Financial Maths summary Updated emc 2020 PDF

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Financial Notes for exa (concise)...

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RECURSION & FINANCIAL MODELLING SUMMARY Recurrence Relation

Rule

Notes

How to recognise

LINEAR GROWTH / DECAY A. Simple Interest Loans / Investments V0 = principal

AND

Vn+1 = Vn + D,

r 100 x V0 D=

n = number of years r = interest rate per year (p.a) D = amount of interest ($) Vn = balance of loan/investment after n years

Vn = V0 + nD

r 100 x V0 where D =

- “simple interest” in the question - includes a percentage. - the change IS constant each time (it does increase or decrease by same amount) - usually yearly

B. Flat Rate Depreciation V0 = initial value

AND Vn+1 = Vn – D r 100 x V0 where D =

Vn = V0 – nD

r 100 x V0 where D =

Vn = value of the asset after n years r = interest rate per year (p.a.)

- “flat rate” in the question - includes a percentage - the change IS constant each time (it increases or decreases by the same amount) -usually yearly

C. Unit Cost Depreciation V0 = initial value

AND

Vn+1 = Vn – D

Vn = V0 – nD

D = cost per unit of use n=number of uses

- decreases at a rate (for example, $0.35 per kilometre) - “unit cost” in the question - decreases by a constant amount (or $100 per week)

GEOMETRIC GROWTH / DECAY A. Compound Interest Loans / Investments V0 = principal

AND r m where R = 1 + 100

Vn+1 = R Vn

(m = number compounding periods per year)

V n = Rn × V 0

r m where R = 1 + 100 V0 = principal (initial value)

r = interest rate per year (p.a.) Vn = value of loan / investment after n compounding periods

- “compound interest” or “compounding” in the question and you are asked to find a recurrence relation or rule

B. Reducing Balance Depreciation V0 = initial value AND Vn+1 = R Vn r m where R = 1 – 100

V n = R n V0

r m where R = 1 – 100 V0 = principal

r = interest rate per year (p.a.) n = number of years (compounding periods) Vn = value of asset after n years

- “depreciate” in the question - something decreases in value by a percentage

COMBINED LINEAR & GEOMETRIC GROWTH / DECAY Reducing Balance Loan V0 = principal

AND r m where R = 1 + 100

Vn+1 = RVn – D

D = regular payment Vn = balance of the loan after n payments r = interest rate per year (p.a.) In reducing balance loans: PV = positive PMT = negative FV = negative (if money is still owed) = zero (if the loan is paid off)

You have borrowed money and will pay it back over time (eg/ home loans or car loans) For the final payment, Step 1 Let n=total length of loan – 1 and solve for FV The FV value found from solving is negative, the borrower owes this. Step 2 Let n = 1 make PV = amount owing which is a positive value and solve for PMT, this is your final payment.

Interest-only loan (reducing balance loan where interest = payment) V0 = principal

AND

You have borrowed money and will never pay down the loan.

Vn+1 = Vn

r /m PMT = ×V 0 100

(to find without the finance solver) The amount owing never changes. In interest only loans: PV = positive PMT = negative FV = negative PV and FV must be equal and opposite N can be any value

The payment is equal to the interest. “Interest only” is in the question.

Annuity (money is invested and gradually withdrawn over a period of time) V0 = principal

and r m where R = 1 + 100

Vn+1 = R Vn – D

D = regular payment received r = interest rate per compounding period In annuities: PV = negative PMT = positive FV = positive (money still in account) = zero (all money is gone)

You have invested money and will receive (or withdraw) a payment over time. Superannuation in the question (or some kind of saving to receive money over time).

Perpetuity (annuity where interest = payment received) V0 = principal

AND

Vn+1 = Vn

PMT =

r /m ×V 0 100

You have invested money and will receive (or withdraw) payments from the investment (to find without the finance solver) indefinitely. The investment amount never changes.

In perpetuities : PV = negative PMT = positive FV = positive PV and FV must be equal and opposite N can be any value

The payment received (or withdrawn) is equal to the interest. Some form of “perpetuity”, “indefinitely” or “ongoing” is in the question.

Compound Interest Investments with Regular Additions to the Principal (Annuity Investment) V0 = principal

and r m where R = 1 + 100

Vn+1 = R Vn + D

D = regular addition to the principal R = interest rate per compounding period

You make an investment and then continue to add more money over time. Saving money over time.

In an annuity investment: PV = negative PMT = negative FV = positive

Other useful rules: LOAN/MORTGAGE

You borrow money and therefore you PAY interest You make regular payments to PAY OFF the loan - zero balance FV = 0

ANNUITY

N = total number of payments made

You invest your money and it EARNS interest Regular payments are made to YOU

I% = the interest rate (must be yearly)

Money eventually RUNS OUT - zero balance

PV = the present value of the loan/ investment PMT = the payment per time period FV = the future value of the loan/investment

P/Y = the number of payments made per year

C/Y = the number of times interest is compounded each year

PERPETUITY

You invest your money and it EARNS interest Regular payments are made to YOU equal to the INTEREST EARNED Principal Amount NEVER CHANGES PV < 0 FV > 0

FV = - PV

Money NEVER RUNS OUT so a perpetuity LASTS FOREVER

AMORTISATION TABLES Reducing Balance Loan and Interest Only Loan Interest paid=

r /m × previous balance of loan 100

Principal reduction= previous payment amount− previous interest paid

Balance of loan=previous balance− previous principal reduction Total cost of loan=∑ of all payments OR

Totalcost of loan=principal+ ∑ of all interest paid

Annuities and Perpetuities Payment number

Payment

Interest

Principal reduction

Balance of annuity

Interest paid=

r /m × previous balance of loan 100

Principal reduction= previous payment amount− previous interest paid 0

0

0

0

6000

Balance of loan=previous balance− previous principal reduction 1

508

15

493

5507

Compound Interest Investments Interest paid=

r /m × previous balance of loan 100

Principal reduction= previous payment made+ previousinterest earned

Balance of investment= previous balance+ previous principal increase...