The Time value of money

1
The Time value of money

• Aswath Damodaran

2
Intuition Behind Present Value

• There are three reasons why a dollar tomorrow is
  worth less than a dollar today
• Individuals prefer present consumption to future
  consumption. To induce people to give up present
  consumption you have to offer them more in the
  future.
• When there is monetary inflation, the value of
  currency decreases over time. The greater the
  inflation, the greater the difference in value
  between a dollar today and a dollar tomorrow.
• If there is any uncertainty (risk) associated
  with the cash flow in the future, the less that
  cash flow will be valued.
• Other things remaining equal, the value of cash
  flows in future time periods will decrease as
• the preference for current consumption increases.
• expected inflation increases.
• the uncertainty in the cash flow increases.

3
Discounting and Compounding

• The mechanism for factoring in these elements is
  the discount rate. The discount rate is a rate at
  which present and future cash flows are traded
  off. It incorporates
• (1) Preference for current consumption (Greater
  ....Higher Discount Rate)
• (2) Expected inflation(Higher inflation .... Highe
  r Discount Rate)
• (3) Uncertainty in the future cash flows (Higher
  Risk....Higher Discount Rate)
• A higher discount rate will lead to a lower value
  for cash flows in the future.
• The discount rate is also an opportunity cost,
  since it captures the returns that an individual
  would have made on the next best opportunity.
• Discounting future cash flows converts them into
  cash flows in present value dollars. Just a
  discounting converts future cash flows into
  present cash flows,
• Compounding converts present cash flows into
  future cash flows.

4
Present Value Principle 1

• Cash flows at different points in time cannot be
  compared and aggregated.
• All cash flows have to be brought to the same
  point in time, before comparisons and
  aggregations are made.
• That point of time can be today (present value)
  or a point in time in the future (future value).

5
Time lines for cash flows

• The best way to visualize cash flows is on a time
  line, where you list out how much you get and
  when.
• In a time line, today is specified as time 0
  and each year is shown as a period.

6
Cash Flow Types and Discounting Mechanics

• There are five types of cash flows -
• simple cash flows,
• annuities,
• growing annuities
• perpetuities and
• growing perpetuities
• Most assets represent combinations of these cash
  flows. Thus, a conventional bond is a combination
  of an annuity (coupons) and a simple cash flow
  (face value at maturity). A stock may be a
  combination of a growing annuity and a growing
  perpetuity.

7
I.Simple Cash Flows

• A simple cash flow is a single cash flow in a
  specified future time period.
• Cash Flow CFt
• _______________________________________________
• Time Period t
• The present value of this cash flow is
• PV of Simple Cash Flow CFt / (1r)t
• The future value of a cash flow is
• FV of Simple Cash Flow CF0 (1 r)t

8
Application The power of compounding - Stocks,
Bonds and Bills

• Between 1926 and 2013, stocks on the average made
  about 9.55 a year, while government bonds on
  average made about 4.93 a year and T.Bills
  earned 3.53 a year.
• If your holding period is one year, the
  difference in end-of-period values is small
• Value of 100 invested in stocks in one year
  109.55
• Value of 100 invested in bonds in one year
  104.93
• Value of 100 invested in T.Bills for one year
  103.53

9
Holding Period and Value
10
Concept Check

• Most pension plans allow individuals to decide
  where their pensions funds will be invested -
  stocks, bonds or money market accounts.
• Where would you choose to invest your pension
  funds?
• Predominantly or all equity
• Predominantly or all bonds and money market
  accounts
• A Mix of Bonds and Stocks
• Will your allocation change as you get older?
• Yes
• No

11
The Frequency of Compounding

• The frequency of compounding affects the future
  and present values of cash flows. The stated
  interest rate can deviate significantly from the
  true interest rate
• For instance, a 10 annual interest rate, if
  there is semiannual compounding, works out to-
• Effective Interest Rate 1.052 - 1 .10125 or
  10.25
• Frequency Rate t Formula Effective Annual Rate
• Annual 10 1 r 10.00
• Semi-Annual 10 2 (1r/2)2-1 10.25
• Monthly 10 12 (1r/12)12-1 10.47
• Daily 10 365 (1r/365)365-1 10.5156
• Continuous 10 expr-1 10.5171

12
II. Annuities

• An annuity is a constant cash flow that occurs at
  regular intervals for a fixed period of time.
  Defining A to be the annuity, the time line looks
  as follows
• A A A A
• 0 1 2 3 4

13
Present Value of an Annuity

• The present value of an annuity can be calculated
  by taking each cash flow and discounting it back
  to the present, and adding up the present values.
  Alternatively, there is a short cut that can be
  used in the calculation A Annuity r
  Discount Rate n Number of years

14
Example PV of an Annuity

• The present value of an annuity of 1,000 at the
  end of each year for the next five years,
  assuming a discount rate of 10 is -
• The notation that will be used in the rest of
  these lecture notes for the present value of an
  annuity will be PV(A,r,n).

15
Annuity, given Present Value

• The reverse of this problem, is when the present
  value is known and the annuity is to be estimated
  - A(PV,r,n).
• This, for instance, is the equation you would use
  to determine your monthly payments on a home
  mortgage.

16
Computing Monthly Payment on a Mortgage

• Suppose you borrow 200,000 to buy a house on a
  30-year mortgage with monthly payments. The
  annual percentage rate on the loan is 8. The
  monthly payments on this loan, with the payments
  occurring at the end of each month, can be
  calculated using this equation
• Monthly interest rate on loan APR/ 12 0.08/12
  0.0067

17
Future Value of an Annuity

• The future value of an end-of-the-period annuity
  can also be calculated as follows-
• This is the equation you would use to determine
  how much money you will accumulate at a future
  point in time if you set aside a constant amount
  each period.

18
An Example

• Thus, the future value of 1,000 at the end of
  each year for the next five years, at the end of
  the fifth year is (assuming a 10 discount rate)
  -
• The notation that will be used for the future
  value of an annuity will be FV(A,r,n).

19
Annuity, given Future Value

• if you are given the future value and you are
  looking for an annuity - A(FV,r,n) in terms of
  notation -

20
Application Saving for College Tuition

• Assume that you want to send your newborn child
  to a private college (when he gets to be 18 years
  old). The tuition costs are 16000/year now and
  that these costs are expected to rise 5 a year
  for the next 18 years. Assume that you can
  invest, after taxes, at 8.
• Expected tuition cost/year 18 years from now
  16000(1.05)18 38,506
• PV of four years of tuition costs at 38,506/year
  38,506 PV(A ,8,4 years) 127,537
• If you need to set aside a lump sum now, the
  amount you would need to set aside would be -
• Amount one needs to set apart now
  127,357/(1.08)18 31,916
• If set aside as an annuity each year, starting
  one year from now -
• If set apart as an annuity 127,537
  A(FV,8,18 years) 3,405

21
Application How much is an MBA worth?

• Assume that you were earning 40,000/year before
  entering program and that tuition costs are
  16000/year. Expected salary is 54,000/year
  after graduation. You can invest money at 8.
• For simplicity, assume that the first payment of
  16,000 has to be made at the start of the
  program and the second payment one year later.
• PV Of Cost Of MBA 16,00016,000/1.08 40000
  PV(A,8,2 years) 102,145
• Assume that you will work 30 years after
  graduation, and that the salary differential
  (14000 54000-40000) will continue through
  this period.
• PV of Benefits Before Taxes 14,000
  PV(A,8,30 years) 157,609
• This has to be discounted back two years -
  157,609/1.082 135,124
• The present value of getting an MBA is 135,124
  - 102,145 32,979
• 1. How much would your salary increment have to
  be for you to break even on your MBA?
• 2. Keeping the increment constant, how many years
  would you have to work to break even?

22
Application Savings from Refinancing Your
Mortgage

• Assume that you have a thirty-year mortgage for
  200,000 that carries an interest rate of 9.00.
  The mortgage was taken three years ago. Since
  then, assume that interest rates have come down
  to 7.50, and that you are thinking of
  refinancing. The cost of refinancing is expected
  to be 2.50 of the loan. (This cost includes the
  points on the loan.) Assume also that you can
  invest your funds at 6.
• Monthly payment based upon 9 mortgage rate
  (0.75 monthly rate)
• 200,000 A(PV,0.75,360 months)
• 1,609
• Monthly payment based upon 7.50 mortgage rate
  (0.625 monthly rate)
• 200,000 A(PV,0.625,360 months)
• 1,398
• Monthly Savings from refinancing 1,609 -
  1,398 211

23
Refinancing The Trade Off

• If you plan to remain in this house indefinitely,
• Present Value of Savings (at 6 annually 0.5 a
  month)
• 211 PV(A,0.5,324 months)
• 33,815
• The savings will last for 27 years - the
  remaining life of the existing mortgage. You will
  need to make payments for three additional years
  as a consequence of the refinancing -
• Present Value of Additional Mortgage payments -
  years 28,29 and 30
• 1,398 PV(A,0.5,36 months)/1.0627
• 9,532
• Refinancing Cost 2.5 of 200,000 5,000
• Total Refinancing Cost 9,532 5,000
  14,532
• Net Effect 33,815 - 14,532 19,283
  Refinance

24
Follow-up Questions

• 1. How many years would you have to live in this
  house for you break even on this refinancing?
• 2. We've ignored taxes in this analysis. How
  would it impact your decision?

25
Valuing a Straight Bond

• You are trying to value a straight bond with a
  fifteen year maturity and a 10.75 coupon rate.
  The current interest rate on bonds of this risk
  level is 8.5.
• PV of cash flows on bond 107.50 PV(A,8.5,15
  years) 1000/1.08515 1186.85
• If interest rates rise to 10,
• PV of cash flows on bond 107.50 PV(A,10,15
  years) 1000/1.1015 1,057.05
• Percentage change in price -10.94
• If interest rate fall to 7,
• PV of cash flows on bond 107.50 PV(A,7,15
  years) 1000/1.0715 1,341.55
• Percentage change in price 13.03
• This asymmetric response to interest rate changes
  is called convexity.

26
Bond Pricing Proposition 1

• The longer the maturity of a bond, the more
  sensitive it is to changes in interest rates.

27
Bond Pricing Proposition 2

• The lower the coupon rate on the bond, the more
  sensitive it is to changes in interest rates.

28
III. Growing Annuity

• A growing annuity is a cash flow growing at a
  constant rate for a specified period of time. If
  A is the current cash flow, and g is the expected
  growth rate, the time line for a growing annuity
  looks as follows

29
Present Value of a Growing Annuity

• The present value of a growing annuity can be
  estimated in all cases, but one - where the
  growth rate is equal to the discount rate, using
  the following model
• In that specific case, the present value is equal
  to the nominal sums of the annuities over the
  period, without the growth effect.

30
The Value of a Gold Mine

• Consider the example of a gold mine, where you
  have the rights to the mine for the next 20
  years, over which period you plan to extract
  5,000 ounces of gold every year. The price per
  ounce is 300 currently, but it is expected to
  increase 3 a year. The appropriate discount rate
  is 10. The present value of the gold that will
  be extracted from this mine can be estimated as
  follows

31
PV of Extracted Gold as a Function of Expected
Growth Rate
32
IV. Perpetuity

• A perpetuity is a constant cash flow at regular
  intervals forever. The present value of a
  perpetuity is-
• Forever may be a tough concept for human beings
  to grasp, but it makes the mathematics much
  simpler.

33
Valuing a Console Bond

• A console bond is a bond that has no maturity and
  pays a fixed coupon. Assume that you have a 6
  coupon console bond. The value of this bond, if
  the interest rate is 9, is as follows -
• Value of Console Bond 60 / .09 667

34
V. Growing Perpetuities

• A growing perpetuity is a cash flow that is
  expected to grow at a constant rate forever. The
  present value of a growing perpetuity is -
• where
• CF1 is the expected cash flow next year,
• g is the constant growth rate and
• r is the discount rate.

35
Valuing a Stock with Growing Dividends

• In twelve months leading into January 2014, Con
  Ed paid dividends per share of 2.52.
• Its earnings and dividends had grown at 2 a year
  between 2004 and 2013 and were expected to grow
  at the same rate in the long run.
• The rate of return required by investors on
  stocks of equivalent risk was 7.50.
• With these inputs, we can value the stock using a
  perpetual growth model
• Value of Stock 2.52 (1.02)/(0.075 ? 0.02)
  46.73

36
Value and Growth!
37
Financial Statement Analysis
38
Questions we would like answered
39
Basic Financial Statements

• The balance sheet, which summarizes what a firm
  owns and owes at a point in time.
• The income statement, which reports on how much a
  firm earned in the period of analysis
• The statement of cash flows, which reports on
  cash inflows and outflows to the firm during the
  period of analysis

40
The Balance Sheet
41
A Financial Balance Sheet
42
The Income Statement
43
Modifications to Income Statement

• There are a few expenses that consistently are
  miscategorized in financial statements.In
  particular,
• Operating leases are considered as operating
  expenses by accountants but they are really
  financial expenses
• R D expenses are considered as operating
  expenses by accountants but they are really
  capital expenses.
• The degree of discretion granted to firms on
  revenue recognition and extraordinary items is
  used to manage earnings and provide misleading
  pictures of profitability.

44
Dealing with Operating Lease Expenses

• Debt Value of Operating Leases PV of Operating
  Lease Expenses at the pre-tax cost of debt
• This now creates an asset - the value of which is
  equal to the debt value of operating leases. This
  asset now has to be depreciated over time.
• Finally, the operating earnings has to be
  adjusted to reflect these changes
• Adjusted Operating Earnings Operating Earnings
  Operating Lease Expense - Depreciation on the
  leased asset
• If we assume that depreciation principal
  payment on the debt value of operating leases, we
  can use a short cut
• Adjusted Operating Earnings Operating Earnings
  Debt value of Operating leases Cost of debt

45
Operating Leases at Boeing and The Home Depot in
1998
46
Imputed Interest Expenses on Operating Leases
47
The Effects of Capitalizing Operating Leases

• Debt will increase, leading to an increase in
  debt ratios used in the cost of capital and
  levered beta calculation
• Operating income will increase, since operating
  leases will now be before the imputed interest on
  the operating lease expense
• Net income will be unaffected since it is after
  both operating and financial expenses anyway
• Return on Capital will generally decrease since
  the increase in operating income will be
  proportionately lower than the increase in book
  capital invested

48
RD Expenses Operating or Capital Expenses

• Accounting standards require us to consider RD
  as an operating expense even though it is
  designed to generate future growth. It is more
  logical to treat it as capital expenditures.
• To capitalize RD,
• Specify an amortizable life for RD (2 - 10
  years)
• Collect past RD expenses for as long as the
  amortizable life
• Sum up the unamortized RD over the period.
  (Thus, if the amortizable life is 5 years, the
  research asset can be obtained by adding up 1/5th
  of the RD expense from five years ago, 2/5th of
  the RD expense from four years ago...

49
Capitalizing RD Expenses Boeing
50
Boeings Corrected Operating Income
51
The Effect of Capitalizing RD

• Operating Income will generally increase, though
  it depends upon whether RD is growing or not. If
  it is flat, there will be no effect since the
  amortization will offset the RD added back. The
  faster RD is growing the more operating income
  will be affected.
• Net income will increase proportionately,
  depending again upon how fast RD is growing
• Book value of equity (and capital) will increase
  by the capitalized Research asset
• Capital expenditures will increase by the amount
  of RD Depreciation will increase by the
  amortization of the research asset For all
  firms, the net cap ex will increase by the same
  amount as the after-tax operating income.

52
The Statement of Cash Flows
53
The Financial perspective on cash flows

• In financial analysis, we are much more concerned
  about
• Cash flows to the firm or operating cash flows,
  which are before cash flows to debt and equity)
• Cash flows to equity, which are after cash flows
  to debt but prior to cash flows to equity

54
Fundamentals of Valuation
55
Discounted Cashflow Valuation Basis for Approach

• where,
• n Life of the asset
• CFt Cashflow in period t
• r Discount rate reflecting the riskiness of
  the estimated cashflows

56
Two Measures of Cash Flows

• Cash flows to Equity Thesea are the cash flows
  generated by the asset after all expenses and
  taxes, and also after payments due on the debt.
  This cash flow, which is after debt payments,
  operating expenses and taxes, is called the cash
  flow to equity investors.
• Cash flow to Firm There is also a broader
  definition of cash flow that we can use, where we
  look at not just the equity investor in the
  asset, but at the total cash flows generated by
  the asset for both the equity investor and the
  lender. This cash flow, which is before debt
  payments but after operating expenses and taxes,
  is called the cash flow to the firm

57
Two Measures of Discount Rates

• Cost of Equity This is the rate of return
  required by equity investors on an investment. It
  will incorporate a premium for equity risk -the
  greater the risk, the greater the premium.
• Cost of capital This is a composite cost of all
  of the capital invested in an asset or business.
  It will be a weighted average of the cost of
  equity and the after-tax cost of borrowing.

58
Equity Valuation
59
Valuing Equity in a Finite Life Asset

• Assume that you are trying to value the Home
  Depots equity investment in a new store.
• Assume that the cash flows from the store after
  debt payments and reinvestment needs are expected
  will be 850,000 a year, growing at 5 a year
  for the next 12 years.
• In addition, assume that the salvage value of the
  store, after repaying remaining debt will be 1
  million.
• Finally, assume that the cost of equity is 9.78.
  

60
Firm Valuation
61
Valuing a Finite-Life Asset

• Consider the Home Depot's investment in a
  proposed store. The store is assumed to have a
  finite life of 12 years and is expected to have
  cash flows before debt payments and after
  reinvestment needs of 1 million, growing at 5
  a year for the next 12 years.
• The store is also expected to have a value of
  2.5 million at the end of the 12th year (called
  the salvage value).
• The Home Depot's cost of capital is 9.51.

62
Expected Cash Flows and present value
63
Valuation with Infinite Life
64
Valuing the Home Depots Equity

• Assume that we expect the free cash flows to
  equity at th Home Depot to grow for the next 10
  years at rates much higher than the growth rate
  for the economy. To estimate the free cash flows
  to equity for the next 10 years, we make the
  following assumptions
• The net income of 1,614 million will grow 15 a
  year each year for the next 10 years.
• The firm will reinvest 75 of the net income back
  into new investments each year, and its net debt
  issued each year will be 10 of the reinvestment.
• To estimate the terminal price, we assume that
  net income will grow 6 a year forever after year
  10. Since lower growth will require less
  reinvestment, we will assume that the
  reinvestment rate after year 10 will be 40 of
  net income net debt issued will remain 10 of
  reinvestment.

65
Estimating cash flows to equity The Home Depot
66
Terminal Value and Value of Equity today

• FCFE11 Net Income11 Reinvestment11 Net Debt
  Paid (Issued)11
• 6,530 (1.06) 6,530 (1.06) (0.40) (-277)
  4,430 million
• Terminal Price10 FCFE11/(ke g)
• 4,430 / (.0978 - .06) 117,186 million
• The value per share today can be computed as the
  sum of the present values of the free cash flows
  to equity during the next 10 years and the
  present value of the terminal value at the end of
  the 10th year.
• Value of the Stock today 6,833 million
  117,186/(1.0978)10
• 52,927 million

67
Valuing Boeing as a firm

• Assume that you are valuing Boeing as a firm, and
  that Boeing has cash flows before debt payments
  but after reinvestment needs and taxes of 850
  million in the current year.
• Assume that these cash flows will grow at 15 a
  year for the next 5 years and at 5 thereafter.
• Boeing has a cost of capital of 9.17.

68
Expected Cash Flows and Firm Value

• Terminal Value 1710 (1.05)/(.0917-.05)
  43,049 million

Year Cash Flow Terminal Value Present Value
1 978 895
2 1,124 943
3 1,293 994
4 1,487 1,047
5 1,710 43,049 28,864
Value of Boeing as a firm Value of Boeing as a firm Value of Boeing as a firm 32,743