Net Present Value
Last Time
We spent the time developing our basic approach to DCF analysis.
We discussed:
The importance of a financial market to the economy and why investors receive interest (compensation) for saving/lending.
The usefulness of the price from this market for decision making concerning real investments.
Now we want to complicate things.
Valuing Streams of Structured Future Cash Flows
Now we are going to discuss the valuation of certain highly structured cash flow streams.
The resulting valuation formulas are useful for simplifying the analysis of certain situations.
Pay attention to the exact timing of the cash flows, the formulas don’ t work unless you get this right.
Drawing diagrams of the cash flows can be useful.
These formulas can make life easier and so are worth understanding.
Perpetuity
A stream of equal payments, starting in one period, and made each period, forever. Forever??
Please, please remember, this gives the value of this stream of cash flows as of time 0, one period before the first payment arrives.
Growing Perpetuity
A growing perpetuity is a stream of periodic payments that grow at a constant rate and continue forever.
The present value of a perpetuity that pays the amount C1 next period, grows at the rate g indefinitely when the discount rate is r is:
Annuities
An annuity is a series of equal payments, starting next period, and made each period for a specified number (3) of periods.
If payments occur at the end of each period (the first is one period from now) it is an ordinary annuity or an annuity in arrears.
If the payments occur at the beginning of each period (the first occurs now) it is an annuity in advance or an annuity due.
Valuing Annuities
We can do a lot of grunt work or we can notice that a T period annuity is just the difference between a standard perpetuity and one whose first payment comes at date T+1.
The present value of a T period annuity paying a periodic cash flow of C, when the discount rate is r, is:
If we have an annuity due instead, the net effect is that every payment occurs one period sooner, so the value of each payment (and the sum) is higher by a factor of (1+r).
Or we can add C to the value of a T-1 period annuity.
Annuity Example
Compute the present value of a 3 year ordinary annuity with payments of $100 at r = 10%.
or,
Annuity Due Example
What if the last example had the payments at the beginning of each period not the end?
Or,
Or,
Growing Annuities
A stream of payments each period for a fixed number of periods where the payment grows each period at a constant rate.
Example
What is the present value of a 20 year annuity with the first payment equal to $500, where the payments grow by 2% each year, when the interest rate is 10%?
Application: Retirement Planning
You have determined that you will require $2.5 million when you retire 25 years from now. Assuming an interest rate of r = 7%, how much should you set aside each year from now till retirement?
Step 1: Determine the present equivalent of the targeted $2.5 million.
PV = $2,500,000/(1.07)25
PV = $2,500,000/5.42743 = $460,623
Step 2: Determine the annuity that has an equivalent present value:
Retirement Planning cont
Now suppose that you expect your income to grow at 4% and you want to let your retirement contributions grow with your earnings. How large will the first contribution be? How about the last?
A College Planning Example
You have determined that you will need $60,000 per year for four years to send your daughter to college. The first of the four payments will be made 18 years from now and the last will be made 21 years from now. You wish to fund this obligation by making equal annual deposits at the end of each of the next 21 years. You expect to earn 8% per year on the deposits.
Step 1: Determine the t = 17 value of the obligation.
Step 2: Determine the equivalent t = 0 amount.
College Planning cont
Step 3: Determine the 21-year annuity that is equivalent to the stipulated present value.
Present Value Homework Problem
Your child will enter college 5 years from now. Tuition is expected to be $15,000 per year for (hopefully) 4 years (t=5,6,7,8).
You plan to make equal yearly deposits into an account at the end of each of the next 4 years (t=1,2,3,4) to fund tuition. The interest rate is 7%.
How much must you deposit each year?
What if tuition were growing by 2% each year over the 4 years?
Think about: How to decide whether/when to refinance your house?