Present Value
What is Present Value?
Present value is what a future sum of money is worth today, after discounting for the interest that could be earned in the meantime.
Because money available now can earn interest, a dollar today is worth more than a dollar in the future. Present value is used to compare investments and value bonds. A higher interest rate lowers present value.
Present Value: a worked example
A promise pays you $1,100 one year from now and the interest rate is 10%. Present value = $1,100 ÷ 1.10 = $1,000, so paying anything above $1,000 today leaves you worse off than simply depositing the money. Now suppose the interest rate rises to 25%. Present value = $1,100 ÷ 1.25 = $880, because $880 deposited at 25% grows to 880 × 1.25 = $1,100 by itself. The promised payment never changed; only the cost of waiting did, and the value today fell by $120.
The mistake students make with present value
The usual error is subtracting the interest instead of dividing by it: students take $1,100, knock off 10%, and report $990. Check it. $990 growing at 10% becomes 990 × 1.10 = $1,089, not $1,100, so $990 is too low. Discounting is the exact inverse of compounding, which means dividing by (1 + r), not subtracting r. It is tempting because 'take off 10%' sounds like the opposite of 'add 10%', but the two percentages are taken from different bases.
Present Value questions
Why is a dollar today worth more than a dollar next year?
A dollar today is worth more than a dollar next year because today's dollar can be lent, deposited, or invested and grow before that future date arrives. Waiting therefore carries an opportunity cost equal to the interest given up. That is the whole logic behind discounting: a future payment gets scaled down by however much a dollar could have grown in the meantime.
Why do bond prices fall when interest rates rise?
Bond prices fall when interest rates rise because a bond is a set of fixed future payments, and present value discounts those payments at the current rate. A bond promising $1,050 in one year is worth $1,050 ÷ 1.05 = $1,000 at a 5% rate, but only $1,050 ÷ 1.10 = about $954.55 once the rate reaches 10%. The payments did not change; the discount did.
How do you find present value more than one year out?
Present value over several years uses the exponent n in (1 + r)ⁿ, which compounds the rate instead of multiplying it. At 10% over three years, (1.10)³ = 1.331, so a payment of $1,331 arriving in three years has a present value of $1,331 ÷ 1.331 = $1,000. Using 1 + (0.10 × 3) = 1.30 instead would give about $1,024, which overstates the value.
Formula / Example
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