Arbitrage (Levered to Levered Firm) - MM approach 🤑🤑🤑

Arbitrage - Arbitrage is a trading strategy that involves taking advantage of price discrepancies in different markets or within the same market to generate a risk-free profit. Arbitrageurs aim to exploit temporary inefficiencies in pricing by simultaneously buying and selling assets, securities, or derivatives to capture the price difference.

The above answer is generated by chat GPT 🫣, That is, nonetheless, the gist of it.

The aim is to prove that there is an arbitrage opportunity if the values of 2 firms which are similar in all aspects except their capital structure differ. If arbitrage opportunity exists, i.e. if an investor can earn a return without assuming additional risk then a rational investor will carry on that process. The assumption is that it is a perfect market.



For CAP-II students, you are generally given a question to prove arbitrage in case of a levered and a unlevered firm, but it can also be proved in case of unlevered to unlevered or levered to levered firm. In this article, we are exploring the arbitrage that exists when 2 levered firm that are similar in every aspect except for their capital structure differ in value.





Here the total capital is same i.e. 300000 in both cases but the value of Firm B is less by Rs 50000. Both are levered firm. As per MM approach, the value of these 2 firms cannot differ. If they do, arbitrage opportunity exists and equilibrium is immediately restored.

So the onus is on us to prove that arbitrage opportunity actually exists if we move from higher value to lower value firm (A to B in this case)



Alternative 1 - Investing in B in the same proportion as in A.



Points to Remember -

• The Risk must not be changed while shifting from A to B
• Yearly Income Must not be Changed.
• Immediate Cash Saving must arise

Suppose an investor holds 10% shares of Firm A,

Step 1: Sell Shares of A , Amount Received = 10000*15*10% = 15000

Step 2: Borrow debt @15% of Rs (200000*10%) 20000, Amount Received = 20000, Total amount is 15000+20000 = 35000

Step 3: Buy 10% shares of Firm B, Amount Required = 5000*10*10% = 5000

Step 4: Buy 10% Debentures of firm B, Amount required = 25000, Total Amount saved = 35000-5000-25000=5000



Hence Immediate Cash Savings = 5000,



But our work is not done yet, we have proved that there is a cash saving while shifting from high value to low value firm, but, we also have to check if the yearly income is same, because if there is cash saving but proportionate decrease in yearly income, there will be no arbitrage.







Hence we see that income each year has not changed and in this process cash saving of Rs. 5000 is generated. The risk has not differed but a return is generated. This is what arbitrage is, generating a return without assuming extra risk.



Alternative 2 - Investing 100% amount in B.



Points to Remember -

• The Risk must not be changed while shifting from A to B
• Yearly Income Must be increased.
• Immediate Cash Saving must not arise

Suppose an investor holds 10% shares of Firm A,

Step 1: Sell Shares of A , Amount Received = 10000*15*10% = 15000

Step 2: Borrow debt @15% of Rs (200000*10%) 20000, Amount Received = 20000, Total amount is 15000+20000 = 35000

Step 3: Invest 100% amount in B. Rs 35000 must be invested in such a way that equal % is invested in equity shares and debt to nullify financial leverage.



The equation becomes something like this

x% of 50000 + x% of 250000 = 35000

Solving for x, we get x= 11.67%,



Step 4: Invest 11.67% in Firm B, Amount required = 50000*10*11.67% = 5830

Step 5 : Invest 11.67% in debenture of Firm B, Amount required = 29170



Cash Saving = 35000-5830-29170 = 0, We have invested 100% of the amount, hence there is no cash saving, but the additional return comes in the form of increased income every year.



Hence we can see that there is increase in income by Rs 750.75.

Lets cross check the cash saving of Rs. 5000 In alternative A and increased income of Rs 750.75 in alternative B.

ROI of Firm B = 45000/300000 = 15%

Invest 750.75 at 15% till eternity, value of such investment = 750.75/15% = 5000 (There is a little bit of approximation error but we can let go 😁)



Explanation about Risk - We need to maintain same level of risk in Firm A and Firm B, By Risk we mean the leverage risk. We know that leverage Risk is the risk of shareholders and to introduce risk (DFL) we borrow personal loan and to eliminate the risk (DFL) we buy debenture at the same % as that of shareholding. The introduction and elimination of risk is just for a shareholder and not for all shareholders. So by raising a personal loan of 20000 @15%, we are at the same risk level as that in Firm A. But when we buy equity shares we are automatically assuming extra financial leverage risk, and to eliminate that risk whatever % of shares we buy in Firm B, we need to buy Same % of shares in Firm B. Thus, the risk in both firms for the shareholder is same.