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The Time Value of Money

If you could receive $10,000 now or a year from now, other things equal no doubt you would choose to take the money now.  In other words, the value of money in hand is viewed as greater than the value of a claim on the same amount at some time in the future.  Why should that be so?

There are a number of reasons:  (1) A claim on the delivery of money in the future has a risk of default.  (2) Price inflation reduces the real value of money over time.  (3) Money in hand can be invested for a longer period.  Any of these explains why lenders normally demand a premium for the money they surrender.  The premium could be a one-time fee, or periodic payments.

One form of investment with periodic payments is an annuity. The annuitant pays a lump sum now to receive n payments at regular intervals over some period of time.  There is no return of principal.  The amount paid for the annuity should equal the present value of the periodic payments, which is determined by:

PV = A*(1-1/(1+i)^n)/i

where A = value of a single payment, i = assumed interest rate over the period of the payments.

A perpetuity is a bond that has no maturity and pays interest forever.  It is similar to an annuity but is marketable, so the owner can sell it any time.  Whoever owns the bond will receive periodic payments, A, for as long as he holds it.  In effect, n = infinity for a perpetuity.  As long i > 0, meaning the value of money held diminishes with time, the annuity equation above shows that the present value of a perpetuity is finite, and equal to:

PV = A/i

A well-known example of a perpetuity is the British consol (short for consolidated annuities).  It was sold by the government in the mid-18th century to convert its outstanding issues of redeemable government bonds into a single bond.  Over the years the consol has gone through several modifications, and today pays 2.5% of its face value per year to the owner.  That is, a consol with a face value of 100 pounds pays 2.5 pounds per year.  Since the interest rate on long bonds is now about 5%, the market value of the British consol is currently only about one-half its face value.

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