1. The Mathematics of Finance
Chapter 6

2. The Mathematics of Finance
Two principal concepts:
Present value
Future value
Future Value and Present Value calculations, for our purpose, are inverse mathematical operations.

3. The Mathematics of Finance
Future Value (FV) is an amount as the result of a planned set of deposits (with interest), where the per period interest rate (effective rate) is constant.
The amount in an account is measured at some future time.
Present Value (PV) – is an amount that must be deposited/invested today (or equivalently, the purchase price of a Financial Asset) to make a planed set of withdrawals, such that the account balance (with interest) goes to zero immediately after the last withdrawal.
The interest rate is the “effective” rate (per period). We presume it to be constant.
The “effective” rate - the rate of growth of wealth as the result of interest per period

4. Calculations in the Mathematics of Finance
FV is $C (once) invested for one period, where r is the effective rate per period.
FV = C + r * C = C (1 + r)
PV of $C (once) after one period where r is the effective rate per period.
(How much you have to put in your account to receive $C at the end)
PV = C / (1 + r)

5. Calculations in the Mathematics of Finance
FV of $C (once) after n periods, if r is the effective rate per period
FV = C (1 + r)n
Rule of “ 7 “:
If you invest at 10% it will take ~ 7 years to double your money.
PV of $C (once) after n periods if r is the effective rate per period
PV = C / (1 + r)n

6. Instances where it is easy to calculate the IRR
IRR for one period investment (once)

7. Instances where it is easy to calculate the IRR
IRR per period for an investment that pays $C after n periods (once)

8. Calculations in the Mathematics of Finance
An annuity is a set of n equal payments of $C each, one per period, where r is the effective rate per period.
FV of an annuity
Case 1: FV of an annuity where your account balance is measured right after last deposit

9. Calculations in the Mathematics of Finance
Case 2: FV of an annuity where your account balance is measured 1 period after last deposit Case 3: measure your account k ≥ 0 periods after the last deposit k = 0: Case 1 k = 1: Case 2

10. Calculations in the Mathematics of Finance
PV of an annuity
Case 1: 1st payment is in one period - Ordinary Annuity
Case 2: 1st payment is received immediately – Annuity Due

11. Calculations in the Mathematics of Finance
PV of accelerated or deferred annuity (relative to ordinary)
k = 1: Ordinary annuity (Case 1)
k = 0: Annuity-due (Case 2)
k > 1 – differed relative to ordinary
0 < k < 1 – accelerated relative to ordinary

12. Calculations in the Mathematics of Finance
Perpetuity is an indefinite into the future set of per period payments of $C each.
PV of perpetuity
Case 1: PV of Ordinary Perpetuity
Case 2: PV of Perpetuity Due

13. Instances where it is easy to calculate the IRR
IRR for an Ordinary Perpetuity
IRR for a Perpetuity Due

14. Calculations in the Mathematics of Finance
Growing Perpetuity:
C – 1st payment
Each next payment is (1+g) greater than the previous
PV of a Growing Perpetuity
g > 0 – growth
g = 0 – no growth (regular perpetuity)
g < 0 – declining perpetuity

15. Calculations in the Mathematics of Finance
Case 1: PV of an Ordinary Growing Perpetuity
Case 2: PV of a Perpetuity Due
PV is finite number only if g < r
g ≠ r

16. Instances where it is easy to calculate the IRR
IRR for an Ordinary Growing Perpetuity
IRR for a Growing Perpetuity Due

17. IRR
IRR – hypothetical discount rate that makes NPV = 0.
General IRR:

18. Rates of Return The Holding-Period Rate of Return:
PV – “Expenditure” on an investment
FV – “Benefit”(or Wealth) of the investment at the end of the holding period including reinvested intermediate payments

19. Nominal & Effective Rates of Interest
The effective rate of interest is the rate at which an investment account grows over the holding period
In all PV and FV calculations we always use the effective rate
The nominal rate of interest represents the way in which interest is calculated and added to investment account
Also known as:
Quoted rate
Contract rate
APR (Annual Percentage Rate)
A nominal rate of interest is always accompanied by a compounding period.

20. Effective Rates of Interest
Effective to effective (short to long)
If r is the effective rate for a sub-period (short), then effective rate for a holding period composed of n sub- periods (long) is:
geometric multiplication
power > 1

21. Nominal & Effective Rates of Interest
Effective to effective (long to short)
If r is the effective rate for a holding period composed of n sub-periods is:
geometric division
power < 1

22. Nominal & Effective Rates of Interest
If i is the per annum rate compounded m times per year, then, the effective rate for one compound period is:
also
If i is the per annum rate compounded m times per year, then, the effective rate for n compound periods is:

23. Nominal & Effective Rates of Interest
Continuous compounding (n = ∞)
If i is the per annum rate compounded continuously, then, the effective annual rate (EAR) is:

24. Calculations in the Mathematics of Finance
Growing Annuity
PV of a Growing Annuity
FV of a Growing Annuity

25. Calculations in the Mathematics of Finance
Growing Annuity
g = r
PV of a Growing Annuity
FV of a Growing Annuity

26. Calculations in the Mathematics of Finance
Mortgage
Mortgage - is a mortgage loan secured by real property
Is a French Law term meaning “death contract”
Typical mortgage:
25 years
Monthly payments
Every payment has 2 parts:
Interest
Principal

27. Mortgage Amortization Table
years
Payment
Interest
Principal
Balance
1,000 1
374.11 60
314.11
685.89 2
41.15
332.95
352.93 3
21.18

28. Mortgage
Interest vs. Principal

29. Mortgage
Principal Repayment If a mortgage makes n payments and the mortgage rate is i% / annum compounded ( and paid) m times per year , then the principal repayment at the jth payment is:

30. Mortgage Canadian Mortgage Market:
Rates are stated APR compounded semi-annually
Rate of interest is periodically reset to market rate
Term – length of time over which the rate is fixed
Interest on mortgage payments (on your home) is not tax deductible
Typical Canadian Mortgage :
25 years – Amortization period
5 years term