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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 πŸ˜­πŸ˜‚

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

\[ \begin{aligned} \text{FV} &= \text{PV} \times \underbrace{(1+r)^t}_{\text{Compound Factor}} \\ \implies \text{PV} &= \text{FV} \times \underbrace{\frac{1}{(1+r)^t}}_\text{Discount Factor} \end{aligned} \]

Multiple Cashflows

\[ \begin{aligned} \text{FV} &= \sum_{t=0} c_t (1+r)^t \\ \text{PV} &= \sum_{t=0} \frac{c_t}{(1+r)^t} \end{aligned} \]

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 βœ… βœ…
Occurs every time period βœ… βœ…
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

\[ \text{EAR } = \left( 1 + \frac{\text{APR}}{m} \right)^m - 1 \]

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
Last Updated: 2024-05-14 ; Contributors: AhmedThahir

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