Chapter Eleven Asset Markets

Different prices in period 1 and 2

Different prices in period 1 and 2 Price of consumption in period 1 is 1 Price of consumption in period 2 is p2, e.g. p2=p1(1+ p) , where p is inflation Consumer consumes c1 in period 1 What is the consumption level in period 2?

Different prices in period 1 and 2

Valuing Securities A financial security is a financial instrument that promises to deliver an income stream. E.g.; a security that pays $m1 at the end of year 1, $m2 at the end of year 2, and $m3 at the end of year 3. What is the most that should be paid now for this security?

Valuing Securities The security is equivalent to the sum of three securities; the first pays only $m1 at the end of year 1, the second pays only $m2 at the end of year 2, and the third pays only $m3 at the end of year 3.

Valuing Securities The PV of $m1 paid 1 year from now is The PV of $m2 paid 2 years from now is The PV of $m3 paid 3 years from now is The PV of the security is therefore

Valuing Bonds A bond is a special type of security that pays a fixed amount $x for T years (its maturity date) and then pays its face value $F. What is the most that should now be paid for such a bond?

Valuing Bonds

Valuing Bonds Suppose you win a State lottery. The prize is $1,000,000 but it is paid over 10 years in equal installments of $100,000 each. What is the prize actually worth?

Valuing Bonds is the actual (present) value of the prize.

Example 1 Suppose that you invest $100 in an asset yielding an interest rate 10 where the interest is paid once a year. Now suppose that the interest is paid a) monthly. b) daily. c) continuously. How much will you get after 10 years?

Example 1 100*(1,1)^10 = 259,37 100*(1+0,1/12)^120 = 270,70 100*(1+0,1/365)^3650 = 271,79 100*EXP(10*0,1) = 271,82

Valuing Consols A consol is a bond which never terminates, paying $x per period forever. What is a consol’s present-value?

Valuing Consols

Valuing Consols Solving for PV gives

Valuing Consols E.g. if r = 0.1 now and forever then the most that should be paid now for a console that provides $1000 per year is

Assets An asset is a commodity that provides a flow of services over time. E.g. a house, or a computer. A financial asset provides a flow of money over time -- a security.

Assets Typically asset values are uncertain. Incorporating uncertainty is difficult at this stage so we will instead study assets assuming that we can see the future with perfect certainty.

Selling An Asset Q: When should an asset be sold? When its value is at a maximum? No. Why not?

Selling An Asset Suppose the value of an asset changes with time according to

Selling An Asset Value Years

Selling An Asset Maximum value occurs when That is, when t = 50.

Selling An Asset Value Max. value of $24,000 is reached at year 50. Years

Selling An Asset The rate-of-return in year t is the income earned by the asset in year t as a fraction of its value in year t. E.g. if an asset valued at $1,000 earns $100 then its rate-of-return is 10%.

Selling An Asset Q: Suppose the interest rate is 10%. When should the asset be sold? A: When the rate-of-return to holding the asset falls to 10%. Then it is better to sell the asset and put the proceeds in the bank to earn a 10% rate-of-return from interest.

Selling An Asset The rate-of-return of the asset at time t is In our example, so

Selling An Asset The asset should be sold when That is, when t = 10.

Selling An Asset Value Max. value of $24,000 is reached at year 50. slope = 0.1 Years

Selling An Asset Value Max. value of $24,000 is reached at year 50. slope = 0.1 Sell at 10 years even though the asset’s value is only $8,000. Years

Selling An Asset What is the payoff at year 50 from selling at year 10 and then investing the $8,000 at 10% per year for the remaining 40 years?

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Selling An Asset What is the payoff at year 50 from selling at year 10 and then investing the $8,000 at 10% per year for the remaining 40 years?

Selling An Asset So the time at which an asset should be sold is determined by Rate-of-Return = r, the interest rate.

Arbitrage Arbitrage is trading for profit in commodities which are not used for consumption. E.g. buying and selling stocks, bonds, or stamps. No uncertainty  all profit opportunities will be found. What does this imply for prices over time?

Arbitrage The price today of an asset is p0. Its price tomorrow will be p1. Should it be sold now? The rate-of-return from holding the asset is I.e.

Arbitrage Sell the asset now for $p0, put the money in the bank to earn interest at rate r and tomorrow you have

Arbitrage When is not selling best? When I.e. if the rate-or-return to holding the asset the interest rate, then keep the asset. And if then so sell now for $p0.

Arbitrage If all asset markets are in equilibrium then for every asset. Hence, for every asset, today’s price p0 and tomorrow’s price p1 satisfy

Arbitrage I.e. tomorrow’s price is the future-value of today’s price. Equivalently, I.e. today’s price is the present-value of tomorrow’s price.

Arbitrage in Bonds Bonds “pay interest”. When the interest rate paid by banks rises, what will happen with the market prices of bonds?

Arbitrage in Bonds A bond pays a fixed stream of payments of $x per year, no matter the interest rate paid by banks. At an initial equilibrium the rate-of-return to holding a bond must be R = r’, the initial bank interest rate. If the bank interest rate rises to r” > r’ then r” > R and the bond should be sold. Sales of bonds lower their market prices.