2- MATHS Flashcards
rational investors will only make investments that compensate for
risk that returns wont materialise
loss of interest and purchasing power of money invested
opportunity cost
if decision to invest in A means surrendering option B - then benefits and returns of B are opportunity cost of A
FV
FV = PV x (1+r)^n
PV
PV = FV / (1+r)6n
NPV = sum of the discounted vals of all cash ourflows and inflows
annuity discount formula
1/r * (1-1/(1+r)^n)
annuity discount with growth
1/r-g * (1-(1+g)/(1+r)^n)
compound interest rate
r= (FV/PV)^1/n -1
Calculates the compound interest rate given an initial investment of PV grows to a value of FV after n accrual periods.
rearrange PV fomula for n
find n given PV, FV and r
n = [log(FV)-log(PV)]/log (1+r)
Calculates the number of periods required to get to a future value (FV) given an initial value (PV)and a rate of interest (r)
adv and disadv on NPV method for investment decision
ADV
-discounts future CF to present day value to find true value
- shows monetary profit
-assumes reinvestment unlike IRR which assumes CF reinvested @ IRR rate
-shows total return
- works for inflows and outflows
- good measure of profitability
- factors in various risks
DISADV
- future CF can be uncertain (e.g. divi)
- true cost of cap not known until investment made (can skew results)
-if CF estimates are unrealistics - NPV calcs may be inaappropriate
-accuracy of discount rate used
-sunk costs ignored i.e. dillligence costs
- no allowance made for non quant factors
- not helpful for investments with different life spance
uses and limitations of IRR
-useful for comparing projects - choose one with higher IRR
- should choose investment with higher NPV @relevant cost of capital
- negative cFs result in multiple IRRs
- assumes all CFs invested @ IRR rate
discounting perpetuities
perpetuity = annuity making regular payments which begin on fixed date and continue indefinitely
PV = C/r
C = regular CF
PV = C/(r-g)
for growing perpetuity
continuous compounding
FV = Pe^rt
P= principal amount
e= log base e
r= rate
t= no. of periods(y)
e.g. amount in 3 years given continual compound of 15% for 6k investment
9408
AER vs APR
APR = rate for mortgages, credit cards and personal loans - denominated as APR
AER = rate applied to savings acc with financial institutions
APR
APR =( (r+f) /principal)/y *365
r= interest paid over life of loan
f= fees
y= no. of says in loan term
balance transfer fees, arrangement fees, early redemption fees and late payment fees not included
typical/representative APR = actual rates set depending on credit score. bank offers this typical APR to at least 51% of potential customer
mortgages quote both headline and APR - headline doesnt include admin fees
APR across globe
applied in UK, US, EU, Canada, parts of E Eur, Asia and Asutralia
definition not consistent - in US included and excluded charges only defined broadly but in EU stringently set out
Elsewehre advertised rate is nominal - so actual cost is higher than advertised
caps on lending rates may form part of monetary policy to lower overall cost of credit - China lowered cap in 2020 as part of broad crackdown on usury and to lower financing costs for small bis
EAR
equiv annual rate - like APR used for borrowing money but specifically in form of overdraft
doesnt include fees for going overdrawn but indicates fees for remaining voerdrawn for the year
EAR = (1+r/n)^n -1
r = nominal rate on APR basis
n = number of periods - if interest applied daily then n= 265
AER
Annual equiv rate - convertes interest payments which are more or less frequent than a year to an annual equivalent rate
- savings rate with fin. institutions
AER = (1+r)^12/n -1
AER = (1+r) ^365/y -1
r = rate of interest for each time period
n = no. of months in time period
y= no. of days in time period
real returns
1 + real = (1 +nom) / (1 + inflation)
IRR
= annualized effective compound rate which can be earned on invested capital
IRR = discount rate that makes NPV =0
Interpolation - will generally be given R1 and R2
if not use 2/3 *profit/outlay to get estimate then go either side of that
N1= NPV @ R1 (pos)
N2 = NPV @ R2 (neg)
IRR = R2 + N1/(N1 - N2) X (R2-R1)
IRR interpolation formula
N1= NPV @ R1 (pos)
N2 = NPV @ R2 (neg)
IRR = R2 + N1/(N1 - N2) X (R2-R1)
uses and limitations of IRR
BENE
- method for eval investments with initial CF followed by later CF
- useufl for bonds where GRY = IRR - if this exceeds cost of financing then project worthwile
LIMITATIONS
-Interpolation only an estimate, since it assume NPV changes linearly with interest
- smaller interval for interpolation = more accurate IRR
-if some -ve Cf and some +ve CF, IRR equation will give rise to more than one ROR
- higher IRR doesnt necessarily mean higher NPV @ relevant cost of capital
- assumes reinvestment @ IRR rate which is unrealistic
-bas for comparing projects with large differences in timing of CF or scale of invesmtent
arithmetic annualization
You invest £1,000 in the FTSE for 3 years. FTSE returns are:
Year 1 + 11%
Year 2 - 5%
Year 3 +8%
arithmetic avag = 11-5+8/3 = 4.67%
compound return of
1,000 x 1.04673 = £1,146.74
but actual return is different
BUT the actual return is different
Year 1: £1,000 gain 11% =£1,110
Year 2: £1,110 loss 5% = £1,054.50
Year 3: £1,054.50 gain 8% = £1,138.86
ra = [(1+r1)(1+r2)(1+r3)…(1+ri)]1/n – 1
ra = [(1+0.11)(1-0.05)(1+0.08)]1/3 - 1
= [1.13886] 1/3 – 1
= 1.0443-1
= 4.43%
This then gives a compound return over 3 years of 1,000 x 1.04433 = £1,138.87
geometric annualised return
ra = [(1+r1)(1+r2)(1+r3)…(1+ri)]^1/n – 1
