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
