Impact of the Risky Situation on Present Value

 An investment that is more risky must offer higher returns (if only potential), than safer investments.  If it did not, why would anyone invest in it? 

One impact on the present value is that the the required interest rate will be increased by the ‘risk premium’, which is the additional rate on top of the risk-free rate.  If the risk free rate was for example 2%, and the risky investment offers a risk premium of for example, 8%, then the total interest rate of the investment would be 2% + 8% = 10%.

The simple formula for present value:

CV = PV (1 + i)^t

needs to be adjusted to include the risk premium:

CV = PV (1 + i + rp)^t

Where:

CV = cash value in the future 
PV = cash value in the present (most commonly known as present cash value) 
i = interest rate of a given time period (for example, the interest rate for 1 year) 
rp = risk premium 
t = the number of time periods to consider

Example: What is the present value of an investment that will give us $12,000 one year from now, if the risk-free interest rate was 2% and the risk premium was 8%?

CV = PV (1 + i + rp)^t  12,000 = PV (1 + 0.02 + 0.08)^1  PV = 12,000 / (1.10)  PV = $10,909

What does $10,909 mean?

1. $10,909 today, if invested for one year at the combined risk-free and risk premium rate would return $12,000 in one year.
2. $12,000 one year from now is worth $10,909 today if a comparable investment is available today.

For the sake of comparison, what is the present value of $12,000 -- to be received one year from now -- without the risk premium?

CV = PV (1 + i + rp)^t  12,000 = PV (1 + 0.2 + 0.0)^1  PV = 12,000 / (1.02)  PV = $11,534

Present Value of Future Cash in a Risky Situation

The most basic way of determining the present value of a future cash flow considers only 3 things:

1. How much cash are we talking about?
2. What is the interest rate that you could invest the cash in if you had it today instead of in the future?
3. How long is the future?  1 year?  2 years?

Why would you be receiving that future cash flow anyway?  Sometimes it’s cash that came from you.  Sometimes it’s cash as payment for services you rendered.  Consider the following three scenarios:

1. You put in $10,000 in a term deposit.
2. You lend $10,000 by buying a corporate bond.
3. You provide a service to a company which now owes you $10,000

In each case, the other party has an obligation to pay you back your money.  Is the chance that you will get paid the same?  It’s almost certain you will get your money back from the term deposit.  Even if the bank collapses, the $10,000 is most certainly covered by insurance.

The corporate bond is at risk if the company that issued the bond goes bankrupt.  How likely this is depends on who the company is.  A bond issued by an IBM is less risky than one issued by a smaller startup.

The company who owes you $10,000 may decide not to pay you at all.

Since the risk of not being paid in these scenarios is not different, there are two key things to note:

1. The value of the future cash flow should not be the same. 
2. You would normally want to be compensated for taking more risk.

The most basic means of computing the future cash flow, as outlined above, is not adequate for computing future cash flows that have a certain amount of risk in them.  The formula needs to incorporate the risk factor.

Present Value of Future Cash

 A thousand dollars in your hands today has more value than a thousand dollars promised one year from now. Why?

There's many reasons, but let's consider three.

First, if the cash was in your hands today, you could place it in an interest bearing facility, such as a time deposit.  In one year, that cash will earn additional money for you.  You could not do this with the thousand dollars promised to you 1 year from now.

Second, if you had the money with you today, any opportunity you come across between today and one year from can be acted upon.  If you don't have that money, you will end up not being able seize the opportunity.  Besides opportunities, you could also experience an urgent need for mpmey, as in the case of emergencies.

Third, for a long as the money is not in your hands, you are under the risk of not being paid your money.  In the case of a seller who sold a thousand dollars worth of merchandise to a buyer, should the buyer end up bankrupt, or for some reason unable or unwilling to pay the  money, the seller would be left with a thousand dollar loss.

So a thousand dollars today is worth more than a thousand dollars promised one year from now.  But how much more?

We can answer the question in two ways.  We can figure out how much a thousand dollars today will be worth one year frpm now. Conversely, we can  figure out how much a thousand dollars one year from now, is worth today.

The method of calculation can be as complicated as there are factors to consider, such as the risk of not being paid, the inflation rate, the possible opportunities to be foregone, and so on.  The simplest and most simplistic approach is to just consider the interest rate by which we can deposit the money if we had it in our hands today. This can be computed using the formula:

CV = PV (1 + i)^t

Where:

CV = cash value in the future
PV = cash value in the present (most commonly known as present cash value)
i = interest rate of a given time period (for example, the interest rate for 1 year)
t = the number of time periods to consider