Financial Concepts

Present Value and Stocks

(corresponds with Chapter 21: Equity Markets)

Students should be able to

Calculate the present value of a stream of nominal payments.

Calculate the real present value of a stream of real payments. Calculate Geometric Sum

Use the Gordon Growth model of the dividend yield to value stocks.

Use the graphical tools to model the effect of hypothetical events on stock market prices.

Nominal Interest Rate

Put one $1 into the bank, get $(1+i) in the future. 1+i is the gross nominal interest rate, iis the net

nominal interest rate. Put $1 in the bank for T periods and after T

periods you will have

$(1 ) (1 ) (1 ) (1 ).....(1 )

$(1 )Ti i i i i

i

Deposit & Lending Interest Rates in Hong Kong

Nov-1979 Nov-1983 Nov-1987 Nov-1991 Nov-1995 Nov-1999 Nov-2003

20

18

16

14

12

10

8

6

4

2

0

HK% pa

Savings Deposit Rate HSBC's Best Lending Rate: P...

Real Interest Rates

The real interest rate or goods interest rate is the amount of goods you will be able buy in one period if you lend enough money to buy one good today.

If you deposit $Pt in the bank at interest rate 1+i,you will have $(1+i)Pt which can be used to buy goods.

The real interest rate is

1

$ 1 t

t

PiP

1

11

t

t

PP

ir

Real Interest RatesNominal vs. Real

-4

0

4

8

12

16

86 88 90 92 94 96 98 00 02 04

HSBC Best Lending RateEx Post Real Interest Rate (12 mnth moving average)

Real Interest Rates:Ex Ante vs. Ex Post

Real interest rate is based on future prices which are not known when decisions to save and borrow are made.

Ex Ante Real Interest Rate is calculated using the expectation of the future price level. Problem: Cannot be calculated without a theory of how

savers and borrowers form expectations. Ex post real interest rate is calculated using the

actual price level. Problem: May only approximate the ex ante rate actually

used by borrowers and savers and can only be calculated for the past.

Present Value

A payment of cash today is of greater value than an equal sized payment in the future, since a payment today can be placed in the bank to earn interest which will allow extra spending in the future.

The Present Value of a future payment is the size of a payment today that would allow spending in the future equal to the size of the future payment.

Present Value

Consider a payment that will be received in j periods, PQt+j. In theory, the present value of PQ is PVt(PQ,j) such that if we received PVt(PQ,j) today we would have spending power at time t+j equal to PQ.

Present values are often used to calculate how much should be paid for some asset which will produce income in the future.

(1 ) ( , ) ( , )(1 )

t jjt j t t j

PQPQ i PV PQ j PV PQ j

i

Real Present Value vs. Nominal Present Value

Assume constant inflation, so Pt+j = (1+π)jPt.

The value in current dollars of a future payment in current dollars is discounted by the nominal interest rate (raised to the number of periods to be waited).

Divide both sides by the current price level to get the constant dollar value.

To derive the constant dollar (real) present value of a future payment measured in constant dollars, discount by the real interest rate.

(1 )t j

t j

PQPV

i

(1 ) (1 )

(1 )(1 )(1 )

t j t jtt j j

t

jt j t j

jjt j

PQ PPVV P i

PQ QirP

Stream of Payments

Rather than the present value of one future payment, you might want to calculate the present value of a stream of payments since most assets produce income over time.

The present value of a stream of income is the sum of the present values of each payment.

The present value of {PQt+1,PQt+2, ….PQt+j} is

1 2

2 ....1 1 1

t jt tj

PQPQ PQ

i i i

Geometric Sums

A geometric series follows the pattern x, x2, x3, …,xj .

A geometric sum is the sum of a geometric series x + x2 + x3 + ….xj

A formula to solve for the sum of a geometric series is

2 3 ... (1 )1

j jxx x x x x

x

Constant Payment Assets

Some assets pay-off a constant value in every time period. We can use geometric sums to calculate their

present value.

2 3

2 3

2 3

11 1 1

1 111

...(1 ) (1 ) (1 ) (1 )

1 1 1 1 1[ ... ],(1 ) (1 ) (1 ) (1 ) 1

[ ... ] (1 )1

(1 ) (1 )1

j

j

j j

j jii i

i

D D D D

i i i i

D xi i i i i

xD x x x x D x

x

DD

i

Total Returns

The returns to an asset in any period are the ratio of the payment associated with that asset today plus the price it could be sold for today relative to the price for which it could be bought in the previous period.

Net Returns are the sum current yield and capital gain.

1

1 t tt

t

P DR

P

1

1 1

t t tt

t t

D P PR

P P

Equity A share of common equity entitles its owner

to an equal share of non-retained earnings called dividends.

Shareholders are also entitled to a vote in the election of the board that will control the firm.

Shareholders are effectively the owners of the firm but enjoy limited liability: if a firm goes bankrupt, shareholders are not liable beyond the value of the stock.

Hong Kong Stock Market

(Million HK$ 2001) USAMarket Capitalization 5208375Number of Companies 852Turnover 2545669GDP 1234967Market Cap as % of GDP 421.74% 158.05%Turnover as % of Market Cap 48.88% 102.31%

Hang Seng Index & Dividend Index

0

4000

8000

12000

16000

200002

3

4

5

6

7

86 88 90 92 94 96 98 00 02 04

DIVYIELD HANGSENG

Dividends and Stock Prices

When investors buy a stock at time t, there is some minimum return the would require to induce them to purchase req.

The price will be bid up until expected return is equal to the required return.

1 1 1 111

eqt t t tt eq

t

P D P Dr P

P r

Recursive Substitution

Projecting forward, we can calculate a formula for the price.

Repeating a large number, N, of times

1 1 2 2 1 2 2

1 2,1 1 1 1

t t t t t t tt t teq eq eq eq

P D P D D P DP P P

r r r r

3 3 1 2 2 1 2 3 3

2 2 2 3,1 1 11 1 1

t t t t t t t t tt t teq eq eqeq eq eq

P D D P D D D P DP P P

r r rr r r

4 4 1 2 3 3 1 2 3 4 4

3 2 3 2 3 4,1 1 11 1 1 1 1

t t t t t t t t t t tt t teq eq eqeq eq eq eq eq

P D D D P D D D D P DP P P

r r rr r r r r

1 2 3 4

2 3 4 ...1 1 1 1 1 1

t t t t t N t Nt N Neq eq eq eq eq eq

D D D D D PP

r r r r r r

Stock Price

Repeating a large number, N, of times

1 2 3 42 3 4 ...

1 1 1 1 1

1

t

t t t t t NNeq eq eq eq eq

t NNeq

P

D D D D D

r r r r r

P

r

Equity Value

A stock owner who holds shares for N periods will receive payments of the dividends over those N periods plus the sale price in N periods.

The stock price is the present value of dividends plus resale price using the required return as the interest rate.

Firms may potentially last forever, so N could be infinitely large.

Pricing Stocks

Assume a constant growth, g, rate for dividends, so that Dt+k = (1+g)jDt.

Set N=∞, so effectively the investor never sells the stock and the value of the stock is the present value of the dividends.

2 3

2 3

(1 ) (1 ) (1 ) (1 )...

1 1 1 1

Nt t t t N

t Neq eq eq eq

g D g D g D g DP

r r r r

Stock Price

Set

If N = ∞, x < 1 then xN = 0

2 3

(1 )

1

( ... ) 11

eq

N Nt t t

gx

rx

P D x x x x D xx

1

(1 )(1 )1

(1 )1 11

eqt t

t t eq eq

eq

gg D Dx rP D

gx r g r gr

Dividend Yield

The current yield of a stock is the dividend yield. The dividend yield is equal to the required return

minus the growth of the dividend in the future. When required returns are high, investors will be

willing to pay a low price for a given future pay-off.

When dividend growth is high, investors believe they will achieve large capital gains and are willing to pay a relatively high price for the stock.

Stock Prices

P

1t

t

Dg

P

eqr

P*

Stock Prices: Interest Rates Rise

P

1t

t

Dg

P

eqr 'eqr

P

P*

Stock Prices: Growth Rates Rise

P

1t

t

Dg

P

eqr

P*

P**

Required Return

One alternative to owning stocks is putting money in the bank. Stocks should pay a return that is at least as great as bank interest rates.

But stock returns are unpredictable and riskier than bank deposits.

Decompose the required return on stocks into two parts, the interest rate and the risk premium: req = i + eqp

Stock Returns

The average stock return on the Hang Seng Index over the past 20 years has been about 15%.

The average stock return on Hang Seng Prime has been about 8%

However, Hong Kong stock market is extremely volatile.

Hong Kong Stock Market Returns

-80

-40

0

40

80

120

86 88 90 92 94 96 98 00 02 04

Stock Return on Hang Seng Index (12 Month Moving Average)

Decompose Growth

Growth in dividends are limited by growth in output (since capital income is an approximately constant share of output)

Growth in nominal output can be divided into growth in prices (inflation) and growth in real output.