ECG590I Asset Pricing. Lecture 2: Present Value 1

Transcription

1 ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide between paying 1$ now or 1$ one year from now, then you would rather pay one year from now. This section is about computing the value today of something in the future.

2 ECG59I Asset Pricing. Lecture 2: Present Value Definition Present value: the value today of an amount to be paid at a specific date in the future. 2.2 Simple case One payment of face amount (principal) X, one period in the future. First, suppose you had V now and could save it at interest rate R. In one period, you would have X = V (1 + R). Rearrange terms to get V = X 1+R. V is the present value of X. Two periods in the future: X = V (1 + R 1 )(1 + R 2 ) = V 2 (1 + R i ) i=1

3 ECG59I Asset Pricing. Lecture 2: Present Value 3 from which we get: V = X [ 2 1 (1 + R i )] = X i=1 [ 2 ] (1 + R i ) 1 i=1 Many periods: V = X [ I ] (1 + R i ) 1 i=1 Special case when R 1 = R 2 = = R I = R: V = X (1 + R) I

4 ECG59I Asset Pricing. Lecture 2: Present Value More complicated cases interim payments V = C + C R 1 + C 2 (1 + R 1 )(1 + R 2 ) + = C I + (1 + R 1 )(1 + R 2 ) (1 + R I ) X + (1 + R 1 )(1 + R 2 ) (1 + R I ) [ ] I i I (1 + R j ) 1 + X (1 + R j ) 1 C i i= j= i= provided that we define R.

5 ECG59I Asset Pricing. Lecture 2: Present Value Continuous time Compounding: 1 + R (1 + R)(1 + R) = (1 + R )2 1 + R (1 + R n )n lim n (1 + R n )n e R (1 + R) t [(1 + R n )n ] t = (1 + R n )nt lim n (1 + R n )nt = e Rt Present value of X to be paid at time T : V = Xe RT Continuous-time vs. discrete-time interest rates: Once per annum: 1 + R = e R n times per annum: (1 + R n )n = e R

6 ECG59I Asset Pricing. Lecture 2: Present Value 6 Evaluating a continuous, constant, cash flow, R constant: V = = X = X = X Xe Rt dt e Rt dt [ 1 ] T R e Rt [ 1 R e RT + 1 ] R = X R (1 e RT ) Note that V X R as T : perpetuity.

7 ECG59I Asset Pricing. Lecture 2: Present Value 7 Time-varying X: V = X t e Rt dt. Can t say more until the function X t is specified. For example, if X t = Ae Bt, then V = = = T o Ae Bt e Rt dt Ae (R B)t dt A R B (1 e (R B)T ) A if R > B R B if R B

8 ECG59I Asset Pricing. Lecture 2: Present Value 8 Time-varying R, constant X: V = (explanations follows). General case: V = (explanation follows). Xe R t R sds dt X t e R t R sds dt

9 ECG59I Asset Pricing. Lecture 2: Present Value Deriving (and understanding) continuous discounting Integrating factor We know how to compute derivatives. For example, d exp(x 7 ) dx = 7x 6 exp(x 7 ) We then know that, up to an integrating constant, 7x 6 exp(x 7 )dx = exp(x 7 ) The integral exp(x 7 )dx might be hard to compute, but it would be much easier if we could somehow multiply exp(x 7 ) by 7x 6. In this example, 7x 6 is the integrating factor. More on this in a few slides.

10 ECG59I Asset Pricing. Lecture 2: Present Value Continuous discounting Earnings on an initial stock of wealth dv t dt = RV t i.e., continuous compounding at rate R. To help you see this: R = dv t/dt V t V t+ǫ V t V t

11 ECG59I Asset Pricing. Lecture 2: Present Value 11 What is V at time T? To find out we solve the differential equation: dv t dt RV t =. The integrating factor is e Rt (how do we know that that s the integrating factor? We just know, or we wait a few more slides): dv t dt e Rt V t Re Rt = d dt (V te Rt ) =. So we get = [ ] d dt (V te Rt ) dt V T = [V t e Rt ] T = V T e RT V = V e RT

12 ECG59I Asset Pricing. Lecture 2: Present Value 12 Now suppose there also is a time-varying income or dv t dt = RV t + X t dv t dt RV t = X t which is just a forced (or non-homogeneous) linear ODE with the same integrating factor: dv t dt e Rt V t Re Rt d dt = X t e Rt ( Vt e Rt) = X t e Rt.

13 ECG59I Asset Pricing. Lecture 2: Present Value 13 So that we have [ d dt V T e RT V = ( Vt e Rt)] dt = V T = V e RT + e RT X t e Rt dt X t e Rt dt X t e Rt dt.

14 ECG59I Asset Pricing. Lecture 2: Present Value 14 Note that if V =, we have V T = e RT = X t e Rt dt X t e R(t T) dt which is the future (time T) value of the cash flow X t. Equivalently, V T e RT = is the present value of the cash flow X t. X t e Rt dt

15 ECG59I Asset Pricing. Lecture 2: Present Value 15 Finally, suppose R is time varying: dv t dt = R tv t + X t dv t dt R tv t = X t. The integrating factor is e R t R sds so that we can write or dv R t t dt e Rsds V t R t e R t Rsds = X t e R t R sds d dt (V t e R ) t R sds = X t e R t Rsds.

16 ECG59I Asset Pricing. Lecture 2: Present Value 16 So that [ d dt (V t e R t R sds) ] dt = [V t e R ] t T R sds = V T e R T R sds V = X t e R t R sds dt X t e R t R sds dt X t e R t R sds dt. The RHS is the present value of {X t } when R varies over time.

17 ECG59I Asset Pricing. Lecture 2: Present Value Ordinary Differential Equation (ODE) and finding the integrating factor How do we know what the integrating factor is? Once you re familiar with the topic, you tend to just know what it is. But it does not mean that there are no formal ways to find it. The remaining slides give a general solution for the integrating factor for a specific class of ODE. Don t get lost in the details. This course is not about solving ODEs

18 ECG59I Asset Pricing. Lecture 2: Present Value 18 Consider a first-order first-degree ODE, which has the form which can be written as dy dx = F(x, y) M(x, y)dx + N(x, y)dy = dy dx = M(x, y) N(x, y) If this equation has a unique solution, it can be written as U(x, y) = c. This is a general solution (allow implicit functions). If we take the differential on both sides du = U U dx + x y dy =

19 ECG59I Asset Pricing. Lecture 2: Present Value 19 Which implies dy dx = U/ x U/ y So that U/ x U/ y = M N Or U/ x M = U/ y N Denote these ratios by µ, U x = µm and U y = µn

20 ECG59I Asset Pricing. Lecture 2: Present Value 2 Next, substitute back into the equation for du: U/ x M du = µmdx + µndy = µ (Mdx + Ndy) = U/ y Mdx + N Ndy = du = U U dx + x y dy = Conclusion 1: multiplying the differential equation M(x, y)dx + N(x, y)dy = by µ give us an exact differential equation. We call µ an integrating factor.

21 ECG59I Asset Pricing. Lecture 2: Present Value 21 Before being able to say what µ is, we need an additional result: If the differential equation M dx + N dy = is exact, then by definition, there is a function U(x, y) such that But we also have that Mdx + Ndy = du du = U U dx + x y dy Therefore, U/ x = M and U/ y = N.

22 ECG59I Asset Pricing. Lecture 2: Present Value 22 For a sufficiently smooth function U, M y = 2 U y x = 2 U x y = N x Thus, M/ y = N/ x if the differential equation is exact.

23 ECG59I Asset Pricing. Lecture 2: Present Value 23 We are ready to find µ, i.e. the integrating factor If the differential equation becomes exact after being multiplied by µ, i.e. µmdx + µndy, then we have (µm) y = (µn) x Next, suppose that µ is a function of x only (with a symmetric result if it is a function of y alone). In this case µ M y dµ µ = µ N x + N dµ dx = 1 ( M N y N ) dx x = f(x)dx by hypothesis µ = e R f(x)dx

24 ECG59I Asset Pricing. Lecture 2: Present Value 24 Conclusion 2: If M(x, y)dx + N(x, y)dy = and if ( 1 M N y N ) = f(x), x then e R f(x)dx is the integrating function.

25 ECG59I Asset Pricing. Lecture 2: Present Value 25 For the first example, we can find the integrating factor using these results. Remember we had Which can be written dv t dt = RV t RV t dt + dv t = We see that, remembering that we want to get V t (take y = V t ) x = t, y = V t, M(x, y) = RV t, N(x, y) = 1 So we get 1 N ( M y N ) x = 1 ( R ) = R 1 Hence, µ = e R Rdt = e Rt.

26 ECG59I Asset Pricing. Lecture 2: Present Value 26 If you want more, have a look at the fine notes on ordinary differential and difference equations written by Professor John Seater. You ll find them on Professor Seater s website: jjseater Or on the website of this course.