Financial Math Fundamentals

Discounted Cash Flows

Welcome to our session on discounted cash flow, one of the most important concepts in finance. In this session, we will cover the following:

- The time value of money
- Calculating present values using the discounted cash flow methodology
- -Calculate the present value of an annuity
- -Calculate the present value of constant and growing perpetuities

The basic idea

Let’s start with the basic idea. The concept of discounted cash flows relates to what someone today would be willing to pay to receive a cash flow in the future. The underlying principle is that the sooner somebody is to receive a cash flow in the future, the more valuable it is today.

The discount rate

In order to determine the value today of a future cash flow, we need to “discount” future cash flows by a discount rate. This discount rate is often referred to as the required rate of return. The higher the discount rate, the more we discount the future value to arrive at the cash flow’s present value. This is why the discount rate is also referred to as the rate of return – the less we pay today to receive a cash flow in the future, the larger our return is.

The discount rate is determined by two factors.

- Firstly a risk premium – they higher the risk that we may not receive the future cash flow, the higher the risk premium.
- The second factor is a principle called the time value of money.

The time value of money

Here’s a question for you – which is more valuable to you, $100 today or $100 in one year’s time? To help you answer this, ask yourself which of these two options you’d prefer.

- If you are like most rational people, you’d prefer $100 today.
- From a finance theory point of view, the rationale for this is that you could take your $100 today, invest it, and receive more in the future.

This idea that money has different values to us today, depending on when we receive it, is known as the time value of money.

Time value of money

A simple example will help illustrate this fact.

Let’s say you had $100 today, and you invest it for three years in a bank, receiving an interest rate of 10% every year. Interest is paid to at the end of each year, and you choose to leave this interest in the bank.

- after 1 year, you would have $110
- after 2 years, you would have $121
- and after 3 years, you would have $133.10

Another way to think about this is, if interest rates were 10%, you wouldn’t care if you received $100 today or $133.10 in 3 years time – they both have the same value to you today.

Time value of money

We can extend the previous simple example to derive a formula linking the future value of a cash flow with its present value, where the interest rate is r. To increase the present value of $100 by 10%, we simply multiply the present value by 1+10% which we write as 1.10. We can do this for each of the 3 years, which is equivalent to multiplying $100 by 1.10 to the power of 3. This gives us a future value of $133.10 and provides us with an equation linking the present value with the future value.

Pop quiz

- Question: If you were able to earn interest at 5% and you started with $100, how much would you have after 2 years?

Discount factor

When we start with the present value and calculate the future value of a cash flow, this is known as compounding. When we know the future value and want to work backward to determine its present value, this is known as discounting, hence the term discounted cash flows.

We can rearrange the previous equation so that we are calculating the present value rather than the future value. Rearranging this equation we can isolate a term called the discount factor. It is this discount factor which we multiply the future cash flow by to determine its present value.

NPV calculation

Now let’s look at a very common application of discounted cash flows, called Net Present Value, or NPV analysis. Managers are often required to make decisions on whether a particular project is profitable. The typical project consists of an initial investment, in this case, 1000, and a stream of cash flows resulting from the investment. The cash flow at the end of year 1 is 400, at the end of year 2 is 600 and at the end of year 3 is 200

The Net Present Value (NPV) methodology sums the present value of the initial investment (which is 1000) and the present values of all the future cash flows. We can see the calculation of the year 2 discount factor. That is, 1 divided by 1 plus the discount rate of 10%, all to the power of 2. If the NPV is positive, the project is profitable; if it is negative, the project is unprofitable. In this example, the NPV is positive at 9.77, so this project is indeed profitable.

Demonstration

We can use Excel to do our NPV analysis

Firstly, let’s write our required rate of return in cell B1. Remember, this is r in our discount factor formula. The number we are using is 10%.

- Set up an excel worksheet with 4 rows. These 4 rows will be time, future value, discount factor, and present value respectively.
- Populate the time and future value rows using the information on the previous slide
- Now we need to do the calculation for the discount factor. I’m going to link my formula with cell B1, absolutely reference this by pressing F4, then simply drag this across. The formula reads = 1/(1+$B$1)^(B3).

Calculating each cash flow’s present value is now simply a matter of multiplying the future value by the discount factor and I can calculate the NPV for this project by summing together all the present values. If the required rate of return rises to 11%, what is the new NPV of the project? Would you invest in this project or not? Why?

As you can see, when the required rate of return of the project rises to 11%, our NPV becomes negative, so we would not go ahead and invest in this project. Look at what has happened to each of the present values….they have become smaller. Think about it this way – if we want to increase our required rate of return, and the future cash flows are fixed, we need to pay less for those cash flows today, that is, the present values will be smaller.

Exercise

Click on the attachment link entitled “NPV”. Once you’ve had a go, check your attempt with the attached “NPV solution”. Good luck!

Annuities

An annuity is a series of equal payments in equal time periods and guaranteed for a fixed number of years. Usually, this fixed interval is 1 year, which is why it is called an annuity, but the time period can be shorter, or even longer. Understanding annuities is crucial for understanding loans, and investments that require or yield periodic payments. While we can calculate the present values of each future cash flow of an annuity to arrive at it’s over present value, we can also calculate an annuity factor which makes our task a lot easier.

Annuity DCF example

Let’s look at an example.

- Say we had a 3-year annuity, receiving $100 starting one year from today, for each of the next 3 years. How much would we be willing to pay today to receive this annuity if interest rates were 10%?
- Using our discount factor formula where r is 10%, we can work out each of the 3 discount factors. Remember the formula is 1/(1+r)^n
- We can then easily calculate the present value of the annuity by first multiplying the future values by their discount factors, and then by adding all the present values together.
- So we would be willing to pay $248.60 to receive this annuity. We also say the fair value of this annuity is $248.60

Annuity Factor

- If you add the discount factors, what do you notice?
- Our three discount factors we get when we calculate the previous annuity are 0.909, 0.826, and 0.751 for years 1, 2 and, 3 respectively.
- If we add the discount factors, we get what is known as the annuity factor.
- This is very useful, as it provides a short cut to calculate the present value of an annuity.
- Notice if we multiply our annuity payment of 100 by the annuity factor we get the same answer as before, namely 248.6.
- We can also use annuity tables to work out our annuity factor to save even more time.

Annuity tables

- Annuity factor tables can easily be used to save us from having to add the individual discount factors to derive annuity factors.
- Typically, as in the case of this table, the number of years the annuity lasts for, or n, is placed down the left-hand side of the table.
- Along the top row of the table are the various discount rates or required rates of returns.
- As you can see from this annuity table, if we went down to the third row and across to the tenth column, the annuity factor is 2.4869.

Demonstration

Let’s now look at a practical application of annuities. Annuities are often sold as investments. Investors pay a lump sum today to receive a fixed amount at fixed time intervals in the future. You can find a PDF copy of the annuity tables under the attachments tab.

- This particular annuity pays investors $15,000 every year for 10 years and offers a return of 7%.
- To find the present value, or the price of this annuity, set up an Excel spreadsheet with the discount rate in cell B1.
- Set up 4 columns, Time, Future Value, Discount Rate, and Present value respectively.
- Time will go form 1 to 10, and the future value is 15,000 each of these 10 years.
- We need to insert the discount factor formula, which reads = 1/(1+$B$1)^(A4). We can now fill this formula down.
- Finally, we can calculate our present value by multiplying the Future Value by the Discount factors. Summing these gives us a present value, or price of the annuity, of $105,354.
- Adding the discount factors we get an annuity factor of 7.0236, which when we multiply by our annual payment of $15,000, gives us the same price of $105,354.

Perpetuities

A perpetuity is an investment that has no definite end, or a stream of cash payments that continues forever. Even though we receive a payment “in perpetuity” we can still calculate the present value of this stream of cash flows, as eventually the future cash flows are received so far into the future, their present value to us today is zero.

Real-life perpetuities examples are:

- War Loans, bonds issued by the UK government with no redemption date.
- Income from real estate can be considered a perpetuity if we assume we will receive rent from it into the foreseeable future.
- The Dividend Discount Model is a way stock analysts value a company based on the dividends they pay, assuming that the company will continue to pay dividends into the foreseeable future.

We will look at 2 types of perpetuities – constant and growing perpetuities.

Constant perpetuity – demo

A constant perpetuity pays the same amount, year after year, forever.

Question:

What is the present value of a constant perpetuity that pays $10 every year forever?

Hint:

If a bank paid you 10% interest once per year, how much would you have to deposit to earn a constant $10 per year forever?

The below is my script for the annotation:

- Let’s say we deposit 100 today.
- If interest rates are 10% in one year’s time I would have 110. I could withdraw 10, leaving 100.
- I leave this 100 in the bank for another year, and at the end of year 2, I would again have 110.
- I could withdraw 10, leaving 100 and so on.
- In other words, I have created a stream of 10 every year forever, starting with 100. Therefore the present value of an annuity if interest rates were 10% paying 10 every year is 100.

Constant perpetuity

- So, from our example, we can see that it is possible to calculate the PV of a cash flow which will occur in perpetuity.
- The present value of a constant perpetuity is the payment we are to receive into perpetuity divided by the discount rate or the required rate of return.

Growing perpetuity

- A growing perpetuity is again a stream of cash flows that start one year from today and go on forever. However, the amount that we receive grows by a constant growth rate, g.
- We can calculate the present value of a growing perpetuity by using the following formula:
- The present value is the first payment we receive one year from today, divided by the discount rate minus the constant growth rate.

Demonstration

- As a demonstration on the use of a growing perpetuity, put yourself in the position of an analyst trying to value a dividend-paying stock.
- Let’s assume that the price of the stock is equal to the present value of all the future dividends you will receive from the stock. This is a fair assumption to make if you intend to buy and hold the stock for a long period of time.
- Let’s now assume that the dividend you will receive 1 year from today is $5, but the company’s dividend policy leads you to believe this will grow at a steady rate of 1% per year. Let’s insert these two variables in cells B1 and B2 respectively.
- Finally, let’s say your required rate of return is 10%. Let’s insert this value into cell B3.
- In cell B5, I am going to type my growing perpetuity formula, which will read = 5/(B3 – B2).
- This gives us a present value of $55.56. If the stock is trading below this figure, this represents a buying opportunity for us.

Exercise

Click on the attachment link entitled “Prize Draw exercise”. Once you’ve had a go, check your attempt with the attached “Prize Draw solution”. Good luck!

Summary

That concludes our session introducing discounted cash flows. The key messages to take away are:

- To find the present value of a future cash flow we use the Discounted Cash Flow methodology
- CF is based on the concept of the time value of money
- Future cash flows are discounted using a discount factor
- DCF can be used to calculate the net present value of a stream of cash flows, including annuities and perpetuities