Week 2: Financial Mathematics - Part 1

Question 1

What is the time value of money?

Answer 1

Financial concept that explicitly recognises that $1 received today is worth more than $1 received in the future. Meaning, the value of money can’t remain the same when time changes.

. Financial managers compare the marginal benefits and marginal costs of investment projects.
. Projects usually have a long-term horizon: the timing of benefits and costs matters.
. The time value of money: a dollar received today is worth more than a dollar received in the future.

Question 2

What is the future value?

Answer 2

The value of an investment made today measured at a specific future date accounting for interest earned over the life of the investment.

Question 3

What is the present value?

Answer 3

The value today of a cash flow to be received at a specific date in the future, accounting for the opportunity to earn interest at a specified rate.

Question 4

What is interest?

Answer 4

What is interest (rate)?
- Cost price of money.

Interest is usually expressed as a % per annum (p.a.)

We will be considering two types of interest:
Simple and compounding.

Question 5

What is a lump sum?

Answer 5

A single payment made a particular time, as opposed to a number of smaller payments or instalments.

Question 6

What is simple interest?

Answer 6

Interest paid only on the initial principal of an investment, not on the interest that accrues in earlier periods.

Question 7

What is the principal?

Answer 7

The amount of money borrowed on which interest is paid.

Question 8

What is compound interest?

Answer 8

Interest earned both on the initial principal and on the interest earned in previous periods.

Question 9

What is the simple interest formula?

Answer 9

I = P x r x t

I - interest
P - principle
r - interest (per period of time)
t - time (term of the loan or investment)

Question 10

Simple interest terminology.

Answer 10

Present value (PV) – The current value of future cash flows. In practice, it is known as principal (of borrowing or lending).

Future Value (FV) – The amount an investment is worth after one or more periods of time. In practice, it is known as principal plus interest at maturity.

Time to maturity (t) – the duration (usually in years) of the interest rate arrangement.

Nominal interest rate (i) – quoted interest rate per annum.

Interest – the monetary return on saving/investment.

Question 11

Golden rule - aligning interest rate (r) and time (t)

Answer 11

It is a must that both r and t are expressed in the same time frequency.

Frequency —- Interest Rate (%)(r) —- Time (t)
Annual —- 12% (A rate) —- 10 (A periods)
Semi-annual —- 12/4 = 6% (SA rate) —- 10x2 = 20 (SA periods)
Quarterly —- 12/4 = 3% (Q rate) —- 10x4 = 40 (Q periods)
Monthly —- 12/12 = 1% (M rate) —- 10x12 = 120 (M periods)

Question 12

The equation for future value.

Answer 12

FV = PV × (1 + r)n

FV = future value of an investment
PV = present value of an investment (the lump sum)
r = interest rate per period (typically 1 year)
n = number of periods (typically years) that the lump sum is invested.

Question 13

Other variations.

Answer 13

You can calculate the interest that an investment earns know as YIELD.

r = 1/t x(FV/PV - 1)

You can calculate the time it takes to achieve you investment target (term of the investment).

t = 1/r x (FV/PV - 1)

Question 14

What is discounting?

Answer 14

The process of calculating present values.

Question 15

Compound interest formula.

Answer 15

FV = PV (1 + r)^n

Question 16

Terminology - compound interest.

Answer 16

In addition to PV, FV, t and i

Compounding frequency (m) – number of compounding periods per year. For example, if interest is paid annually, m = 1; if interest is paid every month, m = 12.

Interest interval (period) – the time period when interest is calculated and paid, which can be in any type of time unit (hour, day, week, month). Number of interest interval is defined as n = m × t.
–When m = 1, n = t × 1 = t.

Interest rate per period (r) – the interest rate per interest interval, defined as i/m.
–When m = 1, r = i/1 = i.

Question 17

Compounding more frequently than annually.

Answer 17

Compound semiannually.

• interest is paid twice per year, m = 2
• nominal interest rate, i = 8% = 0.08
• interest rate per period, r = i/m = 0.08 x 2 = 0.04
• time to maturity, n = t x m = 2 x 2 = 4

Question 18

Nominal yield - compound interest.

Answer 18

Implied rate of rate of a long-term borrowing/lending.

r = (FV/PV) ^1/n - 1

Question 19

Time to maturity - compound interest.

Answer 19

How long the investment target will be achieved.

n = natural log (FV/PV) ➗ natural log (1 + r)

Question 20

Effective interest rates - compound interest.

Answer 20

The nominal interest rate does not provide a true indication of the ‘effective’ interest rate when we compare interest rate arrangements with different compounding frequency.

Effective Annual Rate (EAR, ie)
–We need to know how much interest rate the arrangement implies, effectively
–Effective interest rate provides a ‘standardized’ measurement of interest rate, after considering different compounding frequency.
–‘Equivalent’ interest rate if compounded annually

•10% p.a. compound monthly is indifferent from ? % p.a. compound annually.

ie = (1 + i/m) ^m -1

Where:
–ie = effective annual rate (EAR)
–i = nominal interest rate
-m = compounding frequency

Question 21

Future value of cash flow streams.

Answer 21

Financial managers frequently need to evaluate streams of cash flows that occur in future periods.

Though this is mechanically more complicated than computing the future or present value of a single cash flow, the same basic techniques apply.

Two types of cash flow streams are possible: the mixed stream and the annuity.

Question 22

What is a mixed stream and what is an annuity?

Answer 22

mixed stream - A series of unequal cash flows reflecting no particular pattern.

annuity - A stream of equal periodic cash flows over a stated period of time.

Question 23

Multiple cash flows equation.

Answer 23

Practice using this.