The Time Value of Money

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1 The Time Value of Money This handout is an overview of the basic tools and concepts needed for this corporate nance course. Proofs and explanations are given in order to facilitate your understanding and will not be part of an exam if they go beyond the material covered in class. Why do we need to discount? One fundamental issue in nance is to determine today's value of future cash ows, e.g. the value of a share, a treasury bond or in general of any investment project. Intuitively it makes perfect sense to assume that a certain amount of money tomorrow should be worth less than the same amount today. Receiving it today makes us better or at least as well o as receiving it tomorrow, since it gives us the opportunity to invest and to receive interest payments for future consumption:. Example Imagine you can choose either project A, which yields 00 e today or project B which yields 00 e in one year. Let's compare the projects after one year: if you chose project A you get 00 ( + r) e, where r is the interest rate on a riskless asset. 2 Since option B gives you exactly 00 e we can say that the future value of project A is higher than the future value of project B. In order to be indierent between the two projects you would want project B to pay exactly 00 ( + r) e too, because this would make its present value equal to 00 e..2 Risk F uture V alue = P resent V alue ( + Interest Rate) () [ ] P resent V alue = F + r } uture {{ V alue} (2) } {{ } F uture P ayoffs Discount F actor So far we assumed payos to be riskless, in reality though most of the time future payos will be risky and therefore discounting with the riskless rate is not appropriate. Given two projects with the same future payos, one risky the other one riskless, every normal 3 investor will prefer the riskless payo over the risky one. Therefore we have to discount future cash-ows with a discount rate which takes the project's riskiness into account. E.g. there might be no tomorrow... 2 The existence of a riskless asset is an assumption which is satised approximately in most nancial markets. It is comparable with the textbook's assumption of well functioning capital markets. 3 Strictly speaking every risk averse investor, i.e. every investor with a concave utility function.

2 2 FORMULAS 2.3 Example The present value of a lottery which pays 00 e with probability and 0 else can be 2 computed by discounting the expected cash-ow with an appropriate discount rate. We can nd such a discount rate by looking at an alternative investment at the capital market with an equivalent risk prole. Once such an investment is found we can use its return as above to compute the present value: + r ( ) Capital markets and the opportunity cost of capital As you have seen above the basic idea behind any present value calculation is to use the opportunity cost of capital to discount any future cash ows. The idea is simple: if you can nd a payo on the capital market which is similar to your project regarding its risk-return prole it should have at most the same price. If not you would rather go to your bank and invest on the capital market than continuing with your project. Keep in mind that the underlying assumptions are that capital markets work well, i.e. price the future cash ows correctly and that any frictions like transaction costs can be neglected. Both assumptions might not be entirely justied in many cases but are a good approximation in general. (3) 2 Formulas 2. The net present value formula To calculate todays value of a future cash ow a multi-period version of the calculations above can be applied: T C t N C 0 + (4) ( + r t ) t Typically (but not necessarily) C 0 is a cash outow, e.g. the setup costs of an investment project. Keep in mind that the discount rate might be time varying although it is typically assumed to be constant over time. T denotes the period of the last cash-ow, if cash ows continue innitely you have to use the formula for a the sum of a geometric series. 2.2 Geometric Series A result used frequently in nancial mathematics is the formula for the sum of a geometric series: T aρ t = a ρt ρ, (5) t=0 with ρ. Moreover as T goes to innity t= where the absolute value of ρ has to be less than one. ρt lim a n ρ = a ρ, (6) 2

3 2 FORMULAS Perpetuities You might want to evaluate the PV of a constant stream of cash ows one period from now, e.g. in order to calculate the value of a corporation as in Chapter 5 of the textbook. Using the formula for the sum of a geometric series with ρ = we get C ( + r) + C ( + r) r ( + r) i + r r (7) Notice that the underlying assumptions are that the stream of cash ows C and the discount rate r do not change over time Growing perpetuities Imagine the constant stream of cash ows you want to evaluate is growing at the constant rate g, i.e. C 2 ( + g), C 3 2 ( + g) ( + g) 2 and so on. By the same logic as before we get C C( + g) + ( + r) ( + r) r ( + g) i ( + r) i + r +g r g (8) which is the present value of a growing perpetuity. 2.4 Annuities An annuity pays you a constant sum each year for a xed number of periods T starting from next period, e.g. to calculate the present value of a lease contract. The formula for the present value of an annuity can be computed as follows: C ( + r) + C ( + r) + + C 2 ( + r) T + r T ( + r) ( ) T i + r r r( + r) } {{ T (9) } T year annuity factor 2.4. Annuity due If the stream of payments starts today we have C + C ( + r) + C ( + r) + + C 2 ( + r) T T ( + r) = ( + r) C T = ( + r) (P V of an annuity) (0) i + r ( + r) i 3

4 3 BOND MARKET Growing annuities As with the perpetuities we can calculate the value of a growing annuity as 4 C C( + g) + ( + r) ( + r) + + C( + g)t 2 ( + r) T T + r ( + g) i ( + r) i + r ( +g +g ) T ( r g ) ( + g) T. () r g ( + r) T Equivalent annual costs You might be asked to calculate the payment per period equivalent to a certain amount today, e.g. in order to calculate the rent you have to charge in order to cover your expenses for a house you have just bought. The formula follows directly from the annuity formula above: annuity payment (T year annuity factor) (2) P V annuity payment = (T year annuity factor) (3) Note that the PV might be a present value you have to compute before or an up-front payment Example Say we have bought a house for e which will collapse in 5 years and we know that the appropriate discount rate is 5 %. The rent we have to charge to cover our expenses is then according to the formula above 00000/( ) = (+0.05) 5 3 Bond Market Governments, municipalities, companies who need to raise cash for long-term investment issue bonds to borrow money. Investors who hold bonds in their portfolios lend their money to those debtors expectating to receive cash ows representing interest and repayment of principal. Bonds can also be thought as long-term loans. The present value formula helps us valuing bonds: C ( + r) + C 2 ( + r) Principal+C T 2 ( + r) T where C t represent the coupon payments at time t (Note that C T is the coupon payment at maturity) 5, r is the return oered by similar securities in the same risk class. The discount rate that delivers the actual bond price (quoted in the market) is called yield to maturity. This term should not be confused with r which is the opportunity cost of capital that an investor would have gained if s(he) invested her money in another security with the same risk. One has to be careful with the timing of coupon payments while valuing 4 Pay attention: The formula in Figure 3.5 on page 47 in the textbook is wrong! 5 Coupons are quoted as a percentage of the face value (e.g. 00 in Germany, 000 in U.S), i.e. the amount investors receive as principal at maturity. Don't confuse the face value with the quoted price of the bond. 4

5 5 THE TERM STRUCTURE OF INTEREST RATES 5 bonds, typically bonds pay coupons annually (as in Germany, Japan) or semiannually (as in U.S) 6, but rates are quoted in annual terms (See carefully the valuation of German versus U.S bonds in slides). 4 Bonds and Interest Rates Remember that there is a close link between interest rates oered in the market and bond prices. Lets think for a second: if bonds provide a stream of sure cash ows in the future, why should the price of the bonds change? The only parameter that can change in the present value formula is the interest rate, that is used to discount the stream of future cash ows. Recall that this interest rate is the opportunity cost of capital, once you lend your money to a debtor by holding the bond, you are commited to receive a certain interest 7, but once the interest rates change in the market, lets say they go up(down), you lose the opportunity to hold another bond with higher interest payments, hence your bond is traded at a discount (you are lucky to hold a bond that pays higher interest then prevailing market rates, hence your bond trades at a premium), hence the price of the bond will fall(rises). Notice that another important dimension is the horizon until maturity, since changes in interest rates are likely to have a greater eect on the value of distant cash ows, hence the price of long term bonds is subject to interest rate risk to greater extend. This fact brings us to the following denitions: Duration : ( P V (C )) V + (2 P V (C 2)) V + (3 P V (C 2))... V where V is the total value of the bond. Notice that duration weights the present value of each cash ow(cf) by the time needed to obtain the CF (from today's point of view). It desribes therefore the average time to payment of a bond. A related term is V olatility(%) = duration + yield each bond's volatility is the slope of the curve relating the bond price to the interest rate. 5 The Term Structure of Interest Rates So far we made the simplifying asssumption that one period, two period,..., n-period spot rates are the same in the present value formula, instead with a single rate r, the right present value formula would be ( + r ) + ( + r 2 ) ( + r T ) T where r is the one-period spot rate, r 2 is the two-period spot rate 8, etc... Rather than discounting each cash ow at a dierent intest rate, we could use a single rate, yield to maturity, that would provide the same present value ( + y) + ( + y) ( + y) T 6 Or sometimes they do not pay any coupons, as in zero-coupon government bonds. 7 This rate is set and do not change during the life of the bond, i.e. until maturity. 8 Note that all spot rates are quoted in annual terms. 5

6 6 EXPECTATION HYPOTHESIS 6 even though such a rate would provide the same present value, it neglects the variations over dierent spot rates. If we plot the spot rates over dierent periods, i.e. interest rates of stripped bonds, we obtain the term structure of interest rates (See gure 4.5 on page 69). Typically, the term structure is upward-sloping, i.e. long term rates of interest are higher than short term rates, but not necessarily so. Let's try to see whether we can nd a plausible explanation for the slope of the term structure: 6 Expectation Hypothesis Suppose that the prevailing one-year spot rate is r and the two-year spot rate is r 2, such that r 2 > r. You can either invest your money in one-year Treasury strip and such that you would earn ( ) e at the end of each year for each dollar you invested. If you accept to invest for two years, you would earn the two-year spot rate of r 2 and by the end of two years, each dollar would be worth ( 2 ) 2 e. The extra return you would earn for keeping your money invested for two years is f 2 = ( + r 2 ) 2 /( + r ) which is called the forward interest rate. The question you should ask is how should you decide whether to invest your money today for a two years horizon at an annual rate of r 2, or invest today for one year at an annual rate of r and invest again for another year at the end of the rst year at the then prevailing rate. The problem is that you do not know today which rate will prevail at the end of the rst year for a one year(e.g.one year Treasury strip) investment maturing at the end of the second year. You can only have an expectation about this rate, let's call it r 2 following the book's notation. Now, we can turn to the expectation hypothesis: it asserts that in equilibrium the expected payo from investing in a two-year Treasury strip should be the same as the payo from investing in one-year Treasury strips in two succesive years. In other words, the expected spot rate r 2 should be equal to the forward rate f 2. Claim The expectation hypothesis implies that the only reason for an upward-sloping(downwardsloping) term structure is that investors expect short term rates( r 2 ) to increase(fall). Proof. Suppose that the two-years spot rate r 2 (one year spot rate r ) exceeds the one year spot rate r (two-years spot rate r 2 ), i.e. an upward-sloping(downward-sloping) term structure, then the forward rate f 2 (r ) exceeds r 2 (the forward rate f 2 ) since Since ( + r 2 )( + r 2 ) = By denition ( + r )( + f 2 ) ( + r 2 ) > ( + r ) ( + f 2 ) > ( + r 2 ) f 2 > r 2 > r By Expectation Hypothesis f 2 = r 2 r 2 > r 2 > r i.e. expected short term rates should be higher than r.the opposite case in parentheses can be shown in a similar way following the footnote on page 7. Notice that the expectation hypothesis is just an equilibrium theory, that might explain the shape of the term structure, but its validity should be tested with empirical data. Like any other theory, it is a simplied version of reality and leaves out some essential elements that turn out to be important in reality, e.g. risk, ination. 6

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