2- MATHS

Question 1

rational investors will only make investments that compensate for

Answer 1

risk that returns wont materialise
loss of interest and purchasing power of money invested

Question 2

opportunity cost

Answer 2

if decision to invest in A means surrendering option B - then benefits and returns of B are opportunity cost of A

Question 3

FV

Answer 3

FV = PV x (1+r)^n

Question 4

PV

Answer 4

PV = FV / (1+r)6n

NPV = sum of the discounted vals of all cash ourflows and inflows

Question 5

annuity discount formula

Answer 5

1/r * (1-1/(1+r)^n)

Question 6

annuity discount with growth

Answer 6

1/r-g * (1-(1+g)/(1+r)^n)

Question 7

compound interest rate

Answer 7

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

Question 8

find n given PV, FV and r

Answer 8

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)

Question 9

adv and disadv on NPV method for investment decision

Answer 9

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

Question 10

uses and limitations of IRR

Answer 10

-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

Question 11

discounting perpetuities

Answer 11

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

Question 12

continuous compounding

Answer 12

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

Question 13

AER vs APR

Answer 13

APR = rate for mortgages, credit cards and personal loans - denominated as APR

AER = rate applied to savings acc with financial institutions

Question 14

APR

Answer 14

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

Question 15

APR across globe

Answer 15

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

Question 16

EAR

Answer 16

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

Question 17

AER

Answer 17

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

Question 18

real returns

Answer 18

1 + real = (1 +nom) / (1 + inflation)

Question 19

IRR

Answer 19

= 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)

Question 20

IRR interpolation formula

Answer 20

N1= NPV @ R1 (pos)
N2 = NPV @ R2 (neg)

IRR = R2 + N1/(N1 - N2) X (R2-R1)

Question 21

uses and limitations of IRR

Answer 21

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

Question 22

arithmetic annualization

Answer 22

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

Question 23

geometric annualised return

Answer 23

ra = [(1+r1)(1+r2)(1+r3)…(1+ri)]^1/n – 1