1. Valuation of standardized cash flow streams – Chapter 4, Section 4.4 Module 1.4 Copyright © 2013 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

2. 4-1 4.4 Simplification of Standardized Cash Flow Streams  Perpetuity A constant stream of cash flows that lasts forever  Growing perpetuity A stream of cash flows that grows at a constant rate forever  Annuity A stream of constant cash flows that lasts for a fixed number of periods  Growing annuity A stream of cash flows that grows at a constant rate for a fixed number of periods

3. 4-2 Perpetuity A constant stream of cash flows that lasts forever 0 … 1 C 2 C 3 C

4. 4-3 Perpetuity: Example What is the value of a British consol that promises to pay £15 every year forever? The interest rate is 10-percent. 0 … 1 £15 2 3

5. 4-4 Growing Perpetuity A growing stream of cash flows that lasts forever 0 … 1 C 2 C×(1+g) 3 C ×(1+g) 2

6. 4-5 Growing Perpetuity: Example The expected dividend next year is $1.30, and dividends are expected to grow at 5% forever. If the discount rate is 10%, what is the value of this promised dividend stream? 0 … 1 $1.30 2 $1.30×(1.05) 3 $1.30 ×(1.05) 2

7. 4-6 Annuity A constant stream of cash flows with a fixed maturity 0 1 C 2 C 3 C T C

8. 4-7 Annuity: Example If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% APR compounded monthly on 36-month loans? 0 1 $400 2 3 36 $400

9. 4-8 Important note!  The perpetuity and annuity formulas that were just covered assume the first cash flow occurs one period from today.  Often times, the first cash flow occurs today This is an “annuity due”  Often times, the first cash flow might occur further out in the future  We need to be able handle these occurrences.

10. 4-9 What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%? 0 1 2 3 4 5 $100 $100 $323.97$297.22 4-9

11. 4-10 Growing Annuity A growing stream of cash flows with a fixed maturity 0 1 C 2 C×(1+g) 3 C ×(1+g) 2 T C×(1+g) T-1

12. 4-11 Growing Annuity: Example A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by 3% each year. What is the present value of this retirement account if the discount rate is 10%? 0 1 $20,000 2 $20,000×(1.03) 40 $20,000×(1.03) 39

13. 4-12 Growing Annuity: Example You are evaluating an income generating property. Net rent is received at the end of each year. The first year's rent is expected to be $8,500, and rent is expected to increase 7% each year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%? 0 1 2 3 4 5 $34,706.26

14. 4-13 4.5 Loan Amortization  Pure Discount Loans are the simplest form of loan. The borrower receives money today and repays a single lump sum (principal and interest) at a future time.  Interest-Only Loans require an interest payment each period, with full principal due at maturity.  Amortized Loans require repayment of principal over time, in addition to required interest. This is the form of standard car and home loans

15. 4-14 Pure Discount Loans  Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments.  If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market? PV = 10,000 / 1.07 = 9,345.79

16. 4-15 Interest-Only Loan  Consider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually. What would the stream of cash flows be?  Years 1 – 4: Interest payments of.07(10,000) = 700  Year 5: Interest + principal = 10,700  This cash flow stream is similar to the cash flows on corporate bonds, and we will talk about them in greater detail later.

17. 4-16 Amortized Loan with Fixed Principal Payment  Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year.  Click on the Excel icon to see the amortization table

18. 4-17 Amortized Loan with Fixed Payment  Each payment covers the interest expense plus reduces principal  Consider a 4 year loan with annual payments. The interest rate is 8%,and the principal amount is $5,000. What is the annual payment?  4 N  8 I/Y  5,000 PV  CPT PMT = -1,509.60  Click on the Excel icon to see the amortization table

19. 4-18 Alternatively without using calculator functions:  Consider a 4 year loan with annual payments. The interest rate is 8%,and the principal amount is $5,000. What is the annual payment?  We solve for C by re-arranging our annuity formula:

20. 4-19 Two more financial planning questions to finish module…  Q1: How much do I need to save each month to have $3 million when I retire in 25 years at a rate of 12% APR compounded monthly?  We can solve this a couple of ways: First, find PV of $3 million, and then solve for annuity payment that gives the same PV. OR Directly solve for annuity payment that gives a FV of $3 million (need FV of annuity formula)

21. 4-20 Solving using first method  First, solve for PV of $3 million: Monthly r =.12/12 =.01; periods=12*25=300 PV = $3,000,000/(1.01) (300) = $151,603  Now, find monthly annuity payment that gives us the same PV Recall PV of Annuity: 151603=C/.01[1-1/(1.01) (300) ] 151603(.01)/[.9495] = C C=$1,597

22. 4-21 Solving using second method  Recall PV of Annuity:  So FV of Annuity =  Note that the equation for the FV of an annuity is just the PV of annuity formula multiplied by (1+r) T !

23. 4-22 Now, solving using second method:  FV = $3,000,000  3000000 = C/.01[0.9495](19.7885)  3000000 = C(1878.91)  C = $1,597 per month  Important: You have a new formula for the FV of an annuity, and hopefully you see the logic that caused the answers to be the same

24. 4-23 Last example…  Q2: How long will it take you to accumulate $4 million if you save $2000/month at 12% APR compounded monthly?  Now, we need to use PV equations, but solve for time.

25. 4-24 Annuity equations re-arranged for time  PV of annuity re-arranged for time:  FV of annuity re-arranged for time:

26. 4-25 Now we can solve…  Using the second equation on the prior slide  T = ln((3000000*.01)/2000 +1) /ln(1.01)  T = ln(16)/ln(1.01)  T= 2.7726 /.00995  T = 278.64 months or 23.22 years.

27. 4-26 In the next module, we will build spreadsheets that make these calculations much quicker!