Lecture Four Flashcards by Sidra Iqbal

what formula to use for Annuities: when u know what the cashflow is

FV=(1+r)^t-1/r

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what formula to use for Annuities: when u know need to know the impact on future cashflow

P =1/r
MINUS
1/r(1+r)^t

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steps to calculate the FUTURE VALUE (FV) an annuity,
assuming first cash flow occurs in one period’s time

Method 1 (the two step method): FV=PV*(1+r)^t

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example of a scenario assuming first cash flow occurs in one period’s time

if you want to work out what your regular savings will eventually be worth

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steps to calculate the FUTURE VALUE (FV) an annuity, assumes first cash flow occurs in one period’s time

Method 2 (one step method):

FV= C* (1+r)^t-1/r

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example of a scenario assumes first cash flow occurs in one period’s time

if you were saving up for retirement

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Calculate the PV of a perpetuity
assuming the first cash flow occurs in one year’s time

PV=C/r
=C*1/r

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what is a Perpetuity:

A type of annuity that lasts forever (e.g., university endowments

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what is a Annuity

A fixed sum of money paid or received at regular intervals for a specific period.

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Advanced perpetuity (perpetuity due)

is when the cash flows start immediately (i.e., the first payment happens today, at time 0).

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Advanced perpetuity (perpetuity due) formula

PV=C/r+C

C*(1/r+1)

C= fixed cash paymnet
r = Interest rate (expressed as a decimal)

+ C accounts for the first immediate payment.

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example of advanced perpetunity:

You receive $1,000 per year forever, starting immediately.

Interest rate = 10% (0.10).

PV= 1000*(10+1)=11000
OR
PV= 1000 * (1/0.10+1)=11,000

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delayed perpetuity def

the first cash flow starts after a certain number of years.

The present value needs to be discounted back to today using a discount factor (DF).

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Formula for Delayed Perpetuity

PV=(C/r)*DF

C*(1/r-AF)

C = Fixed cash payment
r = Interest rate
DF = Discount factor
AF (Annuity Factor) = Used to discount back to today

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Example for Delayed Perpetuity: You will receive $1,000 per year forever, but payments start in 4 years.

Interest rate = 10% (0.10)

Step 1: Calculate value at year 4 (when payments start)

PVyear4=1000/0.10=10000

Step 2: Discount it back 4 years to today USING THE DF FORMULA

PVtoday=10000*1/(1.10)^4

PVtoday = 10000*0.683
PVtoday=6830

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equation for DF

DF = discounted factor

=1/(1+r)^t

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Advanced Annuity (Annuity Due) def

An advanced annuity (also called an annuity due) means that the first cash flow happens immediately (at year 0).

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Formula for Advanced Annuity (Annuity Due)

PV=(C×AF)+C=

C×(AF+1)

AF = Annuity Factor (which depends on the interest rate and number of years).

The +C accounts for the extra first payment at year 0

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Difference between regular annuity and Advanced Annuity (Annuity Due)

If a regular annuity has a PV of $10,000, an advanced annuity will be worth more because you get an extra immediate payment at year 0

e.g. If you pay at the beginning of the month, that’s an annuity due.
If you pay at the end of the month, that’s a regular annuity.

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Delayed Annuity meaning

delayed annuity means that the first cash flow does not start immediately.

Instead, there is a gap before the payments begin (e.g., first payment at year 3 in this case).

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delayed annuity formula

PV=C×AF×DF

=C×(AF−AF)

AF = Annuity Factor for the full period
DF = Discount Factor (which adjusts the PV because of the delay)

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example question for delayed annuility:

You will receive $2,000 per year for 5 years, but the first payment starts at year 3.
The interest rate is 8% (0.08).
What is the present value today (Year 0

Step 1: Find PV of the Annuity at Year 3
Since the annuity starts in year 3, we first calculate its value at year 3 using the ordinary annuity

PVyear3=C*(1/r-1/r(1+r)^t
Where:

C = 2,000 (cash flow per year)
r = 8% = 0.08 (interest rate)
t = 5 years (number of payments)

PV=2000(1/0.08 - 1/0.08(1+0.08)^5

=PVyear3=16,934

Discount This PV Back to Today (Year 0)
Since this value is at year 3, we need to discount it to today (Year 0):

PVtoday=PVyear3*1/(1+r)^3

=16,934*1/(1+0.08)^3
PVtoday=13,439

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Example: Advanced Annuity

You will receive $5,000 per year for 4 years, but the first payment starts immediately (at year 0).
The interest rate is 6% (0.06).
What is the present value today (Year 0)?

PV=(C×AF)+C
C = 5,000 (cash flow per year)
r = 6% = 0.06 (interest rate)
t = 4 years (number of payments)
AF = Annuity Factor for 4 years at 6%

AF(4years,6%)=3.4651

Step 1: Compute the PV of a Regular Annuity

PV=5,000×3.4651
PV=17,325.5

PVadvancedan= 17,325.5+5000
=22,325.5

PVtoday=22,325.5

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two ways interest rates can be expressed

Annual percentage rate (APR)
and
Effective annual rate (EAR)

what is APR

Annual percentage rate (APR) - an interest rate that is annualised using simple interest

how is interest presented

as an annual %

what is EAR

Effective annual rate (EAR) - an interest rate that is annualised using compound interest

Example of APR: given a monthly interest rate of 1% what is the APR?

Simply multiply the rate given by the number of periods in the year 1%x12months = 12% APR

Formula for EAR

EAR=[1+APR/m]^m -1 m= number of compounding periods in the year

Example for EAR: given a monthly interest rate of 1% what is the EAR

=[1+0.12/12]^12-1 = 0.1268 = 12.68%

Example for EAR: given an APR of 12% and assuming interest is compounded quarterly, what is the EAR

[1+0.12/4]^4-1 = 0.1255 = 12.55%

what is a Amortizing loan

An amortizing loan is a loan where each payment covers: Interest on the remaining balance Principal repayment (reducing the debt over time)

examples of a Amortizing Loan?

Mortgages, car loans, and business loans are amortizing loans. Each payment reduces the loan balance until it reaches zero at the end of the term.

Loan Amortization Formula

C= AF/PV Where: C = Annual or Monthly Payment PV = Loan Amount (Present Value) AF = Annuity Factor (from financial tables) r = Interest Rate t = Number of Payments (Years × Frequency

what is fixed in Amortization loan and what changes

payments remain fixed, but interest and principal change each period. Over time, less interest is paid, and more goes to principal

example of Amortization loan Loan Amount = $20,000 Interest Rate = 7% per year Loan Duration = 5 years Annual Repayment = $4,877.81

AF(5year, 7%) = 4.1002 C=20000/4.1002=4,877.81

How to work out the Monthly Amortization loan

Interest rate: Convert to monthly (APR ÷ 12) No of payments: Multiply years by 12

Example of Monthly Amortization loan Loan Amount = $300,000 Annual Interest Rate (APR) = 6% Loan Duration = 25 years Payment Frequency = Monthly

M IR= 6%/12=0.05% =0.005 Total payment =25x12=300 C=PV/AF AF (300 months, 0.5%) = 155.2068 PV=300000 C= $1,932.90

what do Amortizing loan reduce to

zero

what is inflation

the rate at which prices are generally increasing rate at which prices increase over time. It reduces the purchasing power of money.

what is Nominal interest rate

he rate at which money invested grows eg the interest rate on a bank savings account stated interest rate on savings, loans, or investments. Does not account for inflation.

what is nominal interest rate also known as

money rate

what is real interest rate

he rate at which the purchasing power of an investment grows interest rate adjusted for inflation. Shows how much your purchasing power increase

formula for real interest RATE

(1+n)= (1+r) * (1+i) (1+r) = (1+n)/ (1+i) (As given on the formula sheet) r= (1+n)/(1+i)-1 n = Nominal Interest Rate r = Real Interest Rate i = Inflation Rate

example for real interest when A savings account pays 5% nominal interest. Inflation is 3%. What is the real interest rate?

(1+0.05)/(1+0.03)-1 r=0.019=1.9%

investment vs inflation: You have $1,000 and two options: Spend it now on goods. Invest it in a savings account at 5% interest Inflation is 3%

Invested Value in 1 Year: 1,000×1.05=1,050 Basket of Goods Cost in 1 Year: 1,000×1.03=1,030 Your real gain: 1050/1030-1= 1.9%

How Do Interest Rates Affect Inflation?

High interest rates reduce inflation because borrowing becomes expensive. Low interest rates increase inflation because borrowing is cheaper

what do central banks do

Central Banks (like the Bank of England) adjust interest rates to control inflation.

Multiplicative Effect of Inflation: A coffee shop spent $25,000 on materials last year. Prices will rise by 25% due to inflation. Production will increase by 50%. How much will the shop spend next year?

New cost: (25,000×1.25)×1.5 =46,875

What is a Replacement Cycle?

replacement cycle refers to the frequency at which a business replaces equipment to balance cost and efficiency

(EAA) meaning

Equivalent Annual Annuity (EAA)

what should u do on a replacement cycle

Choose the option with the lowest Equivalent Annual Annuity (EAA).

Formula for Present Value (PV) of an Annuity

PV=CF×AF Where: PV = Present Value of total costs CF = Level cash flow per year AF = Annuity Factor (depends on discount rate & time)

what is EAA

represents the cash flow per period (e.g. per year) which is equivalent to the PV of buying, operating and selling the equipment

EAA Formula

PV COST/AF

info needed for EAA

* Initial purchase cost * Expected useful lives * Annual running costs (assume incurred at the end of each yr) * Disposal proceeds (zero unless told otherwise) Quicker to use tables

What dies a lower EAA mean

better long-term cost savings.

trade offs of a truck company

Trade-offs: Replace Frequently 🚛 Always new, low operating costs, but high purchase costs. Replace Less Often 🚚 Cheaper upfront, but higher maintenance costs over time.