## Introduction

The time value of money is all about the value of money today compared to the future. Alternatively, you could consider the value of a dollar in the future in today’s terms. The concept is often demonstrated using questions such as, “Would you prefer to have $1,000 today or $1,000 in two years time?”. The obvious answer is to take the money today. What about if the question was “Would you prefer to have $1,000 today or $2,000 in five years time?”. A different question altogether would require the computation of which represents the higher value ^{[1]}.

Consequently, money’s time value considerably impacts financial decision-making. An example is whether or not a business should pay cash for a machine today or finance it over a period of time. Comparing the two using appropriate formulae will determine the better financial decision from a value perspective. The value perspective would still need to be balanced against other issues such as liquidity and gearing levels.

#### Some Important Concepts

The time value of money is relevant because of:-

- Opportunity cost – if you receive $1 today you can invest it in an opportunity. If you don’t receive the dollar until a year hence, that opportunity is lost.
- Inflation – Economies generally experience inflation to some degree, so that $1 in 12 months time will have less purchasing power than $1 today
- Certainty – If you get $1 today, you have it. If you have to wait for twelve months, it is not as certain.

#### Rule of 72

The Rule of 72 provides an indication of how long it takes to double your investment given a particular interest or discount rate.

$$ Rule\ of\ 72\ =\frac{72}{Interest\ Rate}$$

So if you were to invest money at 8% the result would be:-

$$Time\ to\ double\ principal\ = \frac{72}{8}\ =\ 9\ years$$

## Time Value of Money

This video from Youtube, and I’m sure you can find others, looks at the Time Value of Money. You could also search for others around the other topics covered in this post.

## Compound vs Simple Interest

Compound interest is where the interest paid on an investment is added to the principal originally invested. At the end of the second period, the interest is calculated on the total principal invested and the new interest is added to the principal again. As a result, the principal invested keeps increasing and so does the interest earned. Alternately, simple interest is where the interest is paid on the principal and returned to the investor. The principal continues to attract interest based on the original investment.

$$FVn=PV×(1+i)n$$

The formula is Future Value after term = Present Value x (1 + interest rate) x term. As an example, the following formula calculates the future value of $1,000 where the investment term is 20 years and the interest rate is 10%.

$$FV_{20} = 1{,}000 \times (1+0.1)^{20}$$

$$FV_{20} = 1{,}000 \times (1.1)^{20}$$

$$FV_{20} = 1{,}000 \times 6.72750$$

$$FV_{20} = $6{,}727.50$$

This calculation is based on the Future Value where compound interest is being earned and added to the original principal over the term. If this was done with simple interest, the total return would be only $3,000. ($1000 original principal plus 20 years of 10% interest, i.e. $2,000). Such an example demonstrates the power and value of superannuation where the investment is compounding over a person’s working life.

#### Future Value Interest Factor

In the example above, the final step involves multiplying the original principal invested by 6.72750. This number is the Future Value Interest Factor (FVIF). You can use this figure to multiply any principal sum for the given interest rate and given period.

#### Compounding Periods

The value of n represents the number of periods, not years. So if interest is paid quarterly, then n for 10 years would be 40. The more frequently interest is paid, the higher the future value of the investment. Bear in mind that this also applies to borrowings and it is not uncommon for lenders to calculate interest daily or monthly.

## Discounting and present value

Calculation of future values is useful, but another useful tool is calculating what the value of money to be received at a future date will be today. One difference in terminology with this calculation is that instead of “interest rate”, the term “discount rate” is used. The formula for calculation is:-

$$PV = FV_n \times {1 \over (1+i)^n}$$

An example would be that you want to know how much to invest now, at an interest rate of 6% compounding annually, so that in 15 years’ time you will have $200,000.

$$PV = 200{,}000 \times {1 \over (1+0.06)^{15}}>)$$

$$PV = 200{,}000 \times {1 \over 2.397}$$

$$PV = 200{,}000 \times 0.4172 = $83{,}440$$

So if we were to invest $83,440 today at 6%, it would return $200,000 in 15 years. This can be proved by working forward to calculate the future value from the present value.

$$FV_{15} = 83{,}440 \times (1 + 0.06)^{15}$$

$$FV_{15} = 83{,}440 \times 2.397$$

$$FV_{15} = $200{,}005.69$$

The $5 variance is simply the result of rounding issues.

## Annuities

In the real world, time and money usually involves inflows or outflows on a periodic basis. Examples are rent, loan payments or superannuation payments. These are called **annuities**. The textbook defines an annuity as “…a series of **equal** dollar payments that are made at the end of equidistant points in time, such as monthly, quarterly or annually, over a finite period of time, such as three years” (Titman et al. 2019, p. 170)^{[2]}. Just as with lump sums, annuities can have future and present values. Annuities can be of two types; those payable at the end of a period, known as **ordinary annuities** or those payable at the beginning of a period, known as an **annuity due**. If an annuity is going to continue forever it is known as a **perpetuity**.

#### Future Value of an Annuity

The future value of an annuity is the value at the end after all payments have been made. It is calculated using the formula below where FV is future value, *n* is the term, PMT is the payment and *i* is the interest rate.

$$FV_n = PMT \times \left[ {(1+i)^n – 1 \over i} \right]$$

The following example shows the future value if you were to save $100 a month over 20 years that pays 6% compounding monthly.

$$FV_n = 100 \times \left[ {(1+0.005)^{240} – 1 \over 0.005} \right]$$

$$FV_{20} = 100 \times \left[{(1.005)^{240} − 1] \over 0.005} \right]$$

$$FV_{20} = 100 \times \left[{{3.31} − 1] \over 0.005} \right]$$

$$FV_{20} = 100 \times \left [{{2.31} \over 0.005} \right]$$

$$FV_{20}= 100 \times $462 = $46{,}200$$

#### Present Value of an Annuity

The present value of an annuity is the value at the beginning of the term. In other words, what would you have to get today that would equal receiving a certain level of payments over a period of time. It is calculated using the formula below the same values apply as for the above formula.

$$PV = PMT \times \left[ {{1 – {1 \over (1 + i)^n}} \over i } \right]$$

So an example of the Present Value of an annuity paying $1,000 a month over 10 years discounted monthly at an annual rate of 12% would be as follows.

$$PV = 1000 \times \left[ {{1 – {1 \over (1 + 0.01)^{120}}} \over 0.01} \right]$$

$$PV = 1000 \times \left[ {{1 – {1 \over 3.30}} \over 0.01} \right]$$

$$PV = 1000 \times \left[ {1 – 0.303\over 0.01} \right]$$

$$PV = 1000 \times 69.70 = $69{,}700$$

## Perpetuities

A cash flow without an end date is a perpetuity. An example are shares where dividends will be paid for as long as you own the shares. Likewise, rent for as long as you own the property. Perpetuities fall into two different categories; level and growing.

#### Level Perpetuities

A level perpetuity is where the payments are fixed over time. To calculate its present value, the formula is:-

$$PV = {PMT \over i}$$

An example is an investment that guarantees you to pay $20,000 per annum forever. If the interest rate was, say, 10%, the present value would be calculated as follows:-

$$PV = {20{,}000 \over 0.1} = $200{,}000$$

Given the present value, the rate and the annual payment, you would still have $200,000 at the end. It should be borne in mind however that $200K in 50 years time will have certainly devalued in purchasing power as discussed earlier.

#### Growing Perpetuities

With a growing perpetuity, the payments received increase at a fixed rate over the time of the perpetuity. The formula is:-

$$PV = {PMT \text{(in the first period)} \over i – g}$$

In this case “g” is the growth rate and it must be less than the discount rate (*i*). Should this not be the case, then the payments would be growing faster than the interest so it couldn’t be sustainable.

As an example of a growing perpetuity, consider someone who is looking at retirement. They believe they will need about $40,000 a year to live on but need to allow for increases in inflation. If your super is in a fund that returns on average 6% with average growth predicted to be around 2%.

$$PV = {40{,}000 \over 6\% – 2\%}$$

$$PV = $1{,}{000,}000$$

#### Estimating the theoretical price of shares

The Discounted Dividend Model can be used to calculate the theoretical price of shares utilising the idea of perpetuity valuation. Let’s say a company proposed to issue a dividend of $5 per annum at a discount rate of 10% in perpetuity. Using this information, you can theoretically work out what amount you could pay for a share.

$$PV = {5 \over 0.1} = $50$$

This means, as an investor, you could pay $50 per share in return for a dividend of $5 per annum in perpetuity.

## Easy Spreadsheet Calculations

Although all the formulae for the calculations have been included, they are much easier to do in a spreadsheet. I’ve written a spreadsheet that will do all the calculations for Future Value, Present Value and both types of Annuities. Feel free to download from the Resources link in the menu.

## Assessment 1

The first assessment is due next Tuesday. Given the tables and charts created in Excel for the purposes of explaining concepts, it is disappointing there are no marks for art work. It has been really enjoyable working through the assessment and using financial ratios once again to compare organisations. The assessment has been completed and lodged and now it’s the waiting game to find out the marks obtained.

## Summary

This module has also brought back a lot of memories. In the 1980s when working in finance, I used to prepare Discounted Cash Flow models for comparisons between lease and hire purchase for clients. It was very much a manual process and a far cry from generating an Excel spreadsheet like you can today. It is far easier today to prepare such analyses and certainly a lot quicker using spreadsheets.

References*Module 4 Time value of money: Introduction to time value of money*2022, Aib.edu.au, viewed 17 September 2022, <https://learning.aib.edu.au/mod/book/view.php?id=113095&chapterid=41273>[↩]- Titman, S, Martin, T, Keown, AJ & Martin, JD 2018,
*Financial management : principles and applications*, Pearson Australia, Melbourne, Vic.[↩]