# The Time Value of Money The time value of money refers to the fact that one dollar today is not worth the same as one dollar tomorrow. The present value of future cash(flows) depends on the discount rate applicable to that specific cashflow. Expressed as an annual interest rate, the (discount) rate (r) determines the time value of money.

Imagine you live on a green planet, where banks offer two options. Both come with an annual interest rate of 10%.
(a) stash your cash in a secure bank account for the desired time, or
(b) to take a loan for as long as you want,
In this green world, \$100 today, are woth \$1,083 in 25 years to you, since making use of the savings account (a) that is what it will grow to over time.
Alternatively, \$1,083 in 25 years are worth \$100 to you today, since you can bring them forward using a loan product (b).

Now imagine you live on the grey planet instead, where the interest rate is only 3%: In this grey world, \$209 in 25 years are worth \$100 to you today, since you can bring them forward using a loan product (b). Alternatively, \$100 today, are woth \$209 in 25 years to you, since making use of the savings account (a) that is what it will grow to over time.

The time value of money is determines by prevailing interest rates, as you can see below. For a more general purpose (and for more efficient calculations), we introduce discount factors below.

Discount factors (df) describe todays (t=0) present value of one unit of money (\$1) in time i with: df(i) = (1+r)(t-i). If we apply an interest rate of 10% p.a., the present value or discount factor for \$1 in one year is df(1) = (1+0.1)(0-1)=0.91. Below we plotted the discount factors for different rates. What becomes clear, the higher the interest or discount rate and the longer the time horizon, the lower the present value of a future dollar.

When valuaing companies and their earnings streams we might want to make use of some formulas. Some of them are discussed below.

Formulas used for Valuation

Valuation of an annuity is not a challenge to us if we know the right formula to use. An annuity is a cashflow stream into perpetuity. Pensioners like to buy annuity insurance products to insure for longevity risk, buying such a product will provide them with a certain cash inflow every year (until they die). The standard annuity consists of cashflows of \$1 annually into perpetuity, since other annuities can be build from that. This standard annuity, with the next payment taking place in one year, has a present value (pv) to us based on the discount rate r being the single input variable, with pv=1/r.

Valuing a growing earnigns stream is not much more difficult for us. For an earnings stream growing with a rate of g p.a., the next cashflow taking place in one year cf1 and the (risk-adjusted) discount rate being k, we can use the following formula pv = cf1/(k-g).

The present value of the terminal value can be derived as follows. The above formula gives us the value of a growing earnings stream starting one year from now. When valuing a company we have to bring this earnings stream forward to today (present value of terminal value). Making use of a discount factor and the formula for a growing earnings stream (terminal value) we can calculate the present value of the terminal value as pv(tv) = dfT x cf(T+1)/(k-g). T is the last year of the detailed period.

I hope you enjoyed my explanations!