1. A. Caggia – M. Armanini Financial Investment & Pricing 2016-2017
Time Value of Money
A. Caggia – M. Armanini
Financial Investment & Pricing

2. Investments & Cashflows
Simple viewpoint …
Investment = the present commitment of money for the purpose of receiving more money later.
General viewpoint …
Investment = a set of cashflows of expenditures (negative cashflows) and receipts (positive cashflows) spanning a period of time
Commitment = IMPEGNO
Expenditures = SPESE, IN USCITA

3. Objective(s) of Investment
The objective of investment is to tailor the pattern of cashflows over time, in order to fit our needs.
This could be done in order to maximise returns.
Often future cash flows have a degree of uncertainty, and part of the design of a cash flow stream may be concerned with controlling that uncertainty, perhaps reducing the level of risk.
L'obiettivo di investimento è di adattare il modello di flussi di cassa nel corso del tempo, per adattarsi alle nostre esigenze
Spesso i flussi di cassa futuri hanno un grado di incertezza, e parte del disegno/progetto dell’andamento (stream = corso) dei flussi di cassa può concernere il controllo di detta incertezza, riducendo il livello di rischio.

4. Cashflow stream
Usually cashflow stream (either positive or negative) occur at known specific dates, such as at the end of each quarter of a year or at the end of each year. The stream can then be described by listing the value and the date of each cashflow. The stream can be deterministic (known in advance) or random (unknown in advance)

5. Cashflow stream
Example: Imagine that you will receive 10 EUR in six months time, another 10 EUR in one year and you will spend 20 EUR in two years time. This cashflow stream ca be represented by two vectors denoting the values of each individual cashflow and when these will happen. Cashflows: C = (+10,+10,-20) Payment dates: D = (0.5, 1.0, 2.0)

6. Example +     − T1 T2 T3 T4 T0= 0 time
cashflow graphical representation +    
time T1 T2 T3 T4 −
T0= 0

7. Example
cashflow graphical representation 30 +    
time − 10 10 10

8. Example
cashflow graphical representation 30 +    
time − 10 10 10

9. Example
cashflow graphical representation 33 + 3 3 3    
time − 30

10. Present and Future Values
Every financial operation generates cash inflows and/or outflows over a certain time horizon
These cashflows represent the amount of money that are expected to be received or paid over time on the back of an investment/debt decision
If the cashflows are scheduled on different maturities, their value can’t be directly compared
To be compared they need to be expressed on same timing conventions:
PRESENT VALUE and FUTURE VALUE

11. Present and Future ValuesPV FV
time = × PV DF FV

12. Present and Future ValuesPV
FV1
FV2
time
DF2 = × + × PV
DF1
FV1
DF2
FV2

13. Present and Future Values
KNOWN CASHFLOWS
UNKNOWN CASHFLOWS

14. Time Value and Interest Rates
Money has a time value because of the opportunity for investing money at some interest rate (which is the “compensation” for not spending now)
The main reasons that concur to explain the time value of time (as well as the level of interest rates) are:
Individual preferences on consumption and savings
Inflation
Uncertainties

15. Interest Rate Determinants
The main determinants of the level of interest rates are:
a real part: compensate the investor for the choice of postponing his/her consumption in terms of higher expected spending power
a nominal part: is added to compensate for the loss of spending power due to the consequences of positive inflation rates
a risk premium part: depends of the uncertainties associated to investing (e.g. likelihood of debtor not repaying the loan)
una componente reale: compensa il finanziatore (che soffre l’indisponibilità della somma prestata al debitore) per la scelta di posticipare le proprie scelte di consumo
una componente nominale: si aggiunge per compensare la perdita di potere d’acquisto per effetto dell’inflazione
una componente di premio per il rischio: dipende dalla probabilità del debitore di onorare gli impegni (x es. probabilità che il debitore non restituisca il debito)

16. Contents Compounding From Present Value to Future Value
Simple Interest
Compound Interest
Continuous Compounding

17. Compounding Compounding = to put together into a whole; to combine.
Interest Rate Compounding = to apply interest rates to investments with a certain frequency
Therefore, in order to define accurately the amount to be paid under a legal contract with interest, the frequency of compounding (yearly, half-yearly, quarterly, monthly, daily, etc.) and the interest rate must be specified.

18. Types of Compounding
Simple Compounding = interest rate is applied on a single period between today (t=0) and maturity (t=T).
Compounding N Periods = interest rate is applied sequentially N times between today (t=0) and maturity (t=T).
Continuous Compounding = interest rate is applied continuously (i.e. infinite number of periods) between today (t=0) and maturity (t=T).

19. Present to/from Future ValuesCF PV FV
time DF
Future
Value
Present
Value

20. Present to/from Future ValuesPV = FV DF CF PV FV
time DF = × PV DF FV

21. Simple Compounding CF PV FV
time

22. Compounding: N periodsCF CF CF PV FV FV FV
time

23. Compounding: ContinuousCF PV FV
time

24. Simple Compounding A is the amount invested at t=0
r is the interest rate applied (annualized)
T is the maturity of the investment (annualized)
V is the final value of the investment at t=T

25. Compounding: N periods
N is the number of equally periods per year, NT is then the total number of periods between today (t=0) and maturity (t=T)

26. Compounding: N periods
r’ is the effective interest rate, i.e. the equivalent yearly interest rate that would produce the same result without compounding
Compounding: Continuous

27. Conversion Formulas
If Rc is a continously compound interest rate, while Rm is the equivalent m times per year compound rate, the formulas relating Rc and Rm are:

28. Annuities
The present value of future cashflows, for which different discount factors apply, can be manually calculated adding the present value of each individual cashflow
There are formulas that allow to simplify the calculation of present and future values, depending on the shape of cashflows. In general, the typical format is the one of an annuity: a stream of inflows of outflows to be due on future dates

29. Annuities Constant: fixed cashflows, all equal
Ordinary: cashflows are paid at regular intervals for a fixed time horizon
In Arrears: cashflows are exchanged at the end of each relevant period
Variable: variable cashflows, change over time
Perpetuity: stream of cashflows never ending
In Advance: cashflows are exchanged at the beginning of each relevant period

30. Amortization
Amortizing a debt means to repay it back over time. There are 3 main types of models:
Reimburse S(1 + i)t of capital S at the maturity n of the debt
Repay periodic interest rate accrues over capital S (Si at any given period) and repay capital S at the maturity n of the debt
Repay both interest rate accrues and capital S over a certain number of periodic installments