# Cost of Capital Math solution

By (M. Shajahan Mina)

## Exercise # 01:

Calculate the explicit costs of debt for each of the following situations, assuming (a) tax rate of 40 percent, (b) coupon rate of interest on debentures is 14 percent, (c) par value of the debenture is Tk 3000 each.

(A)   If debentures are sold at par and floatation costs are 2 percent;

(B)    If debentures are sold at premium of 10 percent and floatation costs are 3 percent;

(C)    If debentures are sold at discount of 5 percent and floatation costs are 3 percent.

Solution    Given

Tax rate (T) = 0.40

Face value of debenture (FV) = 3000

Net sale value of debenture (NSV) = 3000 – 3000 ´ 0.02 = 2940

Interest on debenture (I) = 3000 ´ 0.14 = 420

Number of year or Maturity period (N) = 1 year

Requirement: Cost of debt (Kd) = ?

(A) Debenture sold at par:

Kd = =

= 10.51% [Ans.]

(B) Debenture sold at 10% premium:

NSV = (3000 ´ 1.10) – (3000 ´ 1.10) ´ 0.03 = 3201

Kd = =

= 1.64% [Ans.]

(C) Debenture sold at 5% discount:

NSV = (3000 ´ 0.95) – (3000 ´ 0.95) ´ 0.03 = 2764.5

Kd = =

= 16.91% [Ans.]

## Exercise # 02

The following are the information of a company:

 Type of capital Book value (Tk) Market value (Tk) Specific cost (%) DebtPreference shareEquity share Retained earnings 300000 200000 600000 200000 380000 220000 600000 500000 7% 8% 14% 11.2% 1300000 1700000

Determine the weighted average cost of capital using (A) Book value weights and, (B) Market value weights. How are they different? Can you think of a situation where the weighted average cost of capital would be the same using either of the weights?

Solution (A)    Weighted Averaged Cost of Capital (WACC)

 Type of capital Book value (Tk) Capital ratio Specific cost (Tk) Weighted Average (1) (2) (3) = S(2)  (2) (4) (5) =(3) ´ (4) DebtPreference shareEquity share Retained earnings 300000 200000 600000 200000 0.23 0.15 0.46 0.16 0.07 0.08 0.14 0.112 0.0161 0.0120 0.0644 0.0179 1300000 SWA = 0.1104

So the book value weight is SWACC = 0.1104 = 11.04%   [Ans.]

(B)          Weighted Averaged Cost of Capital (WACC)

 Type of capital Market value (Tk) Capital ratio Specific cost (Tk) Weighted Average (1) (2) (3) = S(2)  (2) (4) (5) =(3) ´ (4) DebtPreference shareEquity share Retained earnings 380000 220000 600000 500000 0.22 0.13 0.35 0.30 0.07 0.08 0.14 0.112 0.0154 0.0104 0.0490 0.0336 1700000 SWA = 0.1084

So the market value weight is SWACC = 0.1084 = 10.84%   [Ans.]

Comment: Higher market value creates higher cost of capital. When the book value and the market value have the same totals, then the weighted average cost of capital would be the same using either of the weights.

## Exercise # 03

A company has on its books the following amounts and specific costs of each type of capital:

 Type of capital Book value (Tk) Market value (Tk) Specific cost (%) DebtPreference shareEquity share Retained earnings 300000 200000 600000 200000 380000 220000 600000 500000 7% 8% 14% 11.2% 1300000 1700000

Determine the weighted average cost of capital using (A) Book value weights and, (B) Market value weights. How are they different? Can you think of a situation where the weighted average cost of capital would be the same using either of the weights?

Solution (A)    Weighted Averaged Cost of Capital (WACC)

 Type of capital Book value (Tk) Capital ratio Specific cost (Tk) Weighted Average (1) (2) (3) = S(2)  (2) (4) (5) =(3) ´ (4) DebtPreference shareEquity share Retained earnings 300000 200000 600000 200000 0.23 0.15 0.46 0.16 0.07 0.08 0.14 0.112 0.0161 0.0120 0.0644 0.0179 1300000 SWA = 0.1104

So the book value weight is SWACC = 0.1104 = 11.04%   [Ans.]

(B)          Weighted Averaged Cost of Capital (WACC)

 Type of capital Market value (Tk) Capital ratio Specific cost (Tk) Weighted Average (1) (2) (3) = S(2)  (2) (4) (5) =(3) ´ (4) DebtPreference shareEquity share Retained earnings 380000 220000 600000 500000 0.22 0.13 0.35 0.30 0.07 0.08 0.14 0.112 0.0154 0.0104 0.0490 0.0336 1700000 SWA = 0.1084

So the market value weight is SWACC = 0.1084 = 10.84%   [Ans.]

Comment: Higher market value creates higher cost of capital. When the book value and the market value have the same totals, then the weighted average cost of capital would be the same using either of the weights.

## Exercise # 04

XYZ Company has debentures outstanding with 5 years left before maturity. The debentures are currently selling for Tk 90 (the face value is Tk 100). The debentures are to be redeemed at 5 percent premium. The interest is paid annually at a rate of 12 percent. The firm tax rate is 40 percent. Calculate the cost of debenture.

Solution    Given

Tax rate (T) = 0.40

Face value of debenture (FV) = 100

Redeemed value (RV) = 100 + 100 ´ 0.05 = 105

Net sale value of debenture (NSV) = 90

Interest on debenture (I) = 100 ´ 0.12 = 12

Number of year or Maturity period (N) = 5 years

Requirement: Cost of debenture (Kd) = ?

Kd = =

= 10.46% [Ans.]

## Exercise # 05

From the following information, determine the cost of equity capital using the CAPM approach.

(A)   Rate of return on risk-free security 10 percent.

(B)    Rate of return on market portfolio of investments is 13 percent.

(C)    The firm’s beta is 1.6, 1.25 and 1.75.

Solution                  Given   Rate of risk-free return (Rf) = 0.10

Rate of of return on market portfolio (Rm) = 0.13

Beta risk (b) = 1.6

Requirement: Cost of equity capital (Ke) = ?

When b = 1.6,

Ke = Rf + (Rm – Rf)b

= 0.10 + (0.13 – 0.10)1.6

= 14.80%  [Ans.]

When b = 1.25,

Ke = Rf + (Rm – Rf)b

= 0.10 + (0.13 – 0.10)1.25

= 13.75%  [Ans.]

When b = 1.75,

Ke = Rf + (Rm – Rf)b

= 0.10 + (0.13 – 0.10)1.75

= 15.25%  [Ans.]

## Exercise # 06

Investors require a 12 percent rate of return on equity shares of company Y. What would be the market price of the shares if the previous dividend (Do) was Tk 2 and investors expect dividends to grow at a constant rate of (i) 4 percent (ii) 5 percent (iii) – 4 percent, (iv) 10 percent?

Solution         Given   Required rate of return on equity (K) = 0.12

Dividend per share (Do) = 2

Growth rate (g) = 0.04

Requirement: Current market price per share (P) = ?

(i) When growth rate (g) = 0.04, then

P = =  = Tk 26  [Ans.]

(ii) When growth rate (g) = 0.05, then

P = =  = Tk 30  [Ans.]

(iii) When growth rate (g) = – 0.04, then

P = =  = Tk 12  [Ans.]

(iv) When growth rate (g) = 0.10, then

P = =  = Tk 110  [Ans.]

## Exercise # 07

A mining company’s iron ore reserves are being depleted, and its cost of recovering a declining quantity of iron ore are rising each year. As a result, the company’s earnings and dividends are declining, at a rate of 8 percent per year. If the previous year’s dividend (Do) was Tk 12 and the required rate of return is 15 percent, what would be the current price of the equity share of the company?

Solution                  Given,  Required rate of return (K) = 0.15

Dividend per share (Do) = 12

Growth rate (g) = – 0.08

Requirement: Current market price per share (P) = ?

P =

=

= Tk 48  [Ans.]

## Exercise # 08

A large-sized chemical company has been expected to grow at 14 percent per year for the next 4 years and then to grow indefinitely at the same rate as the original economy, i.e., 5 percent. The required rate of return on the equity shares is 12 percent. Assume that the company paid a dividend of Tk 12 per share last year. Determine the market price of the shares today.

Solution                  Given   Required rate of return (K) = 0.12

Dividend per share (Do) = 12

Growth rate (g) = 0.05

Requirement: Current market price per share (P) = ?

P = =  = Tk 108  [Ans.]

## Exercise # 09

(A) A company’s debentures of the face value Tk 1000 bear 14 percent coupon rate. Debentures of this type currently yield 15 percent. What is the market price of debentures of the company?

(B) What would happen to the market price of the debentures if it rises to (i) 16 percent, and (ii) drops to 12 percent.

(C) What would happen to the market price of the debentures in situation (A) if it is assumed that debentures were originally having a 15-year maturity period and the maturity period is 4 years away from now.

(D)    Would you pay Tk 900 to purchase debentures specified in situation (C)? Explain

Solution    Given

Face value of debenture (FV) = 1000

Interest on debenture (I) = 1000 ´ 0.14 = 140

Cost of debt/yield (Kd) = 0.15

Requirement: Current market price per share (P) = ?

(A)     P = =  = Tk 933.33  [Ans.]

(B)(i) When Kd = 0.16, then

P = =  = Tk 875  [Ans.]

(B)(ii) When Kd = 0.12, then

P = =  = Tk 1166.67  [Ans.]

(C) For the rest maturity period 4 years,

P =  +  =  +

= Tk 971.45  [Ans.]

(D) Yes, I would like to pay Tk 900 for purchasing the debenture, because its current worth (Tk 971.45) is more than the purchase price.

## Exercise # 09

Assuming that the firm pays tax at 40 percent rate, compute the after tax cost of capital in the following cases:

0

(i)       A 8½ percent preference share sold at par.

(ii)     A perpetual bond sold at par, coupon rate of interest being 9 percent.

(iii)    A ten-year, 8 percent, Tk 1000 par bond sold at Tk 950 less 4 percent underwriting commission on the par value.

(iv)   A preference share sold at Tk 100 with a 10 percent dividend and a redemption price of Tk 110 if the company redeems it in five years.

(v)     A common share selling at a current market price of Tk 120, and paying a current dividend of Tk 9 per share which is expected to grow at a rate of 8 percent.

(vi)   A common share of a company which engages no external financing is selling for Tk 50. The earnings per share is Tk 7.50, of which sixty percent is paid as dividends. The company reinvests retained earnings at a rate of 10 percent.

Calculate the cost of capital assuming the following weightage in the capital structure:

(i)   0.15              (ii) 0.15                   (iii) 0.20

(iv) 0.10              (v) 0.30                   (vi) 0.10

Solution

(i) Kd = K(1 – T) = 0.085(1 – 0) = 0.085 = 8.5%   [Ans.]

(ii) Kd = K(1 – T) = 0.07(1 – 0.40) = 0.032 = 4.2%   [Ans.]

 (iii) Kd = FV = 1000I = 1000 ´ 0.08 = 80NSV = 950 – 1000 ´ 0.04 = 910 T = 0.40 N = 10

=

= 5.97% [Ans.]

 (iv) Kd = RV = 110D = 100 ´ 0.10 = 10NSV = 100 N = 5

=

= 11.43% [Ans.]

 (v) Ke = D = 9g = 0.08P = 120

=

= 16.10%  [Ans.]

 (vi) Ke = P = 50D = 7.50(1 – 0.40) = 4.50g = RB = 0. 40 ´ 0.10 = 0.04

=

= 13%  [Ans.]

(vii) WACC = (0.085 ´ 0.15) + (0.042 ´ 0.15) + (0.0597 ´ 0.20) + (0.085 ´ 0.10)

+ (0.1143 ´ 0.30) + (0.1610 ´ 0.10)

= 8.99%  [Ans.]

## Exercise # 11

A company finances all its investments by 40 percent debt and 60 percent equity. The estimated required rate of return on equity is 10 percent after-taxes and that of the debt is 8 percent after-taxes. The firm is considering an investment proposal costing Tk 40000 with an expected return that will last for ever. What amount (in Taka) must the proposal yield per year so that the market price of the share does not change? Show calculation to prove your point.

Solution     Weighted Averaged Cost of Capital (WACC)

 Type of capital Capital amount (Tk) Capital ratio Specific cost (Tk) Weighted Average (1) (2) (3) = S(2)  (2) (4) (5) =(3) ´ (4) DebtEquity 16000 24000 0.40 0.60 0.08 0.10 0.032 0.060 40000 SWA = 0.092

WACC = 9.20%

So the investment proposal must yield per year = 40000 ´ 0.092 = 3680   [Ans.]

Proof:

ROE = = = 10% [Ans.]

## Exercise # 12

A public limited company has the following capital structure:

Common share (40000 shares)           Tk 4000000

10% Preference share                               1000000

14% Debentures                                        3000000

Total                                                            8000000

The share of the company sells for Tk 200. It is expected that the company will pay next year a dividend of Tk 20 per share which will grow at 7 percent for ever. Assume a 30 percent tax rate.

You are required to:

(A)   Compute weighted average cost of capital based on existing capital structures.

(B)    Compute the new weighted average cost of capital if the company raises an additional Tk 2000000 debt by issuing 15 % debenture. This would result in increasing the expected dividend to Tk 30 and leave the growth rate unchanged, but the price of share will fall to Tk 150 per share.

(C)    Compute the cost of capital if in (B) above growth rate increases to 10 percent.

Solution

(A) Workings for specific costs:

Ke = = = 0.17

Kp = K = 0.10

Kd = K(1 – T) = 0.14(1 – 0.30) = 0.098

Weighted Averaged Cost of Capital (WACC)

 Type of capital Capital amount (Tk) Capital ratio Specific cost (Tk) Weighted Average (1) (2) (3) = S(2)  (2) (4) (5) =(3) ´ (4) Common share10% Preference share14% Debentures 4000000 1000000 3000000 0.50 0.125 0.375 0.17 0.10   0.098 0.085      0.0125 0.03675 8000000 SWA = 0.13425

WACC = 13.425%   [Ans.]

(B) Workings for specific costs:

Ke = = = 0.27

Kp = K = 0.10

Kd = K(1 – T) = 0.14(1 – 0.30) = 0.098

Kd (additional) = K(1 – T) = 0.15(1 – 0.30) = 0.105

Weighted Averaged Cost of Capital (WACC)

 Type of capital Capital amount (Tk) Capital ratio Specific cost (Tk) Weighted Average (1) (2) (3) = S(2)  (2) (4) (5) =(3) ´ (4) Common share10% Preference share14% Debentures 15% Debentures (add.) 4000000 1000000 3000000 2000000 0.40 0.10 0.30 0.20 0.27 0.10   0.098 0.105 0.108        0.01  0.0294         0.021 10000000 SWA = 0.1684

WACC = 13.84%   [Ans.]

(C) Workings for specific costs:

Ke = = = 0.30

Kp = K = 0.10

Kd = K(1 – T) = 0.14(1 – 0.30) = 0.098

Kd (additional) = K(1 – T) = 0.15(1 – 0.30) = 0.105

Weighted Averaged Cost of Capital (WACC)

 Type of capital Capital amount (Tk) Capital ratio Specific cost (Tk) Weighted Average (1) (2) (3) = S(2)  (2) (4) (5) =(3) ´ (4) Common share10% Preference share14% Debentures 15% Debentures (add.) 4000000 1000000 3000000 2000000 0.40 0.10 0.30 0.20 0.30 0.10   0.098 0.105 0.12        0.01  0.0294         0.021 10000000 SWA = 0.1804

WACC = 18.04%   [Ans.]

## Exercise # 13

A company has the following capital structure at 31st July, 1994, which is considered to be optimum:

12% Debentures                                    Tk  3000000

13% Preference share                                 1000000

Equity (100000 shares)                             16000000

Total                                                            20000000

The company’s shares are selling at a current market price of Tk 25 per share. The expected dividend per share next year is 50 percent of the 1994 EPS. The following are the earnings per share figures for the company during the preceding ten years. The past trends are expected to continue.

 Year EPS Year EPS 1985 1986 1987 1988 1989 1.00 1.10 1.21 1.33 1.46 1990 1991 1992 1993 1994 1.61 1.77 1.95 2.15 2.36

The new debentures can be issued at a coupon interest rate of 13 percent. The company’s debenture is currently selling at Tk 96. The new preference share issue will sell a net price of Tk 20, paying a dividend of Tk 2 per share. The company’s marginal tax rate is 30 percent.

(A)   Calculate the after-tax cost (i) of new debt, (ii) of new preference capital and (iii) of common equity, assuming new equity comes from retained earnings.

(B)    Find the marginal cost of capital, again assuming on new common shares are sold.

(C)    How much can be spent for capital investments before new common share must be sold? Assume the retained earnings available for next year’s investment is 50 percent of 1998 earnings.

(D)    What is the marginal cost of capital (cost of funds raised in excess of the amount calculated in part C) if the firm can sell new common share to net Tk 20 a share? The cost of debt and of preference capital is constant.

Solution

(A)(i) Kd =  = = 9.48%   [Ans.]

(A)(ii) Kp =  = = 10%   [Ans.]

(A)(iii) Ke =  = 14.72%   [Ans.]

(B) MCC = (0.15´0.0948) + (0.05´0.10) + (0.80´0.1472) = 13.698%

(C) Total expenditure = = Tk 147500

(D) Here new Ke =  = 0.159

WACC = (0.15´0.0948) + (0.05´0.10) + (0.80´0.159) = 14.642%