1 Present and Future Value


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1 Lecture 8: Asset Markets c 2009 Je rey A. Miron Outline:. Present and Future Value 2. Bonds 3. Taxes 4. Applications Present and Future Value In the discussion of the twoperiod model with borrowing and lending, we showed that the budget constraint can be written as c + c 2 ( + )! +! 2 ( + ) where the! s are endowments of the consumption good at time and 2. A simple extension of this framework says that if a consumer receives income in each of two periods, then the budget constraint is c + c 2 ( + ) m + m 2 ( + ) where the m s are income. This equation is an example of a present value formula. In words, it says that the present value of consumption equals the present value of income. It is useful to review and extend the idea of present value. The key insight is that a dollar today is not equivalent to a dollar tomorrow. The exact tradeo depends on the interest rate, assuming the consumer can borrow and lend at this rate. A few additional comments are useful reminders or extensions.
2 The concept of present value is related to the concept of future value. in one period of a dollar today is The FV F V () ( + r) because you could lend the dollar and earn interest between now and next period. Similarly, the future value of a dollar T periods from now is F V (T ) ( + r) T The present value of a dollar one period from now is P V () ( + r) because if you had (+r) and loaned it out you would have one dollar next period. The PV of a dollar T periods from now is P V (T ) ( + r) T These formulas apply in cases with multiple time periods and multiple payments. For example, the PV of receiving m t in period t where t goes from, say, to T is P V P T m t t ( + r) t We could also have the summation go to in nity; this particular case is useful in many settings. Finally, the concept of present or future value does not require the interest rate to be constant; one simply has to use a more general formula to account for this possibility. For example, the PV of a dollar three periods from now would be P V ( + r )( + r 2 )( + r 3 ) where the r i s are the interest rates in the three di erent periods. This formula gets used frequently in macroeconomics, nance, and other settings. The crucial economic insight related to present value in the context of budget constraints is that, assuming the consumer can borrow and lend at the market interest rate, the consumer should be indi erent about the timing of when income or endowments arise, assuming one is holding the present value of the endowment 2
3 stream constant. Any change in income or an endowment that raises the present value allows the consumer more choices about what to consume by shifting the budget constraint out. This applies even if the consumer has odd preferences, since, unless they are very odd, the consumer always prefers more to less. This point will arise later in discussions of behavioral economics. 2 Bonds The next item to discuss is a particular type of security, or nancial asset, known as a bond. This particular security is common and appears widely in economic models. Plus, analyzing the value of a bond provides a clean illustration of present values. A bond is a piece of paper representing a promise by one party to make certain payments to another party over time in exchange for money from that party now. A bond is essentially a way to borrow money, and bonds are issued routinely by governments and corporations. Consider two examples. In the case of a coupon bond, the borrow receives some amount from the lender at time 0 and promises to make periodic coupon (interest) payments, x, to the lender plus a xed amount, F at a terminal date T: What is the present value of receiving this series of payments assuming the interest rate is r? It is P V x ( + r) + x ( + r) + : : : + x 2 ( + r) + F T ( + r) T So, this is the price at which the bond should trade in an unrestricted market. Equivalently, it is the most a lender should be willing to lend in exchange for this series of payments from the borrower. It is useful to consider a variation on this kind of bond, known as a console or perpetuity. In this case the terminal payment is zero but the coupon payments continue forever. The P V is therefore x P V ( + r) + x ( + r) + : : : 2 x + + r ( + r) + ( + r) + : : : 2 3
4 x + r! ( ) +r x r using the standard formula for the sum of a geometric series and simplifying. A di erent way to see this is to note that the value of the bond must satisfy V r x because lending out V each period would generate V r in return to the lender, so the lender will issue a bond only if the coupon each period equals this same amount. Rearranging this equation gives the same result. The formula might not seem terribly useful in practice because most investments do not make payments forever. It turns out, however, that for even moderate interest rates (e.g., 50%), the present value of payments 30 years or more in the future is su ciently small that ignoring them has a minor e ect on calculations. Thus, imagine that you want to know the present value of making $20,000 per year in mortgage payments on a 30 year mortgage, given an interest rate of 8%. You could use the correct formula, which is built into Excel and various other programs. If you wanted a quick number, however, you could approximate this present value as 20,000 / ,000. The precise number (from Excel) turns out to be $225, Taxes In de ning interest rates and asset returns, we have so far ignored taxes. In many case, however, the interest paid by bonds and other asset returns is taxable, and a rational investor should care about the aftertax return rather than the pretax return per se. 4
5 Consider a bond that pays an interest rate of r. investment of $ now yields $( + r) one period later. In the absence of taxes, an In the presence of taxes, where the tax rate is the percentage of the interest earned that must be paid to the government in taxes, the aftertax yield is $( + r( )). That is, you end up with your original dollar, plus percent of the interest income. The government gets the remaining percent. This means that if one is discounting future cash ows in a setting where the investor must pay taxes on any interest income, the right expression to use in discounting is ( + r( t)), not ( + r). The appropriate expression is a bit more involved in the presence of in ation. In any discounting problem, it is critical to have the numerator and denominator match, meaning one should always discount nominal amounts at a nominal rate and real amounts at a real rate. If this is done correctly, the answers match perfectly. The correct formula for the real interest rate in the presence of taxes is r( ) and the appropriate discount factor is therefore + r( ). It is useful to consider a simple example of discounting nominal versus real cash ows. Assume the cash ows will be C and C 2 in periods and 2. in ation (the rate of growth of the price level) will be, i.e., Assume that P 2 P 2 where P 2 is the price level in period 2 and the price level in period has been normalized to. Let the nominal interest rate be r. earlier, satis es + + r + Then the real interest rate,, as de ned Then if we discount the nominal amounts at the nominal rate, we get 5
6 C + C 2 + r If instead we discount the real amounts at the real rate, we get C + (C 2P 2 ) + Now substitute + r + into the real formula. You get C + (C 2 P 2 ) + + r + C + (C 2P 2 ) + r + C + (C 2P 2 ) + r + P 2 C + (C 2P 2 ) + r P 2 C + C 2 + r which is the same. 6
7 4 Applications of Present Value 4. Depletable Resources Consider a depletable resource like oil. Assume a competitive market for the activity of extracting oil from the ground. Extraction costs are zero. One question of interest is how we should expect the price of oil to change over time. We can analyze this as follows: Note rst that oil is an asset, meaning it lasts from one period to the next. If the owner of a barrel of oil does not sell it today, he can sell it in a later period, and presumably he would like to sell when it will command the highest price, adjusted for the time value of money. To avoid an arbitrage opportunity, therefore, holding the barrel of oil for one period must earn the same nancial return that could be earned elsewhere. This means the price of oil over time must satisfy P t+ P t ( + r). Why is the right condition? Consider what could happen if this condition were not satis ed. If the price were growing faster than the interest rate, owners would want to extract a bit less out and sell next period, since that produces a higher return than selling now and investing this at the rate r. This tends to raise the current price, which pushes price growth downward. If the price were growing slower than the interest rate, owners would want to extract more and invest this at the going interest rate. This would drive down the price and push price growth upward. So, price growth must obey the equation above. What determines the price of oil today? Suppose demand is constant at D barrels a year and that total world supply is S. These assumptions mean that oil will be exhausted in 7
8 T SD years. When this happens, the economy will use an alternate fuel that costs C dollars per barrel. We know therefore that in T years the oil must sell for C dollars. Thus the price at time 0, P 0, must obey: P 0 ( + r) T C, This says the price must grow at rate r for T years and equal C at the end of those T years. Alternatively, we can rewrite this equation as P 0 C ( + r) T This says that the current price must be the discounted value of the price that the alternative will sell for when oil is exhausted. What if there is an oil discovery? Then the amount of supply, S, goes up, so the number of years to exhaustion, T, goes up, which means the price goes down. This makes sense. What is technological progress decreases C; the cost of an alternative fuel? the price goes down now; this also makes sense. Then One general result, therefore, is that future events relevant to the price of oil should a ect the price now. The textbook discusses examples like war in the Middle East that might reduce the supply of oil. A standard noneconomist thing to say is that such events should not a ect the current price of oil, since it takes weeks before oil in the Middle East gets to the United States. The model here, however, shows that prices should adjust immediately. Indeed, it is e cient for prices to rise right away, since this signals everyone that oil has become scarcer, which encourages development of alternative fuels. 8
9 4.2 When to Cut a Forest A di erent application asks when it makes sense to cut down a set of trees that is being grown to provide lumber. Suppose the size of a forest, measured in terms of the lumber it yields, is some function of time, F (t). Assume that the price of lumber is constant. Assume also that initially the forest grows quickly but that the growth rate slows over time. If there is a competitive market for lumber, when should the owner of the forest cut the trees down for timber? The answer must be, when the rate of growth of the forest equals the interest rate. We can see this more formally by noting that the present value of the forest once cut down is P V F (T ) ( + r) T The question is therefore what value of T maximizes this present value. Two o setting pressures operate. On the one hand, allowing more time to by reduces the value of the forest now because it cannot be sold until a later date. On the other hand, the forest grows quickly at the beginning, so the growth in its value initially outweights discounting. The Appendix to Chapter shows that if interest is compounded continuously, then the correct discount factor is e rt The question is therefore what value of T maximizes P V F (T ) e rt and the answer is the value of T such that r F 0 (T ) F (T ) This is a case where using a bit more math makes the problem cleaner. 9
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