Present Value

A dollar later is worth less than a dollar now

Offered $10,000 today or $10,000 in ten years, everyone takes it today, and not just out of impatience: today's $10,000 can be invested and become more than $10,000 by then. Present value (PV) makes that intuition precise by running compound interest backwards. Where future value asks what today's money grows into, present value asks what a future amount is worth now.

The formula is PV = FV / (1 + r)^n, where FV is the future amount, r is the discount rate and n is the number of periods. Worked example: $10,000 arriving in 10 years, discounted at 5%. Since 1.05 to the power of 10 is about 1.629, the present value is $10,000 / 1.629, or about $6,139. Check it forwards: invest $6,139 at 5% and in ten years you have $10,000. The two amounts are the same promise, quoted at different dates.

The discount rate: the price of time and risk

Everything hangs on r, the discount rate. It is the price of two things at once: time, because money tied up later must at least match what it could safely earn elsewhere, and risk, because an uncertain promise deserves a bigger discount than a guaranteed one. A near-certain government payment might be discounted at a low rate; a shaky IOU from a struggling business deserves a high one, which is a formal way of saying it is worth much less today.

Small changes in r produce big changes in value over long horizons. That same $10,000 in 10 years is worth about $8,203 today at 2%, $6,139 at 5%, and only $4,632 at 8%. Whoever picks the discount rate largely picks the answer, which is worth remembering whenever someone shows you a present-value figure.

Where Canadians actually meet present value

Present value sounds academic, but it prices several things Canadians deal with directly:

• A bond's price is simply the present value of its future coupons plus its face value, discounted at current market yields.
• A pension's commuted value is the present value of all the monthly payments your defined benefit pension would ever pay you. The lump sum on your termination statement is not a bonus or a buyout premium; it is your future pension, discounted to today.
• Lottery winners in some formats, and structured settlements after lawsuits or injuries, face the same trade: a smaller lump sum now versus an annuity paid over years. Comparing them fairly means putting both in present-value terms.
• Courts use present value to convert future lost wages or care costs into a single award payable today.

The seesaw: higher rates, lower present values

Because r sits in the denominator, present value and interest rates sit on opposite ends of a seesaw: when rates rise, present values fall, and vice versa. This one relationship explains two things that puzzle a lot of people. Bond prices drop when yields rise, because the same fixed coupons are now discounted more heavily. And commuted values shrink when rates rise, for the identical reason: your future pension payments have not changed, but a higher discount rate makes today's equivalent smaller. Members who saw large commuted values quoted in low-rate years were often surprised at how much smaller the figure became once rates climbed.