─────
PV
―
n
n = number of time periods
FV = Future Value
PV = Present Value
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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.
From The Present Value and Future Value of Money.
| Future Value of a Dollar (FVD) | |
|---|---|
| FV=P(1+i)n | FV = 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 |
| or | | |||||
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. |
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 | |
| or | | 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. | |
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