in this video I discuss four useful valuation formulas perpetuity growing perpetuity annuity and growing annuity the four formulas have many applications in economics finance and business this is a brief summary of these four key valuation formulas a perpetuity is a cash flow C that will be received forever if you can invest at an interest rate R then the present value of these cash flows is merely c ided r for example if you can invest at an interest rate of 10% the present value of $100 cash flow beginning one year from today and continuing forever is $100 divided by 10% or $1,000 note that if I invest $1,000 at 10% it will spin off $100 in interest every year so this valuation makes sense a growing perpetuity begins with a cash flow C and then grows at a constant growth rate denoted G in general the present value of a growing perpetuity is C / R minus G for example if the cash payment of $100 from our prior example grew at 2% growth rate we would receive $100 in the first year $100 $102 in the second year as the the cash flow has grown by 2% and $104.4 in the third year as compound growth in the promised payments yields a cash payment slightly greater than $104 in year three this is a small effect in early years but after 19 20 or 21 years the effect of compound growth on the promised cash payment starts to make a difference the present value of the growing perpetuity would be the First Cash payment of $100 divided by the prevailing interest rate of 10% less the growth in the cash payment of 2% which yields an answer of $1,250 recall that the promise of $100 forever when interest rates are at 10% would cost us $11,000 of course it makes sense that we must pay more than $11,000 for this cash flow since our initial payments of $100 begins growing at 2% in year 2 note that the growth rate for the promised cash flow payment must be less than the prevailing interest rate R for the growing perpetuity to have a sensible valuation a fixed annuity provides a promised cash flow C for a fixed number of periods T note that the cash flow begins in Period one and stops in Period T to calculate the present value of an annuity we need to know the promised cash flow C the prevailing interest rate R and the number of periods over which the cash flow is paid t for example assume we are promised an annual payment of $50,000 per year for 20 years when interest rates are at 10% though we will receive cash payments totaling a million dollars over 20 years we know the present value of these payments is much less than a million dollar in fact the present value of the promised payments is worth only but then begins to grow at a growth rate G in the second period through the last period T thus to value a growing annuity I need four inputs the cash flow promised in the first period C the growth in that initial cash payment G the prevailing interest rate R and the number of periods over which the annuity will be paid T to illustrate the application of the growing perpetuity formula let's assume we are promised a $50,000 payment in one year but after that the payment will grow by 2% such that in the second year it'll be $51,000 in the third year we enjoy compound growth of 2% over two periods and receive a payment of $52,400 while our last promise payment in year 20 enjoys 19 years of compound growth at 2% resulting in a cash payment of $72,800 applying the growing annuity formula yields a valuation of $486,500 for the promised payments of course it makes sense that an annuity that begins with a cash payment of 50,000 and then grows at 2% will be more valuable than a fixed annuity of 50,000 over the same number of years these four valuation formulas perpetuity growing perpetuity annuity and growing annuity are powerful tools that can be used in many settings in business economics and finance I hope you find this review useful