = annuity payment
i = interest rate per time period
n = number of time periods
The Present Value of an Annuity (PVA)
PVA=A *1-1
──────
(1 + i)n
▬▬▬▬▬▬▬▬▬▬
i
Present Value Annuity Payment
A=PV
▬▬▬▬▬
1-(1+i)-n
─────
i		=PV*i
▬▬▬▬▬
1-(1+i)-n		Formula for the monthly payment of a loan.
A = monthly payment, or annuity payment.
PV = present value, or the amount of the loan.
i = interest rate per time period.
n = number of time periods.


Bond Yields

From Bond Yields.

Current Yield Formula for Bonds
Annual Interest Payment
Price of Bond		 = Current Yield
Taxable Equivalent Yield (TEY) Formula for Municipal Bonds
Muni Yield
100% - Your Federal Tax Bracket %		 = Taxable Equivalent Yield (TEY)
Yield-to-Maturity Approximation Formula for Bonds
Annual Interest Payment + (Par Value - Current Bond Price)/Number of Years until Maturity
(Par Value + Current Bond Price)/2		 = Approximate Yield-to-Maturity Yield Percentage


A more accurate calculation of yield to maturity or yield to call or yield to put:

B = Current Bond Price; I = coupon rate of interest; P = par value of bond or call premium;
n = number of years until maturity or until call;
Y = yield to maturity or yield to call, depending on which values of n and P are chosen.
B =I1+I2+ ... +In+P
(1+Y)1(1+Y)2(1+Y)n(1+Y)n

or, expressed in summation, or sigma, notation:

B =n

∑

k=1		Ik+P
(1+Y)k(1+Y)n
Formula for the Interest Rate of a Discounted Bond
i =(Future Value/Present Value)1/n - 1
orFormula for the equivalent interest rate of a discounted bond, expressed as an equation.i = interest rate per compounding period
n = number of compounding periods
FV = Future Value
PV = Present Value
Formula for Calculating Bond Equivalent Yield (BEY)
BEY = 365 x Discount Rate
──────────────────────────
360 - (Discount Rate x Days to Maturity)		To compare bond yields to
money market instruments using a 360-day year,
such as CDs, change the 365 to 360.

From Bond Pricing, Illustrated with Examples

Formula for Calculating Accrued Interest
Accrued Interest = Interest Payment x Number of Days
Since Last Payment
─────────────────
Number of days
between payments


From Volatility Of Bond Prices In The Secondary Market; Duration and Convexity

Formula for Duration (Macaulay Formula)
T
∑
t=1		t * Ct
─────
(1+y)t		D = Macaulay duration
t = time until payment in years
T = total number of payments
Ct = cash flow at time t
y = bond yield until maturity
D =────────────
 T
∑
t=1		Ct
─────
(1+y)t		Note that the denominator is equal to
the  sum of all cash flows discounted
by the yield to maturity which equals
the bond's price, including accrued
interest.

Duration and Convexity

From Duration and Convexity, with Illustrations and Formulas

Bond Value = Present Value of Coupon Payments + Present Value of Par Value

Duration Approximation Formula
Duration estimate formula.P0 = Bond price.
P- = Bond price when interest rate is incremented.
P+ = Bond price when interest rate is decremented.
∆y = change in interest rate in decimal form.
Macaulay Duration Formula
Macaulay duration formula.T = number of cash flow periods.

Where:

Weighted average of a bond's cash flow.wt = weighted average of cash flow at time t.
CFt = Cash flow at time t.
y = yield to maturity
Modified Duration Formula
Modified duration formula.Dm = Modified Duration
DMac = Macaulay Duration
y = yield to maturity
k = number of payments per year
Effective Duration Formula
Effective Duration formula.

∆i = interest rate differential

∆P = Bond price at i + ∆i – bond price at i - ∆i.

Duration Formula for Coupon Bond Selling for Face Value
Duration Formula for Coupon Bond Selling for Face Value

y = yield to maturity

T = years till maturity

Fixed Annuity Duration Formula
Fixed Annuity Duration Formula

y = yield to maturity

T = years till maturity

Perpetuity Duration Formula
Perpetuity Duration Formula

y = yield to maturity

Portfolio Duration = w1D1 + w2D2 + … + wKDK

Convexity Formula
Convexity formula for bonds.

P = Bond price.

y = Yield to maturity in decimal form.

T = Maturity in years.

CFt=Cash flow at time t.

Calculating the Change in Bond Prices with Interest Rates Using Duration + Convexity Adjustment
Calculating the Change in Bond Prices with Interest Rates Using Duration Plus Convexity.

∆y = yield change

∆P = Bond price change

Convexity can also be estimated with a simpler formula, similar to the approximation formula for duration:

1. Convexity Approximation Formula
Convexity = P+ + P- - 2P0
─────────────
2 x P0(Δy)2		P0 = Bond price.
P- = Bond price when interest rate is incremented.
P+ = Bond price when interest rate is decremented.
∆y = change in interest rate in decimal form.

Note, however, that this convexity approximation formula must be used with this convexity adjustment formula, then added to the duration adjustment:

1. Convexity Adjustment Formula
Convexity Adjustment = Convexity x 100 x (Δy)2∆y = change in interest rate in decimal form.

Hence:

Bond Price Change Formula
Bond Price Change = Duration x Yield Change + Convexity Adjustment

Important Note! The convexity can actually have several values depending on the convexity adjustment formula used. Many calculators on the Internet calculate convexity according to the following formula:

2. Convexity Approximation Formula
Convexity = P+ + P- - 2P0
─────────────
P0(Δy)2		P0 = Bond price.
P- = Bond price when interest rate is incremented.
P+ = Bond price when interest rate is decremented.
∆y = change in interest rate in decimal form.

Note that this formula yields double the convexity as the Convexity Approximation Formula #1. However, if this equation is used, then the convexity adjustment formula becomes:

2. Convexity Adjustment Formula
Convexity Adjustment = Convexity/2 x 100 x (Δy)2∆y = change in interest rate in decimal form.

As you can see in the Convexity Adjustment Formula #2 that the convexity is divided by 2, so using the Formula #2's together yields the same result as using the Formula #1's together.

To add further to the confusion, sometimes both convexity measure formulas are calculated by multiplying the denominator by 100, in which case, the corresponding convexity adjustment formulas are multiplied by 10,000 instead of just 100! Just keep in mind that convexity values as calculated by various calculators on the Internet can yield results that differ by a factor of 100. They can all be correct if the correct convexity adjustment formula is used!

The price value of a basis point (PVBP), or the dollar value of a 01 (DV01).

PVBP = |initial price – price if yield changes by 1 basis point|

(Math note: the expression |x| denotes the absolute value of x.)

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