Financial Formulas

A ready reference for financial formulas, enables to look-up the formula and it’s key input variables for ease in calculation and reference.

Following key financial formulas are covered in this post.

1. Rate of return conversion for different periodicity
2. Discounting
3. Loan Amortization
4. Derivative related formulas

Note:

• This post is work in progress and other relevant formulas and details periodically updated.
• All decimal figures are rounded up to 4 decimal places

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1. Rate Conversion

1. Periodic Rate of Return (PRR)

Annual rate is given and the objective is to convert to periodic rate i.e. convert annual rate of return to either monthly, quarterly or half-yearly return. This cannot be simply achieved by dividing the annual rate by the number of periods. The following formulas has to be used.

\[monthly \space / \space quarterly \space / \space biannually = (1+r)^{\frac{1}{n}}-1\] Where

• r = annual rate
• n = shorter period for which rate has to be obtained. Thus
  • n = 12 for monthly
  • n = 2 for bi-annually
  • n = 4 for quarterly

Examples

1. Convert 17% annual rate to quarterly rate equivalent.

• Solution: Using the formula \((1+r)^{\frac{1}{n}}-1\) wherein r = 0.17 and n = 4, the quarterly rate is 0.0400

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2. Effective Annual Rate (EAR)

1. When compounding is done multiple times in a year and we are interested in finding EAR.

Following formula gives the EAR when compounding is done n number of times.

\[EAR = (1+\frac{r}{n})^n -1\] Where

• r = interest rate / compounding period
• n = number of compounding periods
  • n = 12 for monthly
  • n = 2 for bi-annually
  • n = 4 for quarterly

Examples

1. Annual rate of return is 17% which is compounded twice a year. Calculate EAR.

• Solution: Using the formula \((1+\frac{r}{n})^n -1\) the EAR is 0.1772

2. Effective interest rate per compounding period (similar to PRR, however, in this case compounding is done multiple times in the year)

This second formula give the effective interest rate per compounding period.

\[R = (1+\frac{r}{n})^\frac{n}{p} -1\] Where

• p = No of payment periods per year
• r = nominal annual interest rate
• n = No. of compounding periods per year
• R = Rate per payment period

Example

1. Annual rate of return is 17% which is compounded twice a year and payments are made quarterly. Calculate PRR.

• Solution: Using the formula \((1+\frac{r}{n})^\frac{n}{p} -1\) the PRR (with multiple compoundings in a year) is 0.0416

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2. Discounting

1. Present Value (PV)

Describe present value.

\[PV = (1 + r)^{-t}\]

Where

• r = Annual rate of return
• t = Number of years

Examples

1. Calculate the present value of 3 year maturity instrument when the rate per annum is 18%.

• Solution: Using the formula, \(PV = (1 + r)^{-t}\) the PV discount value is 0.6086

2. Present Value (compounded periodically)

In this case the formula is just the putting a negative in the exponent i.e. -nt

\[PV \space factor = (1+\frac{r}{n})^{-nt}\] Where

• r = Annual rate of return
• n = Number of times compounding is done per year
• t = Number of years

1. Calculate the present value of 3 year maturity instrument when the rate per annum is 18% and compounding is done quarterly.

• Solution: Using the formula, \(PV = FV.(1+\frac{r}{n})^{-nt}\) the PV discount value is 0.1121

3. Annuity Formula

Describe annuity.

\[Annuity \space factor = [1 - \frac{(1+r)^{-t}}{r}]\]

Where

• r = Annual rate of return
• n = Number of times compounding is done per year
• t = Number of years

3. Future value (compounded once a year)

\[PV = (1 + r)^{t}\]

Where

• r = Annual rate of return
• t = Number of years

4. Future Value (compounded periodically)

Compounding Interest Rate (when compounding more frequent than on yearly basis)

Examples are monthly, quarterly, bi-annually.

\[FV \space factor = (1+\frac{r}{n})^{nt}\]

Where

• r = Annual interest rate
• n = number times compounding is done on a yearly basis
• t = time in years

Examples

3. Loan Related

1. Loan Amortization Method

\[A = P\frac{r(1+r)^n}{(1+r)^n-1}\] Where

• A = payment amount per period
• P = Value of borrowed capital
• r = interest rate per period (it is important to calculate the interest rate for the period. Thus simply divide the annual rate with the number of times payment has to be made per period)
• n = total number of payments or periods

Example

1. A individual is desirous of taking a loan for USD 50,000. He wishes to settle it in 5 years, with monthly payments. Bank’s determines the interest rate of 14%, based on the risk profile of the individual. Calculate the monthly instalments due from the individual?

• Solution: Using the formula above \(A = P\frac{r(1+r)^n}{(1+r)^n-1}\) and putting in the values the annuity payment per month is USD 1163.41

4. Derivative Related Formulas

1. Forward Rate

\[F = S.ln^{(r_2-r_1).t}\] Where

• F = Forward rate
• S = Spot rate (this is generally the closing rate for the date)
• \(ln\) = Base of the natural logarithms (2.718)
• \(r_2\) = interest free rate of the target currency in which the final settlement will be done.
• \(r_1\) = interest free rate of the host currency in which the transaction is done
• t = Time to final settlement (in years)

2. Fair Value of Derivative

\[FV = A.(F-K).(e)^{-r_2.T}\] Where

• FV = Fair value
• A = Amount in the host currency
• K = Strike price (the rate on which the final settlement will be done)

3. Discounting based Fair Value / Present Value of Derivative

\[PV = A.exp(-r_2.T)\] Where

• exp = exponent

4. Floating Leg of Interest Rate Swaps

\[PV_{floating} = F - \frac{F}{(1+\frac{r}{n})^{nt}}\]

Where

• F = Face value or notional amount
• r = interest rate (decimal)
• n = number of times interest is compounded per year.
• t = time (in years)