Present Value

Present Value - What is it?

A dollar paid to you now is worth more than one paid to you in the future. Why? Because if you had a dollar now you could invest it and earn interest. Simple enough but exactly how much is money worth today when it is paid in the future? To answer that you need to know about discounting.

Discounting converts future payments into today’s money or present value (PV). It’s at the heart of finance because it tells you what future payments or receipts are really worth. It also allows you to directly compare the value of cash flows even when they are payable on different dates.

Discounting

Let’s look at discounting by using a simple example. Suppose you were due to receive $1,000 in 273 days time. What is it worth today? That depend on interest rates. Let’s use a rate of 4.00%.

The discount factor is:

1 / (1 + 0.04 x 273 / 360) = 0.97055

Multiply the $1,000 by the discount factor 0.97055 and the present value is $970.55

What does this tell us? That in present value terms $1,000 paid in 273 days time is currently worth $970.55.

This simple formula can be used for cash flows up to one year in maturity with the day count and interest rate being adjusted accordingly.

What could affect this value?

The amount of money involved
The interest rate used
The number of days involved

Let’s see what happens when the interest rate is altered first to 5% and then to 3%.

1 / (1 + 0.05 x 273 / 360) = 0.96346

0.96346 x $1,000 = $963.46

When the interest rate increases the discount factor gets smaller so the present value of the future cash flow declines.

Now 3%

1 / (1 + 0.03 x 273 / 360) = 0.97775

0.97775 x $1,000 = $977.75

When the interest rate decreases the discount factor gets larger so the present value of the future cash flow increases.

What’s the practical use?

The practical use is as follows. If you expected to receive $1,000 its present value will depend on the interest rate used. If you reduce the rate the PV increases and vice versa. What you experience is called interest rate risk.

At a personal level this may or may not be important but if you work as a trader it certainly is because your portfolio will be marked-to-market (MTM). This means the market value will be reported daily and you can see how much you have made or lost as a result of interest rate risk.

Whilst the valuation of the $1,000 is insignificant dealers are working with numbers that are altogether of a different order and the profit and loss effects can run into millions.

You may well ask what types of transactions give rise to cash flow payments of the nature we’re discussing. Here’s a few; loans, deposits, bonds, swaps, options even foreign exchange forwards. They are all affected by interest rates. Just consider a simple fixed rate bond. When interest rates increase it falls in value. Why? Because the cash flows are worth less in present value terms.

Where does the interest rate come from?

That really depends on the cash flow you are trying to discount. Conventional wisdom normally assumes that the interest rates used are Libor rates (and for longer dates interest rate swap yields). This makes one assumption. That the credit risk related to the cash flow you want to discount is similar to the credit risk in the interbank loan market from which the Libor rates are derived. Let’s see an example of this. Earlier we discounted $1,000 at 4% for 273 days:

We derived the discount factor:

1 / (1 + 0.04 x 273 / 360) = 0.97055

And then used it:

Multiply the $1,000 by the discount factor 0.97055 and the present value is $970.55

But now suppose the $1,000 you expected to receive was due from an unreliable friend who had a history of not paying. It’s unlikely the present value would be $970.55!

How could you take the risk of non-payment into account? This is a complicated subject but in simple terms you could increase the interest rate to reflect the risk you faced. So if you considered 10% was an appropriate rate to discount the money the value would be $929.51. The increased credit risk has reduced the present value.

The message is simple. If you want to find the present value of cash flows that have an increased credit risk you should use interest rates that reflect this risk. If you don’t you will over state the value.

What about time?

Time has a straight forward effect on present value. The more distant a cash flow is the lower the present value. Why? Because the discount factor gets smaller. Similarly the closer a sum of money gets to today’s date the less it is affected by discounting. Here is an example based on a 4% interest rate:

Amount Days Discount factor Present value

$1,000 0 1.00000 $1,000

$1,000 91 0.98999 $989.99

$1,000 182 0.98017 $980.17

$1,000 273 0.97055 $970.55

More than one year

Once you go over one year the discounting process is a bit more complicated. That’s because longer term rates are not based on simple interest. Market practitioners use zero coupon methodology. However the same principal applies; higher interest rates reduce the present value of future cash flows.

First Published by Barbican Consulting Limited 2009