What is present value in finance?+
Present value (PV) is the current worth of a future sum of money, discounted at a rate that reflects the time value of money and the risk of the cash flow. Because money invested today can earn returns, a future dollar is worth less than a present dollar. The formula PV = FV / (1 + r/n)^(nt) converts any future amount to its equivalent today, using the discount rate r and compounding frequency n over t years.
What is the lump sum present value formula?+
PV = FV / (1 + r/n)^(n x t). FV is the future amount, r is the annual discount rate as a decimal, n is compounding periods per year, and t is years. Example: $50,000 receivable in 7 years at 9% annually: PV = 50,000 / (1.09)^7 = 50,000 / 1.82804 = $27,351.32. This means receiving $50,000 in 7 years at 9% is equivalent to receiving $27,351 today.
What is an ordinary annuity and how do I find its present value?+
An ordinary annuity (also called deferred annuity) has equal payments at the end of each period. PV = PMT x (1 - (1+r)^(-n)) / r. For $10,000/year for 10 years at 8%: PV = 10,000 x (1 - (1.08)^(-10)) / 0.08 = 10,000 x 6.71008 = $67,100.81. Common ordinary annuities include bond coupon payments, loan repayments, and pension income streams.
What is annuity due and how does it differ from ordinary annuity in PV?+
An annuity due has payments at the beginning of each period. PV_due = PV_ordinary x (1 + r). Because each payment arrives one period earlier, it has less time to be discounted, so annuity due PV is always higher than ordinary annuity PV by a factor of (1 + r). For $10,000/year at 8%: PV_ordinary = $67,100.81; PV_due = $67,100.81 x 1.08 = $72,468.87. Lease payments, rent, and insurance premiums are typical annuity-due cash flows.
What is a perpetuity and how do I calculate its present value?+
A perpetuity is an infinite series of equal periodic payments. PV = PMT / r, where r is the annual discount rate. A $5,000/year perpetuity at 6%: PV = 5,000 / 0.06 = $83,333.33. Despite infinite total payments, the PV is finite because each distant payment is discounted to near zero. Classic perpetuities include UK consols (government bonds with no maturity) and certain preferred stocks with no redemption date.
What is the Gordon Growth Model for growing perpetuities?+
The Gordon Growth Model (growing perpetuity): PV = PMT / (r - g), where g is the constant annual growth rate of the payment. It requires g to be less than r. Example: a stock pays a $3 dividend expected to grow 4% annually and you require a 9% return: PV = 3 / (0.09 - 0.04) = 3 / 0.05 = $60 per share. This is the Dividend Discount Model (DDM), widely used in equity valuation. It is sensitive to small changes in r and g: a 1% change in either can move the value by 20-25%.
What discount rate should I use for present value calculations?+
The discount rate should reflect the opportunity cost of capital and the risk of the cash flow. Use the current risk-free rate (Treasury yield) for risk-free government cash flows. Use WACC (Weighted Average Cost of Capital) for company capital budgeting. Use required rate of return for equity investments (often estimated via CAPM). For personal planning, use your realistic expected investment return: 6-10% for equities, 3-5% for bonds. Higher risk always warrants a higher discount rate, producing a lower PV.
How do rising interest rates affect present value?+
Rising interest rates reduce present value. At 5%, the PV of $10,000 in 10 years is $6,139. At 10%, it is $3,855. At 15%, it is $2,472. This inverse relationship between rates and PV explains why bond prices fall when interest rates rise: the fixed future coupon and principal payments are now discounted at a higher rate. Long-duration bonds and growth stocks (with far-future cash flows) are most sensitive to rate changes because their cash flows are discounted over longer periods.
How do I calculate PV for non-annual payments?+
For non-annual payments, convert to periodic rates and periods: divide the annual rate by payment frequency, and multiply years by frequency. For $1,000/month for 5 years at 6% annually: periodic rate = 0.06/12 = 0.005, total periods = 60. PV = 1,000 x (1 - (1.005)^(-60)) / 0.005 = 1,000 x 51.7256 = $51,725.56. In the Annuity mode, enter PMT = 1,000, Rate = 0.5% (monthly), Years = 60 (months), treating each "year" as a month.
What is the difference between present value and net present value?+
Present value discounts future cash flows to today. Net present value (NPV) = sum of all discounted future cash flows minus the initial investment. PV answers: what is this future cash worth now? NPV answers: is this investment profitable after accounting for what I pay upfront? If NPV is positive, the investment earns more than the discount rate and should generally be accepted. Use PV for single-payment or annuity evaluations, and NPV for full investment appraisal with multiple periods.
How is present value used in bond pricing?+
A bond's fair price equals the PV of its coupon payments (ordinary annuity) plus the PV of its face value (lump sum) at maturity, both discounted at the market yield. A $1,000 bond with a 5% coupon, 10-year maturity, discounted at 7%: PV_coupons = 50 x (1-(1.07)^(-10))/0.07 = $351.18; PV_face = 1,000/(1.07)^10 = $508.35; Bond price = $859.53. Since the coupon rate (5%) is below the market rate (7%), the bond trades at a discount to face value.