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Investment Management: Exercises and solutions Part 1

Exercise 1 – Total nominal return index and Total real return index The table below presents the evolution of a stock market and the inflation rate on a 3-year period. Compute, according to this table, the following indexes: 1) The total nominal return index (TNRI) (without adjusting the inflation) at the end of each year with an initial level equal to one (i.e., TNRI(0) = 1); 2) The total real return index (TRRI) (after having adjusting the inflation) with the same hypothesis. Year 1 2 3

Capital gain or loss +10,0% -5,8% +6,0%

Dividend yield at the end of year 2,0% 3,3% 2,5%

Inflation rate 3,2% 3,1% 2,7%

Solution Assume that  D(i) represents the dividend yield at the end of year i, with i  {1, 2, 3}, D(1) = 2,0%, D(2) = 3,3%, D(3) = 2,5%.  R(i) represents the capital gain or loss of the market during year i, with i  {1, 2, 3}, R(1) = + 10,0%, R(2) = 5,8%, R(3) = + 6,0%.  Inf(i) represents the inflation rate during year i , with i  {1, 2, 3}, Inf(1) = 3,2%, Inf(2) = 3,1%, Inf(3) = 2,7%.  TNRI(i) represents the Total nominal return index computed at the end of year i, with i  {1, 2, 3} et TNRI(0) = 1.  TRRI(i) represents the Total real return index, computed at the end of year i, with i  {1, 2, 3} et TRRI(0) = 1. 1) For the computation of the Total nominal return index, we have: TNRI (i)  TNRI (i 1) [1  R( i)] [1  D( i)]

We have: TNRI (1)  TNRI (0) [1  R(1)] [1  D(1)] 1 (1 10,0%) (1 2,0%) 1,122 TNRI (2)  TNRI (1) [1  R(2)] [1  D(2)] 1,122 (1 5,8%) (1 3,3%) 1,0918 TNRI (3)  TNRI (2) [1  R(3)] [1  D(3)] 1,0918 (1  6,0%) (1  2,5%) 1,1862

2) For the computation of the Total real return index, we have: TRRI (i ) 

TNRI (i ) [1  Inf (1)] [1  Inf (2)]  ...  [1  Inf (i)]

We have: TNRI (1) 1,122   1,0872 [1  Inf (1)] 1  3,2% TNRI (2) 1,0918   1,0261 TRRI (2)  [1 Inf (1)] [1  Inf ( 2)] (1 3,2%) (1 3,1%)

TRRI (1) 

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TRRI (3) 

TNRI (3) 1,1862   1,0856 [1  Inf (1)] [1  Inf ( 2)] [1  Inf (3)] (1 3,2%) (1 3,1%)  (1  2,7%)

In addition, we can check our computation with the following formula: TRRI (i )  TRRI (i  1) 

[1 R (i )] [1  D (i )] [1  Inf (i)]

Exercise 2 – Inflation-indexed bonds Assume that an in fine in flation-indexed bond has € 1000 as nominal value, 5% as nominal coupon rate, 8 years as maturity, and 100% as nominal repayment rate at the maturity. The inflation rates during the life-period of the bond are presented in the following table: Year Inflation rate

1 2%

2 3%

3 4%

4 3,5%

5 3%

6 2,5%

7 2,8%

8 2,7%

1) Compute the 8 annual coupons indexed to the inflation. 2) Compute the final repayment amount indexed to the inflation. Solution Assume that  Inf(i) represents the inflation rate during year i, with i  {1, 2, 3, 4, 5, 6, 7, 8}.  Couponindex(i) represents the indexed coupon paid at the end of year i, with i  {1, 2, 3, 4, 5, 6, 7, 8}.  Principalindex represents the indexed repayment paid at the end of year 8. An inflation-indexed bond aims to protect the bondholder against inflation. More precisely, the coupon paid at the end of each year should permit the bondholder to have a purchasing power (measured at the issuing date of the bond) equivalent to that of the nominal coupon. In the current exercise, the nominal coupon is € 50, namely 1000 * 5%. 

For Couponind(i), its purchasing power at the issuing date of the bond is: Coupon index(i )  50 Purchasin g Power of Couponindex ( i)  [1  Inf (1)] [1  Inf (2)]  ...  [1  Inf (i )] As a result, we have:

Couponindex( i) 50 [1  Inf (1)] [1  Inf (2)] ... [1 Inf (i )] For i = 2, 3, 4, 5, 6, 7, and 8, the last equation leads to:

Couponindex(i)  Couponindex ( i 1) [1  Inf (i)] 

We have: Couponindex(1)  50 1  Inf (1)  50  (1  2%)  51

Couponindex( 2)  Couponindex(1) [1  Inf (2)]  51 (1  3%)  52,5 Couponindex(3)  Couponindex( 2) [1  Inf (3)]  52,5  (1  4%)  54,6 Couponindex( 4)  Couponindex(3) [1  Inf (4)]  54,6  (1  3,5%)  56,5 Couponindex(5)  Couponindex(4) [1  Inf (5)]  56,5  (1 3%)  58,2

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Couponindex(6)  Couponindex(5) [1  Inf (6)]  58,2  (1  2,5%)  59,7 Couponindex(7)  Couponindex(6) [1  Inf ( 7)]  59,7  (1  2,8%)  61,4 Couponindex(8)  Couponindex(7) [1  Inf (8)]  61,4  (1 2,7%)  63,0 Similarly, the indexed repayment paid at the maturity date of the bond should permit the bondholder to have a purchasing power equivalent to the nominal amount of the repayment of € 1000. 



For Principalindex, its purchasing power at the issuing date of the bond is: Pr incipalindex Pr incipalindex   1000 [1  Inf (1)]  [1  Inf ( 2)]  ...  [1  Inf (8)] As a result, we have:

Pr incipalindex 1000 [1  Inf (1)] [1  Inf ( 2)] ... [1 Inf (8)] We have: Pr incipalindex  1000 (1 2%)(1 3%)(1 4%)(1 3,5%)(1 3%)(1 2,5%)(1  2,8%)(1  2,7%)  1260,5

Exercise 3 – Classical bond and inflation-indexed bond Assume that a traditional in fine bond is assigned BONDTrad, with 5 years as maturity, €100 as nominal value of the capital, and 6% as the nominal rate of annual coupon, assigned C(i) for i  {1, 2, 3, 4, 5}, which is paid at the end of each year, and 100% as the nominal repayment rate of the capital at the maturity date, and the amount of repayment of the capital is assigned Remb. Assume that an in fine inflation-index bond, assigned BONDInd, has the same characteristics as BONDTrad, except that the annual coupon, assigned CInd(i) for i  {1, 2, 3, 4, 5}, and the amount of repayment of the capital, assigned RembInd, are indexed to the inflation. Assume that the inflation rate is 4% per year during the life-period of these bonds. 1) Compute C(1), C(2), C(3), C(4), C(5), and Remb. What is the purchasing power of each of these flows at the issuing date of the bond? If the bondholder’s expected return is 6%, what is the “real” present value (i.e. after inflation adjustment) of the traditional bond? 2) Compute CInd(1), CInd (2), CInd(3), CInd(4), CInd (5), and RembInd. What is the purchasing power of each of these flows at the issuing date of bond? If the bondholder’s expected return is 6%, what is the “real” present value (i.e. after inflation adjustment) of the inflation-indexed bond ? Solution 1) C(1) = C(2) = C(3) = C(4) = C(5) = 100 * 6% = 6 (euros) Remb = 100 (euros) C(2) C (1)  5,7692 (Purchasing power of C(1)) PP(1) = PP(2) =  5,5473 (1 4%)2 1  4% C (3) C(4)  5,1288 PP(3) = PP(4) =  5,3340 3 (1 4%) (1 4%)4 Remb C(5)  4,9316 PP(5) =  82,1927 PP(Remb) = 5 ( 1  4%)5  (1 4%) Real present value of the bond OBLIG = Sum of real present value of all coupons and the repayment Investment Management – Solutions for exercises

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PP(1) PP(2) PP(3) PP(4) PP (5) PP (Remb )      2 3 4 5 (1  6%)5 1  6% (1  6%) (1  6%) (1  6%) (1  6%) = 5,4427  4,9371  4,4785  4,0625  3,6852  61,4192 = 84,0251

=

2) CInd(1) = 6  (1  4%)  6,2400 CInd(3) = 6  (1  4%)3  6, 7492 CInd(5) = 6  (1  4%)5  7, 2999

CInd(2) = 6  (1  4%)2  6, 4896 CInd(4) = 6  (1  4%)4  7, 0192 RembInd = 100 (1  4%)5  121,6653

PAInd(1) = PAInd(2) = PAInd(3) = PAInd(4) = PAInd(5) = 6 PA(RembInd) = 100 Real present value of the bond OBLIGInd = Sum of real present value of all coupons and the repayment PPInd(1) PPInd (2) PPInd (3) PPInd (4) PPInd (5) PP(Re mbInd )      = 2 3 4 (1  6%)5 1  6% (1  6%) (1  6%) (1  6%) (1  6%)5 = 5,6604 + 5,3400 + 5,0377 + 4,7526 + 4,4835 + 74,7258 = 100 Exercise 4 – Patrimony approach for valuating a stock The patrimony approach is a “past method”, according to which the value of a firm should be equal to the value of all its assets, valuated at their historical prices and adjusted with the capital gain or loss since they are acquired. This approach is suitable for real estate firms and investments companies. Let’s take the example of a company entitled “HOLDING”, which holds 100% of the equity of the firm “SUBSIDIARY” (in communication), 50% of the firm “CONTROL” » (in technology), and 20% of the firm “PARTICIPATION” (in computer). The acquisition prices of these three firms by HOLDING and the part of their current capitalizations held by HOLDING are presented in the following table. Assume that HOLDING does not have other participations, and has no debt, and the goodwill resulting from the group synergy is M€ 5 (million euros). Firm

Acquisition price

SUBSIDIARY CONTROL PARTICIPATION

M€ 75 M€ 60 M€ 40

Market value of the part of equity held by HOLDING M€ 100 M€ 90 M€ 30

Re-valuation M€ 25 M€ 30 M€ 10

1) Valuate the equity of HOLDING with historical book value, defined by the difference between sum of the historical acquiring prices of all tangible assets and the sum of the historical acquiring prices of all debts. Historical book value = Historical acquiring prices of assets – Historical prices of debts 2) Valuate the equity of HOLDING with net book value, defined by the historical book value, adjusted by the re-valuation of the assets. Present book value = Historical book value + Re-valuation of assets 3) Valuate the equity of HOLDING with the sum of the net book value and the goodwill resulting from intangible assets such as trademark, clients, expertise, and group synergy.

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Value of equity = Present book value + Goodwill Solution 1) Liability = 0 Acquiring price of asset = 75 + 60 + 40 = 175 (ME) Historical book value = Acquiring price of asset – Liability = 175 – 0 = 175 (ME) 2)

Re-valuation = 25 + 30 – 10 = 45 (ME) Present book value = Historical book value + Re-valuation = 175 + 45 = 220 (ME)

3)

Goodwill = 5 (ME) Present book value with goodwill taken into account = Present book value + Goodwill = 220 + 5 = 225 (ME)

Exercise 5 – Comparative approach for valuating a stock The comparative approach is a “present approach”, according to which the value of a firm should be comparable to that of another firm having similar characteristics. Especially useful for firms at the moment of their IPO, this approach is however influenced by the circumstance of the stock market at the moment of the valuation. Let’s take the example of INTRO, a computer game producer, which asks to its bank to propose a price for its IPO. Assume that the firm LISTED, having gthe same industrial activity and the same size as INTRO, is already listed. Earnings of these two firms for the last fiscal exercise and the capitalization of LISTED are shown in the following table. Firm LISTED INTRO

Capitalization M€ 250 M€?

Earnings M€ 10 M€ 16

1) What is the PER of the firm LISTED? (PER = Capitalization / Earnings) 2) According to the comparative approach, the firm INTRO should have the same PER as its peer LISTED. What should be the capitalization of INTRO under such hypotheses? 3) If INTRO wishes to issue 10 million stocks, what should be the price per share of INTRO proposed by its bank? Solution 1) 2)

PERLISTED 

Capi LISTED 250   25 10 BNC LISTED

CapiINTRO CapiINTRO   PER LISTED  25 BNCINTRO 16 This formula leads to: Capi INTRO  25 16  400 (ME). PERINTRO 

3) The 10 million stocks as a whole are worthy € 400 million, which means that each stock is worthy € 40. The price proposed by the bank is €40 per share. Exercise 6 – Dynamic approach for valuating a stock

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The dynamic approach is a “future approach”, according to which the value of a firm should be equal to what it can create in the future. The Discounted Dividend Model or DDM is one method within the framework of this approach. According to DDM, the price of a stock can be computed as the sum of all dividend flows discounted at the rate equal to shareholders’ expected return. It is relatively easier to apply this method to firms starting already to pay dividends (ex. Danone) than firms having no historical data about the dividend payment (ex. Eurotunnel). In addition, this method is sensitive to hypothesis made on the dividend flows which will be paid in the future and on the discount rate. Let’s take the example of the firm DYNAMIC whose last dividend paid per share is € 10 for the last fiscal exercise. Financial analysts consider that the dividend growth rate for this firm will be constant until the infinite at 6% per year and shareholders’ expected return is 8%. 1) According to these hypotheses, what should be the next dividend to be paid in one year par the stock DYNAMIC, and what is its present value? What should be the dividend to be paid in 2 years and what is its present value? Write down the formula to compute the present value of the dividend to be paid in n years. 2) According to the DDM, what should be the present value of the stock? 3) If the estimation of the dividend growth rate decreases from 6% to 5.75%, what should be the present value of the stock? What is the percentage of this result compared to the one obtained in question 2)? 4) If the estimation of shareholders’ expected return rate increases from 8% to 8.25%, what should be the present value of the stock? What is the percentage of this result compared to the one obtained in question 2)? 5) If the estimation of the dividend growth rate decreases from 6% to 5.75%, and the estimation of shareholders’ expected return rate increases from 8% to 8.25%, what should be the present value of the stock? What is the percentage of this result compared to the one obtained in question 2)? Solution 1) The next dividend to be paid in one year is 10  (1  6%) , and its present value for (1  6%) shareholders whose expected return is 8% is : 10  . The dividend to be paid in (1  8%) two years is 10  (1  6%)2 , and its present value for shareholders whose expected return (1  6%) 2 is 8% is: 10  . The dividend to be paid in n years is : 10  (1  6%)n , and its (1  8%) 2 present value for shareholders with 8% as expected return is :10 

(1  6%) n . (1  8%) n

2) According to the DDM, the price of the stock should be equal to: n

2

10 

(1  6%)  1  6%  1 6%   10     ...  10     ... (1  8%)  1  8%  1  8 % 

 10   10 

n  (1  6%)   1  6%   1  6%   1     ...     ... (1  8%)   1  8%   1  8%  

(1  6%) 1 1  6%  530  10   1  6%  6% (1  8%) 8 % 1 1  8%

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3) If the dividend growth rate is 5.75%, then the price of the stock is: (1 5,75%) 1 1 5,75% 10  10   470  1 5 , 75 %  (1  8%) 8%  5,75% 1 1 8% This result represents 89% of that got in question 2). 4) If shareholders’ expected return is 8.25%, the price of the stock is: (1 6%) 1 1  6% 10  10   471   (1 8,25%) 1 1 6% 8,25%  6%  1  8,25% This result represents 89% of that got in question 2). 5) If the dividend growth rate is 5.75%, and shareholders’ expected return is 8.25%, the the price of the stock is: (1 5,75%) 1 1 5,75% 10  10   423  (1 8,25%) 1  1 5,75% 8,25%  5,75% 1 8,25% This result represents only 80% of the one obtained in question 2) with the initial hypotheses. This means that the DDM is very sensitive to changes in the hypotheses made on the dividend growth rate and shareholders’ expected return on the stock. Exercise 7 – Variance, covariance, and correlation coefficient The following table presents the evolution of the annual returns of three stocks A, B, and C during the last 5 years. Year 1 2 3 4 5

Stock A +24% +30% 0% -18% -5%

Stock B +75% +10% +20% -9% -12%

Stock C -2% -8% 0% +4% +7%

1) Compute the average of the return of A, B, and C, respectively. 2) Compute the (non-corrected) variance and the standard-deviation of the annual return of A, B, and C, respectively. Which one is the riskiest, and which one is the least risky? 3) Compute the covariance of the returns between A and B, the one between A and C, and the one between B and C. 4) Compute the correlation coefficient between the returns of A and B, the one between A and C, and the one between B and C. 5) Which two stocks are the most positively correlated? Which two stocks are the most negatively correlated? Solution Assume that  m(RA) represents the average of the annual return of stock A, m(RB) represents that of stock B, and m(RC) represents that of stock C.

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



Var(RA) and ( RA) represent respectively the variance and the standard-deviation of the annual return of the stock A, Var(RB) and ( RB) those of the stock B, Var(RC) and ( RC) those of the stock C. Cov(RA, RB) and Corr(RA, RB) represent respectively the covariance and the correlation coefficient between the annual return of A and that of B, Cov(RA, RC) and Corr(RA, RC) those of stocks A and C, Cov(RB, RC) and Corr(RB, RC) those of stocks B and C.

1) Computation of the average 1 n m (R A )   R A (i )  (24%  30%  0% 18%  5%) / 5  6,2% n i 1 m (R B ) 

1 n  RB (i )  (75%  10%  20%  9%  12%) / 5  16,8% n i 1

m ( RC ) 

1 n RC (i )  (2%  8%  0%  4%  7%) / 5  0,2% n i 1

2) Computation of the non-corrected variance and the standard-deviation 1 n 2 Var ( R A )   R A (i )  m (R A )  3,3%  (R A )  Var (RA )  18,1% n i 1 1 n 2 Var ( RB )   R B (i )  m (R B )  9,9%  (RB )  Var (RB )  31,4% (riskiest) n i 1 1 n 2 Var( R C )   RC (i )  m( RC )  0,3%  ( RC )  Var( RC )  5,2% (least risky) n i 1 3) Computation of the non-corrected covariance 1 n Cov( R A , RB )   R A (i )  m (R A ) R B (i )  m (R B )  3,6% n i 1 Cov( RA , RC ) 

1 n  R A (i )  m (R A ) RC (i )  m (RC )   0,8% n i 1

Cov( RB , RC ) 

1 n  R B (i )  m (R B ) RC (i )  m (RC )   0,7% n i 1

4) Computation of the correlation coefficient Cov (R A , R B )  0,63 Corr (R A , R B )   ( RA )   ( RB ) Cov (R B , R C )  0, 45 Corr (R B , R C )   ( RB )  ( RC ) 5)

Corr (R A,R C ) 

Cov (R A, R C )  0,86  ( RA)   ( RC )

Stocks A and B are the most positively correlated, while A and C are the most negatively correlated.

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