maths:financial

### Table of Contents

# a brief summary of financial equations

see also:

## Introduction

## Simple Interest

- interest rate per year = (redemption value - principal)/principal X (365/no.days)

## Compound Interest

- compound value = principal (1 + int.rate)^
^{no.periods} - interest rate = exp(ln(compound value/principal)/no.periods) - 1 (????)

## Nominal annual interest rate

- effective interest rate = (1+(Nominal Rate/no.compounding periods/yr))^
^{(no.compounding periods/yr -1)} - ie. if 4 compounding periods per year and effective rate is 5.3543%/yr then nominal rate s 5.25%

## Present and Future values

**Future Value**= present value (1 + int.rate)^^{no.periods}**Present Value**= future value (1 + int.rate)^^{-(no.periods)}- =amount needed to invest now @ int.rate to give a FV @ n periods.

## Annuities

**Annuities:**= equal periodic payments over equal periods of time- ordinary: = payments at end of each period
- annuity due: = payments at beginning of each period
- deferred anuity: = payments are delayed
- perpetuity: = anuity continues forever (ie. n → infinity)
- future value = [ payment (1 + int.rate)^
^{no.periods}- 1] / int.rate- eg. $1000 invested @ end yr for 5yrs @ 10%pa ⇒ FV = $6105

- present values (PV):
- ordinary annuity = [ payment (1-(1+int.rate)^
^{-(no.periods)})] / int.rate - annuity due = PV (ord.annuity) * (1+int.rate)
- perpetuity = payment / int.rate (as n → infinity )

## Capital Investment Assessment Methods:

**Payback period:**- if equal annuity: = init.investment /annual cash flow
- if mixed then work out manually

**Average Rate Of Return:**= av. after tax profits / av. investment- = % return on average investment
- rarely used as no account of time value of money!

**Discounted Cash Flow Methods:**- recognises time value of money;
- appropriate discount factor is max. rate of return required (RROR) by firm to invest in such a project.
- a) NPV = (PV cash flows discounted at RROR) - initial investment (II)
- if < 0, then reject project!
- = sum (cash flowi * (1+k)^-i - II, k = RROR, i = cash flow period;

- b) IRR = internal rate of return
- = discount rate that equates PV cash flows with initial investment
- ie. solves for k by making II = PVcash flows;
- if IRR < RROR then project is rejected.

- c) PI = profitability index = benefit-cost ratio
- = PV cash inflows / initial investment

**Modified Internal Rate Of Return (MIRR):**- = int. rate that relates FV with initial outlay over the period
- = (PV/FV)^n - 1 (??)

## Risk Assessment:

- Measures of risk (the greater value the greater risk, ? in order of importance):
- 1) range
- 2) variance
- 3) st. dev.
- 4) coeff. of variation

- Eg.
**Forecast****Pr(forecast)****Return (R)****Mean Average****Weighted Value**Pr * R (R-mean av.)2 * Pr Pessimistic 0.25 13% 3.25 1 Most likely 0.50 15% 7.50 0 Optimistic 0.25 17% 4.25 1**sum**1.00 15 2 =**variance**1.41% =**st.dev.**

maths/financial.txt · Last modified: 2021/07/24 11:38 by gary1