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maths:financial

a brief summary of financial equations

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