# 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, illustrated with detailed examples, please read the referring page.

## Present Values and Future Values of Money

 FV = P(1 + r)n FV = Future Value of a dollar P = Principal or Present Value r = interest rate per time period n = number of time periods
 PV = FV(1 + r)n PV = Present Value FV = Future Value r = interest rate per time period n = number of time periods
 i = ( FVPV ) 1/n - 1 i = Interest Rate of Discount per time period n = number of time periods FV = Future Value PV = Present Value

or

 FVOA = A × (1 + r)n - 1r
 FVAD = A × (1 + r)n - 1r + A(1+r)n - A
 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
 PVA = A * 1- 1(1 + i)n i
 A = PV1-(1+i)-ni = PV * i1-(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.

 Nominal Yield = Annual Interest PaymentPar Value
 Current Yield = Annual Interest PaymentCurrent Market Price of Bond
 Taxable Equivalent Yield = Muni Yield100% - Your Federal Tax Bracket %
 Approximate Yield-to-Maturity % = Annual Interest+(Par Value - Bond Price)/Years till Maturity(Par Value + Bond Price)/2

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

 Bond Price = C1(1+Y)1 + C2(1+Y)2 + ... + Cn(1+Y)n + P(1+Y)n C = coupon payment per period P = par value of bond or call premium n = number of years until maturity or until call or until put is exercised Y = yield to maturity, yield to call, or yield to put per pay period, depending on which values of n and P are chosen.

or, expressed in summation, or sigma, notation:

 B = n ∑ k=1 Ik(1+Y)k + P(1+Y)n
 i = (Future Value/Present Value)1/n - 1 i = interest rate per compounding period n = number of compounding periods FV = Future Value PV = Present Value

or

 Interest Rate Per Term Number of Terms per Year BEY = Face Value - Price PaidPrice Paid × Actual Number of Days in YearDays Till Maturity
 Accrued Interest = Interest Payment × Number of Days Since Last PaymentNumber of days between payments
 T ∑ t=1 t × Ct(1 + y)t D = T ∑ t=1 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 Note that the denominator = the sum of all cash flows discounted by the yield to maturity, which = the bond's price.

## Duration and Convexity

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

 Duration = P- − P+ 2 × P0(Δy) P0 = Bond price. P- = Bond price when interest rate is incremented. P+ = Bond price when interest rate is decremented. Δy = change in interest rate in decimal form.
 Macaulay Duration = T ∑ t=1 t × wt T = number of cash flow periods. t =time in years wt = weighted average of cash flow at time t CFt = Cash flow at time t y = yield to maturity

Where:

 wt = CFt / (1 + y)tBond Price = Present Value of Cash FlowBond Price t =time in years wt = weighted average of cash flow at time t CFt = Cash flow at time t y = yield to maturity
 Modified Duration = DMac 1 + y/k DMac = Macaulay Duration dP/P = small change in bond price dy = small change in yield y = yield to maturity k = number of payments per year
 Effective Duration = − ΔP/PΔi Δi = interest rate differential ΔP = Bond price at i + Δi − bond price at i - Δi.

The formula for the duration of a coupon bond is the following:

 Coupon Bond Duration = 1 + yy − (1 + y) + T (c − y)c [(1 + y)T− 1] + y y = yield to maturity c = coupon interest rate in decimal form T = years till maturity

If the coupon bond is selling for par value, then the above formula can be simplified:

 Duration for Coupon Bond Selling for Face Value = 1 + yy [ 1 − 1 (1 + y)T ] y = yield to maturity T = years till maturity
 Fixed Annuity Duration = 1 + yy − T (1 + y)T − 1 y = yield to maturity T = years till maturity
 Perpetuity Duration = 1 + y y Δi = interest rate differential ΔP = Bond price at i + Δi − bond price at i - Δi.

Portfolio Duration = w1D1 + w2D2 + … + wKDK

• wi = market value of bond i / market value of portfolio
• Di = duration of bond i
• K = number of bonds in portfolio
 Convexity = 1P × (1 + y)2 T ∑ t=1 [ CFt(1 + y)t (t2 + t) ] P = Bond price. y = Yield to maturity in decimal form. T = Maturity in years. CFt=Cash flow at time t.
 ΔPP = -Dm × Δy + (Δy)22 × Convexity Δy = yield change ΔP = Bond price change

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

 Convexity = P+ + P- - 2P02 × P0(Δy)2 P0 = Bond price. P- = Bond price when interest rate is incremented. P+ = Bond price when interest rate is decremented. Δy = 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:

 Convexity Adjustment = Convexity × 100 × (Δy)2 Δy = change in interest rate in decimal form.

Hence:

 Bond Price Change = Duration × 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:

 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. Δy = 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:

 Convexity Adjustment = Convexity/2 × 100 × (Δy)2 Δy = 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 |×| denotes the absolute value of ×.)