Time Value of Money¶
You should never compare money across different time instants. We can only compare at the same instant.
| When we take cashflow ___ in time | Name |
|---|---|
| forward | compounding |
| backward | discounting |
Return for every investment is a compensation
- Time
- Inflation
- Risk
In a finance interview, if you’re not sure of the answer, just say it’s compounding 😭😂
Keywords¶
| Denotion | Expressed as | Value of something at | |
|---|---|---|---|
| Present Value | PV | Currency | \(t = 0\) (not even \(t \approx 0\)) |
| Future Value | FV | Currency | \(t > 0\) |
| Interest Rate Discount Rate Compound Rate Opportunity cost of capital Required return | \(r\) | % | Exchange rate between present & future value |
| Number of Periods | \(n\) or \(t\) | ||
| Timeline | Graphical reprsesentation of the timing of cash flows |
Singular Cashflow Formula¶
Multiple Cashflows¶
where \(c_t\) can be
| Flow Type | |
|---|---|
| Inflow | \(c_t>0\) |
| Outflow | \(c_t<0\) |
Types of Interest¶
If \(P =\) original principal amount
| Type | FV |
|---|---|
| Simple | \(P \times (1+r) \times t\) |
| Compound (Default) | \(P \times (1+r)^t\) |
Types of Cashflows¶
Infinite series of cashflows which has
eg: Preference share in a corporation
| Perpetuity | Annuity | |
|---|---|---|
| Finiteness | Infinite | Finite |
| Term | Forever | |
| Fixed Cashflow | ✅ | ✅ |
| Occurs every time period | ✅ | ✅ |
| Present Value | \(\frac{c}{r}\) | \(\frac{c}{r} \left[ 1-\frac{1}{(1+r)^t} \right]\) |
| Future Value | N/A | \(\frac{c}{r} \left[ (1+r)^t - 1 \right]\) |
Conceptual understanding of long-term loan¶
Every [equal] installment is actually a combination of
- interest payment
- principal repayment
As time goes on, your installment will be constituting: less of interest repayment & more of principal repayment
I missed a few classes¶
Interest Rates¶
APR¶
Annual Percentage Rate
EAR¶
Effective Annual Rate
The actual interest rate you are paying
where \(m =\) interest compounding frequency
This is the value of \(r\) we use when calculating present/future value
Compounding Frequency¶
| \(m\) | |
|---|---|
| Annual | 1 |
| Semi-Annual | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |
| Hourly | 365 * 24 |
| Minutely | 365 * 24 * 60 |
| Second | 365 * 24 * 60 * 60 |
As we go from annual compounding towards more frequent compounding frequency, we are moving from discrete compounding to continuous compounding
Compounding Cycle¶
Frequency of compounding
Let \(m\) be the compounding cycle, ie number of compounding per year $$ \begin{aligned} \text{FV} &= \text{PV} \cdot (1+r)^t \ &= \text{PV} \left(1 + \dfrac{r}{m} \right)^{mt} & \text{(Discrete)} \ &= \text{PV} \times e^{rt} & \text{(Continuous)} \end{aligned} $$
IDK¶
If you are in the middle of time period, and certain cashflows have already been taken, $$ \text{PV}' = \text{PV} \times \dfrac{1}{(1+r)^{t_a/t_b}} $$
- \(t_a=\) Time elapsed in current time period
- \(t_b=\) Time Period