Concept Of Present Value

# CHAPTER 1: THE CONCEPT OF PRESENT VALUE

## Introduction

Present value is a simple concept to grasp. It is derived from the notion that time has value in monetary term. The concept is applied to money or cash flow occurring in the future and present value simply states their current value. The concept is fundamental to finance and adopted in all valuation aspects of financial product including options and structured products.

This book shall look into the concept with respect to financial instruments namely interest rate and foreign exchange without delving into option theory.

## Basic Investment

A simple investment accessible to most of us is fixed deposit. A fixed deposit is a deposit with a fixed return and maturity placed with a financial institution. Unlike a saving deposit, a fixed deposit can’t be withdrawn prior to its maturity. This however is subject to the regulatory policy as well as internal policy of the bank.

#### Example 1-1

Let’s consider an investment of MYR1,000,000.00 in a 1-month fixed deposit at a rate of 3.00%p.a with a financial institution. On placement date the investment amount is debited from our account. On maturity, the investor will receive MYR1,000,000 and the interest due.

Diagram 1-1

The basic formula for calculating interest is:

$$I = P\dfrac{rt}{12}$$

Equation 1-1

CALCULATOR: EQUATION 1-1

where;

$$I$$ = interest
$$r$$ = nominal interest rate
$$P$$ = principal amount invested
$$t$$ = period of investment in terms of months

Using Equation 1-1 to calculate the interest due;

$$I = \textrm{MYR1,000,000 x} \dfrac{\textrm{0.030 x 1}}{12}$$ $$I= \textrm{MYR2,500}$$

The amount returned to the investor on maturity therefore is RM1,002,500.00 being principal and the interest due. We shall refer to the maturity amount as future value (FV) being the sum of the principal and interest due i.e $$FV=P+I$$. Another basic formula can be derived from this to yield;

$$FV = P\left(1+\dfrac{rt}{12}\right)$$

Equation 1-2

If the target future value is known, the initial investment amount required can be easily calculated by rearranging Equation 1-2 to:

$$P= \dfrac{FV}{\left(1+\dfrac{rt}{12}\right)}$$

Equation 1-3

If an investor requires a specified amount on at the end of the second month such as MYR1,005,000 for instance, the required investment amount can be calculated using Equation 1-3. $$P = \dfrac{\textrm{MYR1,005,000}}{1+ \left(\dfrac{\textrm{0.030 x 2}}{12}\right)}$$ $$P= \textrm{MYR1,000,000}$$

CALCULATOR: EQUATION 1-3

Investment that has periodic interest payment requires special attention and minor modification to the formula. Let's assume that our 3.00% nominal rate is paying monthly interest to the account (common to savings accounts, nowadays), How do we calculate the amount needed to be kept in the account to achieve the target maturity amount (MYR1,500,000).

Firstly, Equation 1-1 has to be modified to cater for the periodic interest. At the end of the first month, an amount of MYR2,500 ($$I_1$$) will be credited being the interest due for the period (see above calculation). The balance will increase to MYR1,002,500 ($$FV_1$$) and the interest due at the end of the second month ($$I_2$$) is calculated in the same manner. The balance and the end of the second month ($$FV_2$$) can then be calculated and is $$FV_1 + I_2$$.

\begin{align} FV_2 &= FV_1 + I_2 \\ &= FV_1 + FV_1\dfrac{rt}{12} \\ &= FV_1\left(1+ \dfrac{rt}{12}\right) \\ &= P\left(1+ \dfrac{rt}{12}\right)\left(1+ \dfrac{rt}{12}\right) \\ &= P\left(1+ \dfrac{rt}{12}\right)^2 \end{align}

where $$t$$ is now the interest period, which is one (1) month.

As a general equation, it is often stated as the following:

$$FV= P\left(1+ \dfrac{rt}{12}\right)^2$$

Equation 1-4

We have just introduced another powerful concept - compounding, denoted by $$^2$$ in the formula. We will relook at compounding at a later stage with more examples, its application by various financial products and the underlying assumptions.

If the account doesnt pay monthly interest, one would need MYR1,000,000 to get a balance of MYR1,005,000 in the account after two months when the nominal rate is 3.00% per annum. If the account pays monthly interest, we can rearrange Equation 1-4 to:

CALCULATOR: EQUATION 1-5

$$P= \dfrac{FV}{\left(1+\dfrac{rt}{12}\right)^2}$$

Equation 1-5

We can then calculate the required deposit amount to have a balance of MYR1,005,000 if the account pay monthly interest at the rate of 3.00% p.a.

\begin{align} P &= \dfrac{1,005,000}{\left(1+\dfrac{0.03 \textrm{x} 1}{12}\right)^2} \\ &= \textrm{999,993.78}\\ \end{align}

Equation 1-5 can be generalized for mutiple period as:

\begin{align} P &= \dfrac{FV}{\left(1+\dfrac{rt}{12}\right)^n}\\ \textrm{where } n &= \textrm{number of equal interest period.} \end{align}

Equation 1-6

### Day Count Convention

Interest calculation is subjected to the day count convention used by the currency of the investment and the financial product itself. The numerous convention are explained clearly by Wikipedia. For the purpose of our illustrations, two common conventions for short dated investment will be used, namely:

1. Actual/365 Fixed
2. Actual/360

The term ‘Actual’ replaces the term ’$$t$$’ in Equation 1-1 and refers to the actual number of days of the investment. The term ‘365’ and ‘360’ replaces the term ’12’ in the equation and refers to the number of days in a year. Equation 1-1 can then be modified to:

$$I = P\dfrac{rt}{365 \textrm{ or } 360}$$

Equation 1-7

CALCULATOR: DAY COUNT FACTOR
Previous calculation of interest due uses $$\frac{rt}{12}$$ for simplicity. For 'Actual/365 Fixed' convention, Equation 1-2 is modified to:

$$FV=P\left(1+\dfrac{rt}{365}\right)$$

Equation 1-8

Assuming there are 30 days in the month, the amount due on maturity based on 'Actual/365 Fixed' convention will be;

$$FV=\textrm{1,000,000}\left(1+\dfrac{\textrm{0.03 x 30}}{365}\right)$$

$$FV=\textrm{1,002,465.75}$$

The term $$\frac{t}{360}$$ or $$\frac{t}{365}$$ in Equation 1-7 is referred to as the day count factor which is subjected to the method of calculation described in relevant documents (see Wikipedia for list of source documents).

A calculator is provided to calculate the day count factor based on various conventions. Some of these conventions however are more relevant for bond type instruments/ products as implied by the optional fields 'Maturity', 'Frequency' and 'Next Coupon'. For 'Actual/Actual' convention type, these information are mandatory except for the 'Maturity' field.

Note:

The calculator does not check the validity of the coupon period as provided in 'Start Date' and 'Next Coupon' and assumes that dates provided are valid.