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.
Alternate Compounding Periods • Interest may be “compounded” over periods other than a year. In terms of “bank account” examples, this simply means that interest is credited to the account more frequently than once a year. • We have considered only annual compounding so far. • Caveat: All of the time value of money formulas use the implicit assumption that the compounding interval is the same as the payment interval. e.g.: • Mortgage loans call for monthly payments. • Bonds make coupon payments semiannually. • If this is not true you must adjust the interest rate to match the payment interval.
Alternate Compounding Periods (Cont.) • Let m denote the number of compounding intervals per year, n the number of years, and r be the stated annual rate of interest. • The relation between present and future values is stated as: • FVn = PV(1 + r/m)n×m • e.g., if PV = 1000, r = .12 and m = 1 then FV2 is: • FV2 = 1000(1 + .12)2×1 = $1254.40, • while if m = 4 (quarterly compounding), then • FV2 = 1000(1 + .12/4)2×4 • FV2 = 1000(1 + .03)8 = $1266.77
Example • Find the PV of $500 to be received in 5 years, with: • 12% stated annual rate, annual compounding, • 12% stated annual rate, semiannual compounding, • 12% stated annual rate, quarterly compounding,
Stated And Effective Annual Rates • Notice that the use of more frequent compounding acts as if to (or effectively) increase(s) the stated interest rate. • The Effective Annual Rate (EAR) is the annual interest rate that would produce the same answer, with annual compounding, as is obtained with more frequent compounding. It can be obtained by: • EAR = (1 + r/m)m - 1 • so if r = .12 and m = 4, then EAR = (1.03)4 - 1 = .1255. • The effective annual rate is the rate that if you earned it for a year with annual compounding, you would end up with the same amount of money as you would have under more frequent compounding. • Note the appropriate discount rate in any application is always an effective rate. At times this may also be a stated annual rate.
Example • A bank quotes a mortgage rate of 8% (the stated annual rate), but will compute monthly loan payments using standard time value formulas. This implies monthly compounding. What is the effective annual interest rate on the loan? So the loan effectively costs you 8.30% per year for every dollar you borrow for a year.
Example • A bank quotes a rate of 8% (the stated annual rate), but will compound interest quarterly. What is the effective annual interest rate on the loan? • What is the effective semi-annual interest rate? • Just think in a flexible way about what a period is.
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.
C C C … 0 1 2 3 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 periodbefore the first payment arrives.
C1 C1(1+g) C1(1+g)2 … 0 1 2 3 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:
… … Examples Perpetuity: $100 per period forever discounted at 10% per period $100$100$100 0 1 2 3 PV = C/r = $100/0.10 = $1,000 Growing perpetuity: $100 received at time t = 1, growing at 2% per period forever and discounted at 10% per period $100 $102 $104.04 0 1 2 3 PV = C1/(r –g ) = $100/(0.10 – 0.02) = $1,250
Verification of the Perpetuity Example Answers Place the present value in a bank account, and recreate the payments. Let’s stop at 4 years since “forever” would take a while. Note that the account balance is growing. At what rate? Why must this happen?
2 2 3 3 1 1 0 0 C C C C C 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. C
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,
Example: A five year annuity paying $2000 per year, with r = 5%. • Valuing the payments individually we get: • Using the annuity formula we get:
Alternatively, suppose you were given $8,658.95 today instead of the annuity • Notice that you can exactly replicate the annuity cash flows by investing the present value to earn 5%. • This again demonstrates that present value calculations provide a literal equality, in that the future cash flows can be converted into the present value and vice versa, if (and only if) the selected discount rate is representative of actual capital market conditions.
C1 C1(1+g) C1(1+g)T-2 C1(1+g)T-1 … 0 1 2 T-1 T Growing Annuities • A stream of payments each period for a fixed number of periods where the payment grows each period at a constant rate.
500 500(1.02) 500(1.02)18 500(1.02)19 … 0 1 2 19 T=20 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%?
A Valuation Problem What is the value of a 10-year annuity that pays $300 a year at the end of each year, if the first payment is deferred until 6 years from now, and if the discount rate is 10%? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 300 300 300 • • • • • • 300 The value of the annuity payments as of five years from now is: Now discount this equivalent payment back 5 years to time zero:
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?