The Time Value of Money

Money NOW

is worth more than

money LATER!

What is The Time Value of Money?

• A dollar received today is worth more than a dollar received tomorrow– This is because a dollar received today can

be invested to earn interest– The amount of interest earned depends on

the rate of return that can be earned on the investment

• Time value of money quantifies the value of a dollar through time

Uses of Time Value of Money

• Time Value of Money, or TVM, is a concept that is used in all aspects of finance including:– Bond valuation– Stock valuation– Accept/reject decisions for project management– Financial analysis of firms– And many others!

Obviously, $10,000 today$10,000 today.

You already recognize that there is TIME TIME VALUE TO MONEYVALUE TO MONEY!!

The Interest RateThe Interest Rate

Which would you prefer -- $10,000 today $10,000 today or $10,000 in 5 years$10,000 in 5 years?

TIMETIME allows you the opportunity to postpone consumption and earn INTERESTINTEREST.

Why TIME?Why TIME?

Why is TIMETIME such an important element in your decision?

Formulas

• Common formulas that are used in TVM calculations:*– Present value of a lump sum:

PV = CFt / (1+r)t OR PV = FVt / (1+r)t

– Future value of a lump sum:

FVt = CF0 * (1+r)t OR FVt = PV * (1+r)t

– Present value of a cash flow stream: n

PV = [CFt / (1+r)t] t=0

Variables

• where– r = rate of return– t = time period– n = number of time periods– PMT = payment– CF = Cash flow (the subscripts t and 0 mean at time t

and at time zero, respectively)– PV = present value (PVA = present value of an

annuity)– FV = future value (FVA = future value of an annuity)

Present Value of a Cash Flow Stream

• A cash flow stream is a finite set of payments that an investor will receive or invest over time.

• The PV of the cash flow stream is equal to the sum of the present value of each of the individual cash flows in the stream.

• The PV of a cash flow stream can also be found by taking the FV of the cash flow stream and discounting the lump sum at the appropriate discount rate for the appropriate number of periods.

Example of PV of a Cash Flow Stream

• Joe made an investment that will pay $100 the first year, $300 the second year, $500 the third year and $1000 the fourth year. If the interest rate is 10 percent, what is the present value of this cash flow stream?

1. Draw a timeline:

00 11 22 33 44

??

$100$100 $300$300 $500$500 $1000$1000

??????

i = 10%i = 10%

Example of PV of a Cash Flow Stream

2. Write out the formula using symbols: n

PV = [CFt / (1+r)t] t=0

OR

PV = [CF1/(1+r)1]+[CF2/(1+r)2]+[CF3/(1+r)3]+[CF4/(1+r)4]

3. Substitute the appropriate numbers:

PV = [100/(1+.1)1]+[$300/(1+.1)2]+[500/(1+.1)3]+[1000/(1.1)4]

Example of PV of a Cash Flow Stream

4. Solve for the present value:

PV = $90.91 + $247.93 + $375.66 + $683.01

PV = $1397.51

Effect of discountingNet value of 100 kr in 10 years

Chinese for riskChinese for risk

In most language risk means something dangerous will happen, in finance the definition for risk is both an opportunity and dangerous!

RiskRisk

Tillgång A Tillgång B

1000 900 1500

1000

2000

Risk vs. uncertainty

Cost of capital

There are two capital interests:

• debt

• equity

Discounted cash-flowDiscounted cash-flow

1 )1(tt

t

r

CFP

P = net value of future cashflows CFt = cashflow at time t r = cost of capital

Cost of capitalCost of capital

We got 3 firms A, B and C. All firms do have exactly the same cash-flows.

Firm Cost of capital NPV projection Terminal value Value of firm A 20 % 11,1 Msek 44,2 Msek 55,3 Msek B 15 % 11,6 Msek 55,8 Msek 67,4 Msek C 10 % 12,2 Msek 93,3 Msek 105,5 Msek

CAPMCAPM

• rf = Risk Free rate

• rm = Stockmarket return

• β= Beta value (risk)

)( fmfi rrrR

The argument

• Risk free rate – the least in all investment

• Risk premium, the extra risk taken when investing at the stock market

• Beta value, the relative risk by investing in one individual stock.

Risk Free Rate

• Represents return an investor can achieve on the least risky asset in the market

• Generally based on current yield to maturity on Government bond

• What bond?

– Use YTM on local government bond; or

– a global bond (e.g. Germany) and adjust for country risk (Global CAPM)

• What term? … rule of thumb is to match to the term to explicit period in the projections

Development of Bond market

10 year bond

0,002,004,006,008,00

10,0012,0014,0016,00

jan-

90

jan-

92

jan-

94

jan-

96

jan-

98

jan-

00

jan-

02

jan-

04

jan-

06

Beta

“What Is Beta and How Is It Calculated?”

or….

Beta

• A “coefficient measuring a stock’s relative volatility”

• Beta measures a stock’s sensitivity to overall market movements

Source:UBS Warburg Dictionary of Finance and Investment Terms

Beta – portfolio theory

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Number of Securties in Porfolio

Por

tfolio

Sta

ndar

d D

evia

tion

Specific Risk

Systematic Risk

Specific risk of investments can be eliminated, but systematic risk cannot be diversified away

… Beta measures an equity’s exposure to systematic risk

• In practice, Beta is measured by comparing changes in a stock price to changes in the value of the S&P 500 index over a given time period

• The S&P 500 index has a beta of 1

A Generic Example

• Stock XYZ has a beta of 2

• The S&P 500 index increases in value by 10%

• The price of XYZ is expected to increase 20% over the same time period

Beta can be Negative

• Stock XYZ has a beta of –2

• The S&P 500 index INCREASES in value by 10%

• The price of XYZ is expected to DECREASE 20% over the same time period

• If the beta of XYZ is 1.5 …

• And the S&P increases in value by 10%

• The price of XYZ is expected to increase 15%

• A beta of 0 indicates that changes in the market index cannot be used to predict changes in the price of the stock

• The company’s stock price has no correlation to movements in the market index

Beta

What influences an equity’s beta?

• cyclicality of revenue streams (i.e. relationship to state of the economic cycle)

• level of operational leverage in a company’s cost structure

• level of financial leverage in a company’s capital structure

Company Beta

AMGN 0.82

BRK.B 0.73

C 1.37

XOM 0.10

MSFT 1.80

MWD 2.19

NOK 2.05

PXLW 1.93

TXN 1.70

VIA.B 1.39Source: taken from yahoo.finance.com, except PXLW from bloomberg.com

Beta and Risk

• Beta is a measure of volatility

• Volatility is associated with risk

Risk-Reward CurveRisk

Expected Return

• If beta is a measure of risk, then investors who hold stocks with higher betas should expect a higher return for taking on that risk

Beta and CAPM

The capital asset pricing model:

E(R) = Rf + B(Rm-Rf)

where:E(R) or Re = Expected returnRf = risk free rate of returnB = betaRm = market return

WACC

Weighted average cost of capital:

WACC = (D/V)*Rd*(1-T) + (E/V)*Re

where:D = market value of firm’s debtRd = return on debt securitiesT = tax rateE = market value of firm’s equity securitiesRe = return on equity securities (from CAPM) V = total value of firm’s securities (D + V)

WACC and Beta

• WACC increases as the beta and the rate of return on the equity securities increases (all else constant)

• WACC is used as the discount rate in DCF models

• Therefore, increasing WACC reduces the firms valuation to reflect the increase in risk

How to Calculate Beta

Beta = Covariance(stock price, market index)

Variance(market index)

**When calculating, you must compare the percent change in the stock price to the percent change in the market index**

How to Calculate Beta

• Easily calculated using Excel and Yahoo! Finance

• Use COVAR and VARP worksheet functions

Risk premiumRisk premium

Changes in the economy. Risk premium are higher for more volatile countries. Weaker economy – higher risk premium.

Political risk. Risk for more political instability – higher risk premium.

Structure of the stock market. Some markets are very risky as Sweden due to many and small firms. More volatile market – higher risk premium.

Risk premium

EMRP = (Rm - Rf)

• EMRP reflects the premium investors require for investing in equities rather than risk-free securities

• There are two main approaches to calculate EMRP; these are:

– historic averages - ; and

– forward looking methods (surveys, dividend discount model)

Risk premium

5.2%3.8%200 years (since 1798)

7.8%5.8%72 years (since 1926)

8.2%7.0%60 years (since 1938)

8.1%6.9%50 years (since 1948)

6.3%5.2%40 years (since 1958)

5.2%4.0%30 years (since 1968)

8.5%7.8%20 years (since 1978)

Arithmetic averageGeometric averagePeriod

US EMRP (relative to government bonds

5.2%3.8%200 years (since 1798)

7.8%5.8%72 years (since 1926)

8.2%7.0%60 years (since 1938)

8.1%6.9%50 years (since 1948)

6.3%5.2%40 years (since 1958)

5.2%4.0%30 years (since 1968)

8.5%7.8%20 years (since 1978)

Arithmetic averageGeometric averagePeriod

US EMRP (relative to government bonds

Source: Ibbotson Associates and Grabowski and KingSource: Ibbotson Associates and Grabowski and King

Current consensus

EMRP range is 4%-8%

Historical risk premium

Tidsperiod Aktie – kort statspapper Aktie - StatsobligationerAritmetisk Geometrisk Aritmetisk Geometrisk

1928-2002 7,67 % 5,73 % 6,25 % 4,53 %1962-2002 5,17 % 3,90 % 3,66 % 2,76 %1990-2002 6,32 % 4,69 % 2,15 % 0,95 %

Aritmetisk Geometrisk1919-2002 4,13 3,101928-2002 5,46 4,121962-2002 5,53 4,251992-2002 -1,82 -4,60

Swedish market

US market

CAPM

Total Company Risk

Specific Risk Systematic risk

Litigationindustry eventsLabour force strikeNew rival company

Market events:interest rateGDPinflation

Company sizeNumber of

ObservationsStatistical

significanceLocationGearing

Comparators

Asset Beta

Cyclicality of revenues

Operational leverage

Companyactivities

Max and min cost of capital Sectra January 2003

• Beta 0,99 vs. 2,70

• Risk premium 3,10 vs. 6,97

• Risk free rate 3,61 vs. 4,70

3,61 + 0,99 * 3,10 = 6,68

4,70 + 2,70 * 6,97 = 23,52All other forecasts equal will give expected price per share

118 kr and 26 kr, the price was 44 kr.

WACC… summing up

Risk Free Rate 4.50%

EMRP 5.0%

Beta (Equity) 1.25 6.25%

Cost of Equity 10.75%

Cost of Debt 6.50%

Tax Shield @ 35% (2.28%) 4.22%

Gearing (Debt: Debt+ Equity) 40%

WACC 8.14%