Week 3: Financial Mathematics - Part 2

Question 1

Multiple cash flow question characteristics.

Answer 1

The questions states the:
. Interest rate frequency
. Time interval
. Cash flow interval

Eg.
Interest rate frequency - 10% p.a.
Time interval - annual
Cash flow interval - $30,000 p.a.

Question 2

j.

Answer 2

Value of j ranges from 0 to n.

Question 3

Multiple cash flows - Annuities.

Answer 3

When the periodic cash flows are of the same amount, the pattern is defined as annuity.

Eg. Pension payments, wages, rental.

There are two basic types of annuity, ordinary and annuity due.

These reduce work for you, you don’t have to work out tons of fractions.

Question 4

What is ordinary annuity and annuity due?

Answer 4

ordinary annuity - An annuity for which the payments occur at the end of each period.

Eg. Mortgage repayments and wages

annuity due - An annuity for which the payments occur at the beginning of each period.

Eg. Lease payments and rental

Question 5

What is a perpetuity?

Answer 5

A level cash flow stream that continues forever.

A perpetuity is an annuity with an infinite life; it promises to pay the same amount at the end of every year forever.

PV = PMT ×∑ 1/(1+r)^t

Infinity symbol is ∑ and t=1 is below ∑.

Same as PV = PMT/r
Formula sheet says PV0 = CF/r

Question 6

PMT.

Answer 6

Cash flow interval = CF/m.

Question 7

What is amortisation?

Answer 7

Amortisation is the paying off of debt with a fixed repayment schedule in regular instalments over a period of time.

PV annuity (amortisation)

Question 8

What is sinking find?

Answer 8

Sinking fund is establishing a fund with a fixed payment schedule in regular instalments over a period of time for future capital expenditure.

FV annuity (sinking fund)

Question 9

Multiple cash flow - annuities.

Answer 9

Compared with ordinary annuity, annuity due is received/paid one period in advance.

Hence the extra (1 + r) in the formula.

Question 10

Annuity - other application.

Answer 10

Deferred annuity - first payment made in period d (d>1).

Amount of periodic payment (annuity), such as mortgage - place in all variables and then rearrange formula to solve for PMT.

Time to maturity - place in all variables and then rearrange formula to solve for t (n = t x m).

Question 11

Perpetuity (slides).

Answer 11

• Perpetuity is a special ordinary annuity where the cash flows begins at the end of each period and continues perpetually.
• Perpetuity only has present value, but no future value (infinity).
• PV of perpetuity formula can be obtained from PV of ordinary annuity.

PV = PMT/r

Question 12

Growing perpetuity.

Answer 12

• Cash flows grow at a constant speed and continues perpetually.
• Sam receives a stream of cash flows and the first payment is $1,000 in one year. After that, the annual cash flow will grow at 2% p.a. What is the value of this income stream if the discount rate is 10% p.a.?
• PV of growing perpetuity

PV = CF/r - g, r>g
PV = 1000/0.1 - 0.02 = 12,500

Question 13

Mixed cash flow pattern.

Answer 13

• Some cash flow patterns can be very complicated.
- avoid ‘easy thinking’, some patterns need more than one formula.

• In practice, we care about PV more than FV.
- ‘Present’ is more realistic and easier to compare with alternative choices.

• Methodology to ‘tackle’ mixed cash flow pattern – ‘Divide and Conquer’
- Dismantle a mixed cash flow pattern into stages
- Each stage can be sorted out by available formulae
. PV of lump sum payment, PV of annuities (ordinary, due or deferred)
- Sum up these PVs and obtain the PV of mixed cash flow pattern
- If you need FV, just one step further FV=PV(1+r)^n