# Lecture 4 Note

*2021-07-21*

# Chapter 1 Review for Chapter 2

It is a fact of life that $100 to be received after one year is worth less than the same amount today. The main reason is that money due in the future or locked in a fixed term account cannot be spent right away. One would therefore expect to be compensated for postponed consumption. In addition, prices may rise in the meantime and the amount will not have the same purchasing power as it would have at present. Finally, there is always a risk, even if a negligible one, that the money will never be received. Whenever a future payment is uncertain to some degree, its value today will be reduced to compensate for the risk. (However, in the present chapter we shall consider situations free from such risk.) As generic examples of risk-free assets we shall consider a bank deposit or a bond.

The way in which money changes its value in time is a complex issue of fundamental importance in finance. We shall be concerned mainly with two questions:

- What is the future value of an amount invested or borrowed today?
- What is the present value of an amount to be paid or received at a certain time in the future?

The answers depend on various factors, which will be discussed in the present
chapter. This topic is often referred to as the *time value of money*.

## 1.1 Simple Interest

Suppose that an amount is paid into a bank account, where it is to earn interest. The future value of this investment consists of the initial deposit, called the principal and denoted by \(P\), plus all the interest earned since the money was deposited in the account.

To begin with, we shall consider the case when interest is attracted only by the principal, which remains unchanged during the period of investment. For example, the interest earned may be paid out in cash, credited to another account attracting no interest, or credited to the original account after some longer period.

After one year the interest earned will be \(rP\), where \(r > 0\) is the interest rate. The value of the investment will thus become \(V (1) = P +rP = (1+r)P\). After two years the investment will grow to \(V (2) = (1 + 2r)P\). Consider a fraction of a year. Interest is typically calculated on a daily basis: the interest earned in one day will be \(\frac{1}{365}rP\). After \(n\) days the interest will be \(\frac{n}{365}rP\) and the total value of the investment will become \(V (\frac{n}{365}) = (1+ \frac{n}{365}r)P\). This motivates the following rule of simple interest: The value of the investment at time \(t\), denoted by \(V (t)\), is given by \[ V (t) = (1+tr)P, \]

where time \(t\), expressed in years, can be an arbitrary non-negative real number. In particular, we have the obvious equality \(V (0) = P\). The number \(1+rt\) is called the growth factor . Here we assume that the interest rate \(r\) is constant. If the principal \(P\) is invested at time \(s\), rather than at time 0, then the value at time \(t\geq s\) will be \[ V (t) = (1+(t ā s)r)P. \]

## 1.2 Periodic Compounding

Once again, suppose that an amount \(P\) is deposited in a bank account, attracting interest at a constant rate \(r > 0\). However, in contrast to the case of simple interest, we assume that the interest earned will now be added to the principal periodically, for example, annually, semi-annually, quarterly, monthly, or perhaps even on a daily basis. Subsequently, interest will be attracted not just by the original deposit, but also by all the interest earned so far. In these circumstances we shall talk of discrete or periodic compounding.

In general, if \(m\) interest payments are made per annum, the time between two consecutive payments measured in years will be \(\frac{1}{m}\), the first interest payment being due at time \(\frac{1}{m}\). Each interest payment will increase the principal by a factor of \(1+\frac{r}{m}\). Given that the interest rate \(r\) remains unchanged, after \(t\) years the future value of an initial principal \(P\) will become

\[ V(t) = \left(1+\frac{r}{m}\right)^{tm}P, \]

because there will be \(tm\) interest payments during this period. In this formula
\(t\) must be a whole multiple of the period \(\frac{1}{m}\). The number \((1+\frac{r}{m})^{tm}\) is the *growth factor*.

## 1.3 Streams of Payments

An *annuity* is a sequence of finitely many payments of a fixed amount due
at equal time intervals. Suppose that payments of an amount \(C\) are to be
made once a year for \(n\) years, the first one due a year hence. Assuming that
annual compounding applies, we shall find the present value of such a stream
of payments. We compute the present values of all payments and add them up
to get
\[
\frac{C}{1+r}+\frac{C}{(1+r)^2}+\dots + \frac{C}{(1+r)^n}
\]
It is sometimes convenient to introduce the following seemingly cumbersome
piece of notation:

\[ PA(r,n) = \frac{1}{1+r}+\frac{1}{(1+r)^2}+\dots + \frac{1}{(1+r)^n} \]

This number is called the present value factor for an *annuity*. It allows us to
express the present value of an annuity in a concise form:

\[ PA(r, n)\times C. \] The expression for \(PA(r, n)\) can be simplified by using the formula

\[ a + qa + q^2a +\dots+ q^{nā1}a = a\frac{1-q^n}{1-q} \]

In our case \(a = \frac{1}{1+r}\) and \(q = \frac{1}{1+r}\), hence

\[ PA(r,n) = \frac{1-(1+r)^{-n}}{r}. \]