- 2. 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.
- 3. 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.
- 4. Selling An Asset Q: When should an asset be sold? When its value is at a maximum? No. Why not?
- 5. Selling An Asset Suppose the value of an asset changes with time according to V ( t ) = −1000 + 1000t − 10t 2
- 6. Value Selling An Asset 24000 19000 14000 9000 4000 -1000 0 10 20 30 40 50 60 Years
- 7. Selling An Asset V ( t ) = −1000 + 1000t − 10t Maximum value occurs when V'( t ) = 1000 − 20t = 0 That is, when t = 50. 2
- 8. Value Selling An Asset 24000 Max. value of $24,000 is reached at year 50. 19000 14000 9000 4000 -1000 0 10 20 30 40 50 60 Years
- 9. 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%.
- 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.
- 11. Selling An Asset The rate-of-return of the asset at time t is In our example, V'( t ) . V( t ) 2 V (t ) = −1000 + 1000 t − 10 t . so V'( t ) = 1000 − 20t = 0 V'( t ) 1000 − 20t = . V ( t ) − 1000 + 1000t − 10t 2
- 12. Selling An Asset The asset should be sold when V'( t ) 1000 − 20t = = 0⋅1 V ( t ) − 1000 + 1000t − 10t 2 That is, when t = 10.
- 13. Value Selling An Asset 24000 Max. value of $24,000 is reached at year 50. 19000 14000 slope = 0.1 9000 4000 -1000 0 10 20 30 40 50 60 Years
- 14. Value Selling An Asset 24000 Max. value of $24,000 is reached at year 50. 19000 14000 slope = 0.1 9000 4000 -1000 0 10 Sell at 10 years even though the asset’s value is only $8,000. 20 30 40 50 60 Years
- 15. 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?
- 16. 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? 40 $8,000 × (1 + 0 ⋅ 1) = $362,074 > $24,000
- 17. Selling An Asset So the time at which an asset should be sold is determined by Rate-of-Return = r, the interest rate.
- 18. 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?
- 19. 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 p1 − p0 asset is R= I.e. p0 (1 + R )p0 = p1 .
- 20. Arbitrage Sell the asset now for $p0, put the money in the bank to earn interest at rate r and tomorrow you have (1 + r )p0 .
- 21. Arbitrage When is not selling best? When (1 + R )p0 > (1 + r )p0 . I.e. if the rate-or-return to holding the asset R > r the interest rate, then keep the asset. And if R < r then (1 + R )p0 < (1 + r )p0 so sell now for $p0.
- 22. Arbitrage If all asset markets are in equilibrium then R = r for every asset. Hence, for every asset, today’s price p0 and tomorrow’s price p1 satisfy p1 = (1 + r )p0 .
- 23. Arbitrage p1 = (1 + r )p0 I.e. tomorrow’s price is the future-value of today’s price. Equivalently, p1 p0 = . 1+r I.e. today’s price is the present-value of tomorrow’s price.
- 24. Arbitrage in Bonds Bonds “pay interest”. Yet, when the interest rate paid by banks rises, the market prices of bonds fall. Why?
- 25. 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.
- 26. Taxation of Asset Returns rb is the before-tax rate-of-return of a taxable asset. re is the rate-of-return of a tax exempt asset. t is the tax rate. The no-arbitrage rule is: (1 - t)rb = re I.e. after-tax rates-of-return are equal.
- 27. Financial Intermediaries Banks, brokerages etc. – facilitate trades between people with different levels of impatience – patient people (savers) lend funds to impatient people (borrowers) in exchange for a rate-of-return on the loaned funds. – both groups are better off.