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

Deposit & Lending Interest Rates in Hong Kong

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 $P t in the bank at interest rate 1+i,you will have $(1+i)P t which can be used to buy goods. The real interest rate is

Real Interest Rates Nominal vs. Real

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, PQ t+j. In theory, the present value of PQ is PV t (PQ,j) such that if we received PV t (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.

Real Present Value vs. Nominal Present Value Assume constant inflation, so P t+j = (1+π) j P t. 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.

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 {PQ t+1,PQ t+2, ….PQ t+j } is

Geometric Sums A geometric series follows the pattern x, x 2, x 3, …,x j. A geometric sum is the sum of a geometric series x + x 2 + x 3 + ….x j A formula to solve for the sum of a geometric series is

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

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.

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

Hang Seng Index & Dividend Index

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

Recursive Substitution Projecting forward, we can calculate a formula for the price. Repeating a large number, N, of times

Stock Price Repeating a large number, N, of times

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 D t+k = (1+g) j D t. Set N=∞, so effectively the investor never sells the stock and the value of the stock is the present value of the dividends.

Stock Price Set If N = ∞, x < 1 then x N = 0

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 P*P*

Stock Prices: Interest Rates Rise P P P*P*

Stock Prices: Growth Rates Rise P P*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: r eq = 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

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.