TVM will help you answer simple questions:

1) Would you rather be given 1000€ today, or 1000€ in 1 year?

2) Would you rather be given 1000€ today, or 1050€ in 1 year?

3) Would you rather be given 1000€ today, or 1100€ in 2 years?

4) How much is 1000€ that you will get in 10 years, worth today?

5) How long will it take me to double my investment?

TVM is a crucial concept in finance, investment analysis, and financial decision-making.

Simple interest is calculated on the initial principal (the original amount of money), and the interest is not reinvested or added to the principal.

Simple Interest (SI): FV = Principal (P) x Interest rate (r) x Time (t)

FV = P + (P * r * t)

FV = P + (P * r * d/360)

• FV: Final value

• P = Principal (initial investment)

• r = Annual interest rate (expressed as a decimal)

• t = Time in years (it represents also the number of days (d) for which the interest applies, In finance, a typical year is 360 days long).

Example of Simple Interest

• Let’s say you invest $1,000 at an annual interest rate of 5% for 3 years. The simple interest calculation would be:

𝑆𝑖𝑚𝑝𝑙𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 1,000 × 0.05 × 3 = 150

• So, the total interest earned is $150, and after 3 years, the total value will be:

𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 = 𝑃 + 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 1,000 + 150 = 1,150

Compound interest: takes into account the interest that accrues on both the initial principal and any previously earned interest over time.

• At the end of Period 1: 𝑷 + 𝑷 × 𝒓 = 𝑷(𝟏 + 𝒓) with r interest rate of the period

Beginning of period 1, the new principal is 𝑷(𝟏 + 𝒓)

• At the end of Period 2: 𝑃(1 + 𝑟) + 𝑃(1 + 𝑟) × 𝑟 = 𝑃 1 + 𝑟 (1 + 𝑟) = 𝑷(𝟏 + 𝒓)𝟐

Beginning of period 2, the new principal is 𝑷(𝟏 + 𝒓)𝟐

And so on…

• At the end of n periods: the accumulated amount is FV = 𝑷 𝟏 + 𝒓 𝒕

FV = 𝑃 (1 + 𝑟)^t

• FV: future value

• P: Principal amount

• with r the interest rate of the period

• and t the number of periods

Example 1 of Compound IR (annually)

• Now, letʹs take the same principal of $1,000, but this time the interest is compounded annually at an interest rate of 5% for 3 years.

• FV= 1,000 (1 + 0.05)^3 = 1,000× (1.05)^3 = 1,000×1.157625 = 1,157.63

𝑆𝑜, 𝑎𝑓𝑡𝑒𝑟 3 𝑦𝑒𝑎𝑟𝑠, 𝑡ℎ𝑒 𝑓𝑢𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑤𝑜𝑢𝑙𝑑 𝑏𝑒 $1,157.63.

Example 2 of Compound IR (monthly)

Now, letʹs take the same principal of $1,000, but this time the interest is compounded monthly at an interest rate of 5% for 3 years.

• FV = 1000 (1 + 0,05/12) ^12*3= $1,161.47

𝑆𝑜, 𝑎𝑓𝑡𝑒𝑟 3 𝑦𝑒𝑎𝑟𝑠, 𝑡ℎ𝑒 𝑓𝑢𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑤𝑜𝑢𝑙𝑑 𝑏𝑒 $1,161.47

These formulas allow you to compare the annual return on different investments.

• Investment over one year

(1 + R) = ( 1 + r/n) ^n

• R is the real equivalent annual interest rate,

• r is the annual interest rate,

• n is the number of times interest are compounded in a year

• Investment over n years

(1 + R)^t = (1 + r/n) ^n*t

• R is the real equivalent annual interest rate,

• r is the annual interest rate,

• n is the number of times interest are compounded in a year

• t the number of years