Discounting Greg Mason September 22, 2010
The two meanings of discounting • A bird in the hand is worth two in the bush. What this means, of course, is that something that is certain (received now) is valued more than something that is received later (uncertain). The present value of cash to be received in the future is always less than the nominal value of the future amount. Discounting cash flows – time value of money • Asset values reflect changes to anything that affects their use/enjoyment. The owner of a rural mansion will experience a loss in enjoyment if the adjacent farm starts a hog operation. That loss in enjoyment will result in a reduced value of the property. We say that the smell/noise of the adjacent land use is discounted into the value of the mansion. Discounting asset values
Discounting cash flows – time value of money • $100 promised to you next year is worth less than $100 in your hand now, because • you are uncertain you will be around to receive it • you may not trust those who promise the payment • inflation erodes the value of future payments • you may need the money to meet immediate obligations. • When we adjust our valuation of future payments, we are said to discountthe value of future payments to create a present value that is less than the future nominal value.
How do we measure value across time? • If we are completely trusting, expect to live forever, with inflation at 0% and have all the money we need, the present value of $100 to be received in one year is $100. • If these conditions are not met, then our present valuation of $100 to be received in a year will always be less than $100. • An equivalent way of saying this is that if we are forced to wait a year for the $100, we need to be compensated. • The level of compensation reflects the risks we incur in waiting. • The higher the risk, the more the compensation needed. • An interest rate reflects this compensation and reflects the time price of money.
Future valueSaving – the reward for abstinence We start by using the compound formula for interest and saving: • $100 saved for 1 year at 10% will yield $110 in 1 year $100 x (1+.10) = $110 • $100 saved for 2 years at 10% will yield $121 in 2 years $100 x (1+.10) x (1.10) = $121 $121 = $100 x (1+.10)2 FV = PV x (1+.10)2 • $100 saved for n years at 10% will yield a future value in 2 years FV = $100 x (1+.10)n
Present value • What is the value of $110 received in one year, if the interest is 10%? • We can equate any amount received (or spent) in the future to a present value, as long as we know the interest rate. • $110 received in one year has a present value of $110 / (1+.10) = $100 • $121 received in two years has a present value of $121 / (1+.10)2 = $100 $100 = $121 / (1+.10)2 PV = FV / (1+.10)2 • A future value (FV) to be received in n years has a present value of PV = FV / (1+.10)n
Compare the two formulas $121 = $100 x (1+.10)2 FV = PV x (1+.10)2 FV = $100 x (1+i)n $100 = $121 / (1+.10)2 PV = FV / (1+.10)2 PV = FV / (1+i)n The interest rate “i” equates present and future values.
The higher the interest rate, the more we discount the future and value the present $100 received in one year when interest is 5% is presently worth PV = $100(1+.05) = $95.24 $100 received in one year when interest is 15% is presently worth PV = $100(1+.15) = $86.96 This is intuitive – if interest rates are high, we are less patient in delayed payment, since we could use the cash right now and invest in savings with a high return. We need to be compensated to accept the delay.
Interest rates • Market interest reflects uncertainty and risk, as well as the demand and supply of investment/savings. • Low-risk investments tend to have low interest rates and vice versa. • The extent to which present rewards are preferred over future reward is termed “time preference.” • Risk-averse people want the money now, and need a high interest rate to compensate them for waiting. • Risk aversion/preference also reflects social conditioning and experience. (If external events constantly upset plans, people develop a high time preference … they are less willing to wait for a reward and need high interest rates to save.)
What is a future salary worth? • It is possible to create a present value measure of a stream of future incomes or expenses. • The value of a business or rental property often starts by summing the present value of the future funds to be received. • Funds received next year are discounted less than funds to be received in 10 years.
The present value of a flow of future income • Imagine that you will receive $10,000 a year for the next five years and that i = 5%. • To calculate the present value of this stream of income, we simply calculate the present value of each year and add. $10,000/(1.05)1+ $10,000/(1.05)2+ $10,000/(1.05)3+ $10,000/(1.05)4+ $10,000/(1.05)5 = $9,523+$9,070+$8,638+$8,227+$7,835 = $43,493 • $10,000 received each year for five years has a present value of $42,493 if the interest rate = 5%. • If i = 7%, the PV = $41,002. Present value at end of year 1 Present value at end of year 2 Present value at end of year 5 Present value of the income stream over 5 years
Year 0 is the start of the year (January 01) Year 1 is the end of the year (December 31) The present value of $1,000 received every year for 15 years with i = 2% The present value of $1,000 received at the end of Year 8 with i = 16% The formula in any column is PV=FV/(1+i)n