Time value of money

This is an introductory page in Fixed Income. If you are unfamiliar with any of the terms, you can refer to the Fixed Income Glossary.

The Time Value of Money (TVM) refers to the idea that money available immediately is worth more than the same amount worth some time in the future. This is because the money can earn interest, hence is worth more the earlier that it is received.

For example, if interest rates were 5%, then $100 that is invested today will become $105 in a year. Conversely, $100 received in a year, is only worth $\$100 / 1.05 = \$95.24$ today. Hence, it is better to receive $100 today, than it is to receive $100 in a year.

Suppose that you have $1000 now, and it is in a bank that earns 10% interest per annum. How much would it be worth in 3 years?
In the first year, it would be worth $\$1000 \times 1.1 = \$1100$.
In the second year, it would be worth $\$1100 \times 1.1 = \$1210$.
In the third year, it would be worth $\$1210 \times 1.1 = \$1331$.

The future value (FV) of Principal $P earning interest rate $i \%$ for $n$ times periods is

$$FV = P ( 1 + i \% ) ^ n$$

If you place $1000 in a bank that earns 5% interest per year, how much is it worth in 10 years?

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It is worth $\$1000 \times (1 + 5 \% ) ^ { 10 } = \$1628.89$.

Present Value of a single amount

Suppose you want to retire in 20 years and have $1,000,000. How much money must you place in a bank, earning 10% interest? In other words, what is the Principal $P that will be worth $1,000,000 in 20 years?

From the above equation, we get that

$$P = \$1,000,000 \times ( 1 + 10\% ) ^ { -20} = \$148,643.$$

The Present Value (PV) of a Future Value earning interest rate $i \%$ for $n$ times periods is

$$PV = \frac{ FV } { ( 1 + i \% ) ^ n}.$$

Present value of several amounts

How would we calculate the present value of several payments that are spread across the years? We apply the same approach, treating each individual payment separately, and then summing across the different time periods.

The Present Value (PV) of several Future Values of $FV_t$ at time period $t$, earning interest rate $i \%$ is

$$PV = \sum_{t=0}^n \frac{ FV_t } { ( 1 + i \% ) ^ t}.$$

An annuity is a type of payment in which a set amount of money is received throughout each of the $n$ periods.

For an annuity with constant payments $A$ over $n$ periods, we can apply the Geometric Progression Sum with the constant ratio $\frac{1}{ 1 + i \% }$ to obtain that

$$PV = \sum_{t=1}^n \frac{ A } { ( 1 + i \% ) ^ t} = \frac{ A} { i \% } \left( 1 - \frac{ 1} { (1+ i \%) ^ n } \right).$$

Present Value of a Perpetuity

A Perpetuity is a type of payments of a set amount of money that occur on a routine basis and continues forever.

For a perpetuity with constant payments $P$ indefinitely, we can apply the Infinite Geometric Progression Sum with the constant ratio $\frac{1}{ 1 + i \% }$ to obtain that

$$PV = \sum_{t=1}^{\infty} \frac{ P } { ( 1 + i \% ) ^ t} = \frac{ P} { i \% }.$$

Present Value of a growing Perpetuity

When the perpetual payments grow at a fixed rate, then we can still apply the Geometric Progression to sum up the series to infinity.

For a perpetuity with payments that grow at a fixed rate $g$, IE $P_t = P (1+g\%) ^t$, we can apply the Infinite Geometric Progression Sum with the constant ratio $\frac{1+g \%} { 1 + i \% }$ to obtain that

$$PV = \sum_{t=1}^{\infty} \frac{ P (1 + g \%)^t } { ( 1 + i \% ) ^ t} .$$

In the case that $i > g$, the ratio is less than 1 and hence this sum will converge to

$$\frac{ P } { i - g }.$$

In the case that $i \leq g$, then the ratio is more than 1 and hence the sum diverges to infinity.

This is a common approach used for stock valuation, known as the Gordon Growth model.