Opportunity Cost of Capital. The Risk of a Single Risky Asset. Free assets

Transcription

1 Risk & Retun Oppotunity Cost of Capital Histoical Evidence on Risk and Retun The Risk of a Single Risky Asset The Risk and Retun of a Potfolio of a Risky & Risk- Fee assets

2 Oppotunity Cost of Capital How to detemine the Cost of Capital? The oppotunity cost of capital fo a poject is the expected etun on an investment with simila isk We will define simila isk How do investos decide how much isk they want in thei potfolio? What potfolio povides the optimal tadeoff between isk and etun? Let s look how investos attitudes towad isk ae manifested in the histoy of secuity etuns

3 Histoical Evidence Risk and Retun This table shows the Real Retuns (Inflation adjusted) d) fom 195 though 000: This shows that ove long peiods of time thee has been a tadeoff between isk and expected etun. Stocks have histoically had much highe etuns but they ae y g y also substantially iskie!

4 The U.S. Stock Maket, Stocks vesus Bonds. The table shows the gains fom investing in Stocks elative to Bonds. If you invest in stocks you ealize high gains as well as high losses

5 Risk Pemium ove the wold etun on maket isk fee ate + isk pemium Denmak Belgium Switzeland Spain Canada Ieland Gemany UK Aveage Nethelands USA Sweden South Afica Austalia Fance Japan Italy

6 The U.S. Stock Maket The Dow Jones Industial Aveage fell fom a high of g f f g f in 199 to a low of 41. in 193, a fall of 89.%

7 The U.S. Stock Maket The Dow Jones Industial Aveage did not each its 199 high of 381 again until late 1954, ove 5 yeas late.

8 Annual ates of etun fo % 60% ge etun Pecenta 40% 0% 0% -0% % -60% Yea

9 Histogam of annual ates of etuns to to to -0-0 to to 0 0 to to 0 0 to to to to 60 Numbe of yeas Annual etun

10 The Japanese Stock Maket, Th N kk 5 k d D b O The Nikkei 5 peaked at 38,957 on Decembe 9, On Febuay 11, 003, oughly 13 yeas late, it closed at 8,485.

11 Quantifying Risk and Retun Teminology 1. Realized Retun: The etun investos in a secuity actually ean ove the peiod, measued at the end of the peiod.. Expected Retun: The etun investos s in a secuity expect to ean ove a peiod, measued at the beginning of the peiod. We calculate the expected etun E() by summing acoss the possible ealized etuns (possible events s ) times the pobability of these events P s S E( ) P s s s 1

12 Quantifying Risk and Retun Teminology 3. Retun Vaiance: The expected squaed deviation fom the mean ove a peiod, measued at the beginning of the peiod. We calculate the etun vaiance V() by summing acoss the possible ealized squae deviations times the pobabilities. S V( ) σ Ps s E s 1 )] [ E() ] E ( ) [ ( )] 4. Rt Retun Standad ddeviation: iti The positive squae oot of the vaiance: σ σ

13 Quantifying Risk and Retun - Example Conside an investo who has $50,000 to invest. She has the choice to invest eithe in a isk-fee asset that pays 3% o in Stock A. Stock A will eithe go down by 50% o go up by 100% with equal pobabilities. Calculate the expected etun, expected excess etun and etun standad deviation of the potfolio which h is fully invested in Stock A. Risk fee asset $50,000 $51,500 Risky asset $100,000 $50,000 $5,000

14 Risky asset The two possible ealized etuns ae: 100,000 50, %, 50, ,000 50,000 50, % The expected etun is: 1 1 E( ) 100% + ( 50%) 5% Risk-Fee asset 51, , The only possible etun is: 3% 50,000 Stock A s Excess Retun is defined as its expected etun minus the isk-fee etun: STOCK A RISK FREE 5% 3% %

15 Risky asset The vaiance of the etun is: 1 1 σ (1 0.5) + ( ) 0.56 The standad deviation is: σ σ % The standad deviation tells us oughly how fa above and below the mean we expect to be each peiod What is the vaiance and standad deviation of the etun on the Risk-Fee asset?

16 Potfolio Choice: Risky and Risk-Fee Assets The plot below shows the location of the two assets Which investment should ou investo pick? Ae thee othe possible investments t which hou investo might pefe?

17 Risk and Retun of Potfolios Example (continued): Suppose that ou investo invests $5K in the isk fee asset and $5K in asset A? What etuns could the investo ean? What is the potfolio s expected etun, vaiance of etun and standad deviation of etun? Notation A etun on stock A E( ) expected isky ate of σ f A A etun (stock A) etun standad deviation (stock A) isk fee ate (of etun) w A faction of the potfolio invested in isky asset (stock A)

18 The etun of a potfolio p w A f (if A goes up) 0.5 ( 0.5) (if A goes down) A The potfolio s expected etun + (1- w) E( p) w E( A ) + (1- w) f

19 To calculate the potfolio s etun vaiance ecall the following fomulas fom statistics va(x)[ σx ] va(a x) a E(x va(x) ) - ( E(x) ) va(x + y) va(x) + va(y) + cov(x, y) cov(x, y)[ σ x,y ] E(xy) E(x)E(y) cov(a x,b y) a b cov(x, (, y) cov( x, x) σ x cov(x, y) x, va(x) va( y) ρ y σ x σ y ρ x, y

20 The potfolio s etun vaiance σ p va( p) va(w A + (1- w) f ) va(w va((1- w) cov(w A ) + f ) + A,(1- w) f ) w 0.5 va( A 0.75 ) σ p σp Notice that the standad deviation is popotional to the faction of p p f f he potfolio she invests in the isky asset

21 The Capital Allocation Line The Capital Allocation Line (CAL) epesents all the possible combinations of isk and etun that can be geneated fom holding a potfolio of the isky asset and the isk fee asset. If we invest w in the isky asset then we have, E( ) w E( ) + (1- w) p p A We can now substitute fo w and get the CAL A σ wσ w σ σ p A σp σp E( A ) - f E( + + p) E( A ) 1- f f σp σ A σa σa Rewad Expected etun Risk fee ate + Risk Risk f

22 The Capital Allocation Line Fo ou example: E( p ) σp σ 0.75 p

23 Retun fo Potfolios with Multiple Assets Example: You have a $1M potfolio with $00K invested in Micosoft and $800K in GM. If you expect (annual) etuns of 10% fo Micosoft and 15% fo GM ove the next yea, then what is the expected etun on the potfolio? 00,000 w Micosoft w 0. w GM (1 w) 1,000, E( ) w E + ( ) + (1 w) E ( ) E( p Micosoft GM 0.(10%) + 0.8(15%) 14%

24 Retun fo Potfolios with Multiple Assets A yea late, it tuned out that the ealized (annual) etun on Micosoft was actually 1% and on GM it was -5%. What then is you ealized etun on the potfolio? p w Micosoft + (1 w) GM 0.(1%) + 0.8(-5%) -1.6% How much did you ean on you investment in Micosoft? $00K Micosoft $00K 1% $4K How much did you ean on you potfolio? $1M p $1M ( 0.016) $16K

25 Risk of Potfolios with Multiple Assets Example (continued): Suppose that the (annual) etun standad deviation of these stocks ove the following yea will be is 40%, and the coelation between the etun on Micosoft and GM is What is the standad deviation of the etun on you potfolio? V( ) V w p V w w 0. V V ( Micosoft + (1- w) GM ) ( w ) ( ) ( Micosoft + V (1- w) GM + cov w Micosoft,(1- w) GM ) ( ) (1- w) V( ) w(1 w)cov( Micosoft + GM + Micosoft,GM ) ( Micosoft ) + (1- w) V( GM ) + w(1 w)[ρmicosoft,gmσmicosoftσ ( 0.4) ( 0.4) [ ] σ p GM ]