# ๐ฐ๏ธ TVM (Time Value of Money) Tutorial ๐บ

## โฐ A Simple Introduction [2020 Update] ๐

#### by Eric Smith

### Video: "Level I CFA: Quant The Time Value of Money - Lecture 1"

The following lectures provide a nice overview of this topic:### 1) What is Time Value of Money (TVM)?

Time Value of Money or TVM denotes the general financial concept that cash in hand today is worth more than the same amount of money that may be received in the future. This is because available money always has potential earning power via immediate investment opportunities.For example, money in hand could be deposited in a savings account at once, or utilised to buy bonds, and thereby earn a certain fixed interest rate income as compared to money which would be received at a future date. The incremental gain from the interest is known as time value of money.

Conversely, money not received would lose value due to inflation.

### 2) What is Present Value (PV)?

#### 2.1) Basic Definition

Present Value or PV denotes the current value of an amount of money or sum of income streams that is expected to be received in future.#### 2.2) Example: Harold promises $1000 after 1 Year

Let us look at the following example:- Question: Harold promises to pay us $1000 next year. What is the Present Value? Assume the interest rate is 10%.
- Answer: The PV of this money is $909.09, because if we immediately invested this cash in hand in a fixed deposit at 10% today, we would receive $90.90 after one year. So after one year we would have $909.09 (Present Value) + $90.90 (interest), which comes to $1000 (the amount Harold has promised to pay us).
- Explanation: If Harold gave us the $1000 today, we could have invested it
for one year and obtained a risk-free return of $1000 x 10% = $100.
So the PV of this money must be LESS than the $1000 he has promised us. The PV
is that amount of money which, if we deposited in a bank, would then give us
$1000 after a year.
Hence, in order to determine the present value, we have to move this future
value backwards by one year. So we divide $1000 by 1.10 as follows:
PV of $1000 promised after 1 year = $1000 รท 1.10 = $909.09.

Thus, the PV in this case is $909.09.

#### 2.3) Example: Roberta promises $1000 after 3 Years

Let us look at another example:- Question: Roberta promises to pay us $1000 after 3 years. What is the Present Value? Assume the interest rate is 10%.
- Answer: The PV of this money is $751.31.
- Explanation: To move this promised future payment backwards by three years,
we must divide $1000 by 1.10 three times. So, the PV of $1000 promised after
3 years is calculated as follows:
PV of $1000 promised after 3 years = $1000 รท 1.10 รท 1.10 รท 1.10 = $751.31.

In other words, if we invest $751.31 in a bank fixed deposit at 10% right now, we would have $1000 after 3 years (which is what Erica has promised to pay us).

### 3) What is Future Value (FV)?

#### 3.1) Basic Definition of FV

Future Value of FV is the value of a current asset at a specific date in the future, based on an assumed rate of growth. It shows how much a particular investment can grow over time.#### 3.2) Formula for FV Calculation

Thus, FV of a single annual cash flow for a single investment can be computed using the following formula:

When the compounding frequency is not annual, the following formula should be used:FV = PV(1 + IR) ^{N}where:

- FV = future value of the investment
- N = number of periods
- PV = present value of the investment
- IR = interest rate

Be careful to understand which formula to use.FV = PV(1 + IR/n) ^{nN}where:

- FV = future value of the investment
- N = number of years (not periods as above)
- PV = present value of the investment
- IR = annual interest rate in decimal format
- n = number of compounding periods per year

### 4) What is Interest Rate (IR)?

#### 3.1) Basic Definition of IR

Interest Rate is essentially a reward, fee or rent paid by a borrower for the use of an asset belonging to a lender.#### 3.2) Formula for IR Calculation

IR can be computed using the following formula:

IR = Risk Free Interest Rate + IP + LP + MP + DP where:

- IR = Interest Rate
- IP = Inflation Premium, the premium lenders demand due to the expected annual inflation.
- LP = Liquidity Premium, the premium lenders demand because of the potential lack of liquidity (i.e. less frequent trading) in an instrument.
- MP = Maturity Premium, the premium lenders demand to compensate for higher risks due to longer maturities.
- DP = Default Premium, the premium lenders demand due to the potential for defaults (i.e. the potential that borrowers may renege or go back on their promised commitments).