Term 1 - Topic 2 - Basic Mathematics of Finance

Question 1

First basic principle of Finance: ‘‘[…] value of money’’

Money […] is worth […] than money in the […] because I can use money today to invest and earn interest on it.

Answer 1

First basic principle of Finance: ‘‘Time value of money’’

Money today is worth more than money in the future because I can use money today to invest and earn interest on it.

Question 2

Future Value = […] Value + […]

Since the interest income is equal to the present value multiplied by the interest rate, r

Future Value = Present Value + (…)
or
Future Value = Present Value * (…)

Answer 2

Future Value = Present Value + Interest Income
Since the interest income is equal to the present value multiplied by the interest rate, r

Future Value = Present Value + (r * Present Value)
or
Future Value = Present Value * (1 + r)

Question 3

[…] interest occurs when interest paid on an investment during the first period is added to the principal; then during the next period, interest is earned on the combined (principal + interest) sum and so on.
So you are earning interest on interest as time goes by.

The general formula for finding the future value of any amount is:

FVn = […]

where:
FVn = the future value of an investment at the end of n years
n = number of years (or periods)
r = the annual interest rate
PV = the present value, or original amount invested at the start of the first year.

Answer 3

Compound interest occurs when interest paid on an investment during the first period is added to the principal; then during the next period, interest is earned on the combined (principal + interest) sum and so on.
So you are earning interest on interest as time goes by.

The general formula for finding the future value of any amount is:

FVn = PV(1 + r)^n

where:

FVn =	the future value of an investment at the end 		of n years
n    = 	number of years (or periods)
r     = 	the annual interest rate
PV  = 	the present value, or original amount 			invested at the start of the first year.

Question 4

To compare an amount of money in the future with an amount of money in the present, we calculate the […] value of the future amount of money.
The present value of a delayed payoff may be found by multiplying the payoff by a discount factor which is […] than 1.
Present Value (PV) = Future Value X discount factor
This discount factor is the value today of £1 received in the future.
It is usually expressed as the reciprocal of 1 plus a rate of return:
Discount factor = […]

Answer 4

To compare an amount of money in the future with an amount of money in the present, we calculate the present value of the future amount of money.
The present value of a delayed payoff may be found by multiplying the payoff by a discount factor which is less than 1.
Present Value (PV) = Future Value X discount factor
This discount factor is the value today of £1 received in the future.
It is usually expressed as the reciprocal of 1 plus a rate of return:
Discount factor = 1/(1+r)

Question 5

Discounting: Present Value = […] / […]

General formula to calculate the present value of money earned n years from today:
Present Value= […] n years in the future / […]
PV = FVn / […]

Answer 5

Discounting: Present Value = Future Value / (1+r)

General formula to calculate the present value of money earned n years from today:
PV = Future Value n years in the future / (1 + r)^n
PV = FVn / (1 + r)^n

Question 6

The present value of the property is also its […].
To calculate present value, we discount […] by […] offered by […] in the capital market.
This rate of return is often referred to as the […] rate, […] rate, or […].
It is called the […] cost because it is the return foregone by investing in the project rather than investing in securities.

Answer 6

the present value of the property is also its market price.
To calculate present value, we discount expected payoffs by the rate of return offered by equivalent investment alternatives in the capital market.
This rate of return is often referred to as the discount rate, hurdle rate, or opportunity cost of capital.
It is called the opportunity cost because it is the return foregone by investing in the project rather than investing in securities.

Question 7

The formula for calculating Net Present Value can be written as:
NPV = [...] 
where:
i=Required return or discount rate
t=Number of time periods​

Answer 7

The formula for calculating NPV can be written as
NPV = [Cash flow / (1+i)^t] - Initial investment
where:
i=Required return or discount rate
t=Number of time periods
​

Question 8

Rate-of-Return Rule:
Accept investments that offer rates of return in […] of their […].
i.e. Invest as long as the […] on the […] exceeds the […] of […] on equivalent investments in the capital market.

Answer 8

Rate-of-Return Rule - Accept investments that offer rates of return in excess of their opportunity costs of capital.
i.e. Invest as long as the return on the investment exceeds the rate of return on equivalent investments in the capital market.

Question 9

Net Present Value Rule: Accept investments that have […] net present values.

Answer 9

Net Present Value Rule - Accept investments that have positive net present values.

Question 10

Second basic financial principle:

[…] money is worth more than […] money.

Answer 10

Second basic financial principle:

Safe money is worth more than risky money.

Question 11

Using […] (FVIF) Tables:
A shortcut for calculating future values is by using FVIF tables.
FV = PV(FVIFr,n)

Answer 11

Using Future Value Interest Factor (FVIF) Tables:
A shortcut for calculating future values is by using FVIF tables.
FV = PV(FVIFr,n)

Question 12

A shortcut for calculating […] is by using PVIF tables (often called discount tables)

PV = FV(PVIFr,n)

Answer 12

A shortcut for calculating present values is by using PVIF tables (often called discount tables)

PV = FV(PVIFr,n)

Question 13

Annuities: This is the case where you receive a […] stream of […] sized cash flows C, for a […] period n.

Answer 13

Annuities: This is the case where you receive a constant stream of equal sized cash flows C, for a fixed period n.
More info: https://www.investopedia.com/retirement/calculating-present-and-future-value-of-annuities/

Question 14

The future value of an annuity is calculated as follows:
𝐹𝑉= [...]
where:
PMT= payment (cashflow)
r 	= interest rate
n 	= number of periods

Answer 14

The future value of an annuity is calculated as follows:
𝐹𝑉= 𝑃𝑀𝑇∗[( (1+𝑟)^𝑛 −1 )/𝑟]
where:
PMT	= payment (cashflow)
r 	= interest rate
n 	= number of periods

Question 15

An example of Annuities in daily life is […] payments, which are calculated using annuities formula to ensure that you pay the same amount every month over a period of years so that by the end of that period, you will have paid back the entire loan along with the interest required.

Answer 15

An example of Annuities in daily life is loan payments. Loan payments are calculated using annuities formula to ensure that you pay the same amount every month over a period of years so that by the end of that period, you will have paid back the entire loan along with the interest required.

Question 16

The present value of an annuity is calculated as follows:

𝑃𝑉=(𝑃𝑀𝑇/r) * […]

Answer 16

The present value of an annuity is calculated as follows:

𝑃𝑉=𝑃𝑀𝑇/𝑟 * [ 1 − ( 1/(1+𝑟)^𝑛 ) ]

Question 17

[…] : Is where you receive a […] stream of […] sized cash flows/payments (PMT), with no end.
As n tends to ∞ (no end) an annuity becomes a […].
The present value of an perpetuity is calculated as:
[…]

Answer 17

Perpetuity: Is where you receive a constant stream of equal sized cash flows/payments (PMT), with no end.
As n tends to ∞ (no end) an annuity becomes a perpetuity.
The present value of an perpetuity is calculated as:
𝑃𝑉=𝑃𝑀𝑇/𝑟

Question 18

The general formula for finding the future value of any amount is:
FVn = PV * […]
where:
FVn = the future value of an investment at the end of n years
n = number of years
m = number of compounding periods per year
r = the annual interest rate
PV = the present value, or original amount invested at the start of the first year.

Note: In most of the examples we looked at previously, m was 1 (once per year) so we simply didn’t worry about m

Answer 18

The general formula for finding the future value of any amount is:
FVn = PV * (1 + r/m)^mn
where:
FVn = the future value of an investment at the end of n years
n = number of years
m = number of compounding periods per year
r = the annual interest rate
PV = the present value, or original amount invested at the start of the first year.

Note: In most of the examples we looked at previously, m was 1 (once per year) so we simply didn’t worry about m

Question 19

Nominal vs Real:

Approximate Real Interest Rate
i (nominal) = […]
r = […]

Exact Real Interest Rate
i = r + π + (r x π)
r π + r = i – π
r(π + 1) = i – π
r = (i – π)/(π + 1)

Answer 19

Nominal vs Real:

Approximate Real Interest Rate
i (nominal) = r (real) +π (inflation)
r = i – π

Exact Real Interest Rate
i = r + π + (r x π)
r π + r = i – π
r(π + 1) = i – π
r = (i – π)/(π + 1)