Chapter 13 The Business of Banking Objectives 1. Analyze the strengths and weaknesses in bank balance sheets and income statements. 2. Use T-accounts to document marginal effect of transactions on bank balance sheets. 3. Categorize Bank Risk and Analyze Strategies for managing risk. 4. Measure interest rate exposure with Gap Analysis. Bank Balance Sheets Basic Operation of a Bank General Principles of Bank Management Liquidity Management Asset Management Managing Credit Risk Liability Management Managing Market Risk Capital Adequacy Management Gap Analysis Role of Financial Intermediaries 1. Pooling Savings •Take advantages of economies of scale •Diversify Risks • Safekeeping of Assets 2. Providing Liquidity Reduce transactions costs by allowing depositors to convert assets into cash. 3. Reduce Information Costs •Ameliorate asymmetric information Profits of Banking • Banks collect retail deposits from savers and make loans from depositors. • Profits are earned by banks when they are able to make loans at higher interest rates than they pay depositors. • Net Interest Margin is the average interest rate earned on assets (mainly loans) minus the average interest paid on liabilities (mainly assets). Net Interest Margin HK: Net Interest Margin: Local & Foreign Retail Bank % 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 Jun-1997 Jun-1998 Jun-1999 Jun-2000 Jun-2001 Jun-2002 Jun-2003 Jun-2004 Jun-2005 Sources of Bank Profits • Net interest income Bank of East Asia Net Interest Income Operating Income • US Commercial Banking System. 2003 (in millions) HK$3,596.00 HK$5,496.00 65.43% Net Interest Income Operating Income 2001 (in millions) $210,809 $364,543 57.83% Bank Balance Sheets • All balance sheets have three categories. Assets $198,476,118 Liablities Net Worth $198,476,118 $174,089,449 $24,386,669 $198,476,118 Bank Liabilities 1. Checkable Deposits – Demand Deposits 2. Non-transactions Deposits – • • • Savings Accounts, Money Market Deposit Accounts, Time Deposits, Certificate Deposits. 3. Borrowings – Short-term loans from central bank or other banks. Liabilities of Bank of East Asia • As an international financial center, HK Banking system has large borrowings from foreign banks. This may occur internally on the books of multinational banks. • The more domestically oriented Bank of East Asia in 2001 gets 4.32% of liabilities from borrowings Deposits and Balances of Banks Liabilities Ratio $7,516,565 $174,089,449 4.32% Liabilities of the HK Banking System: 2001 Liabilities & Net Worth Million HK$ % Checkable Deposits Non-Transactions Deposits --NCD's --Time and Savings Deposits Borrowings --Domestic Banks and Institutions --Foreign Banks --Other Debt Instruments 138348 3423510 Liabilities 5531007 90.19% 601817 9.81% Capital Total 2.26% 55.82% 174082 3249428 1969149 2.84% 52.98% 32.11% 388818 1532731 47600 6132824 6.34% 24.99% 0.78% Comparison of HK Banking w/ US Banking • US Banks get higher percentage of funds from transactions deposits. • Banks in HK have somewhat better capitalization than US banks. US Banking 12-2002 Transaction Deposits Non Transaction Deposits Borrowings Other Liabilities Total Liabilities Capital Total Assets Billion US$ 634.1 3467.7 1119.6 557.2 5778.6 498.4 6277 10.97% 60.01% 19.37% 9.64% 92.06% 7.94% Bank Capitalization • Compared to non-financials, banks have low capitalization. • In 1999, ratio of assets to liabilities among top 100 HK nonfinancial corporations was 3-to-1. • Among banks, closer to 1-to-1 HK Balance Sheets 3.5 3 2.5 2 1.5 1 0.5 0 Nonfinancial Banks Assets to Liabilities Bank Capital: Funds put at risk by the owners of the bank. • Tier One (Shareholders Funds) – Share Capital: Money raised by selling equity shares in Primary Markets. – Reserves: Money raised through Retained Earnings. • Tier Two (Loan Capital) – Subordinated Debt: Uncollateralized junior debt. Bank Assets 1. Cash Items • Primary Reserves: Vault cash + Balances at Central Bank. • • The central bank is the bank of banks. Banks keep deposit accounts at the central bank for the purpose of clearing checks and electronic transactions. Other Cash Items • • Amount Due from Other Banks: Deposits at other banks. As an international financial center, loans to and from foreign banks are an important part of balance sheets in HK. Cash Items in Process of Collection Bank Assets Pt. 2 2. Loans - Chief purpose of banks is making loans. Loans are the least liquid of bank assets, but typically pay the highest returns 3. Securities – Banks sometimes hold government and other bonds as an important category of assets. Government securities are very liquid and are held as secondary reserves. Some securities are less liquid and are held as an alternative to loans. 4. Tangible Assets – Land & Buildings, Repossessed Collateral Assets: Hong Kong Banking Industry Assets Primary Reserves --Aggregate Balances --Cash Amount Due From Other Banks --Domestic --Foreign Loans to Customers NCDs Securities Other Assets HK Million % 12686 0.21% 596 0.01% 12090 0.20% 2754497 44.91% 487789 7.95% 2266708 36.96% 2116848 34.52% 135716 2.21% 871853 14.22% 241224 3.93% 6132824 100% Basic Operation of the Bank T-Accounts • T-Accounts are a handy tool for examining the effects that any transaction has on balance sheets. • A bank transaction will change both liabilities and assets (and possibly net worth). The total change in liabilities plus net worth must always equal the total change in assets. Ex: Check Deposit • A Hang Seng customer sells his car for a check drawn on BOC for $100. He deposits the check in his Hang Seng account and Hang Seng sends the check to HKMA for settlement. • Hang Seng Assets Reserves + $100 Liabilities Checkable Deposits +$100 Ex: Bank Lends Funds to Customer • Hang Seng bank lends $100 to a customer by crediting their checking account. Assets Loans + $100 Liabilities Checkable Deposits +$100 • The customer writes a check to a BOC customer. BOC passes the check to HKMA Assets Liabilities Reserves $100 Checkable Deposits -$100 • A bank writes off a loan of $100. • A bank sell equity shares for more than book value Assets -$100 Loans Liabilities & Net Worth -$100 Net Worth Assets Liabilities & Net Worth -$100 Shares +$110 Cash +$10 Net Worth General Principles of Bank Management Banks as Risk Taking Institutions • Banks may specialize in ameliorating effects of asymmetric information. • But there is still asymmetric information between banks and depositors. Banks info advantages are offset in at least 2 ways. 1. Bank Capital – Owners of banks put some of their own funds into banks and these funds are at risk. 2. Liquidity Advantage of Depositors – Depositors can withdraw funds very quickly from banks. Managing Banks: Balance Risks and Returns • Banks must take risks as part of their business. • Often most profitable activities of a bank will generate most risks for the banks. • Bank managers must manage risk return trade-offs. Types of Risk • • • • • Liquidity Risk Credit Risk Interest Rate Risk Market Risk Operational Risk Bank Management 1. Liquidity Management: The decisions made by a bank to maintain sufficient liquid assets to meet the demand by depositors. 2. Asset Management: The acquisition of assets that have a low rate of default and diversification of asset holdings to increase profits. 3. Liability Management: The acquisition of funds at low cost to increase profits. 4. Capital Adequacy Management: A bank’s decision about and acquisition of the amount of capital it should maintain. Liquidity Management • Majority of Bank liabilities are very Liquid. – Checkable deposits & Savings deposits can be withdrawn at any time. Time Accounts & CD’s are typically of short maturity and can be withdrawn with some penalty to depositor. – Most profitable bank assets, loans, are illiquid. • Liquidity Risk: The possibility that depositors may collectively decide to withdraw more funds than the bank has on hand. Liquidity Measures • Loan to Deposit Ratio – Ratio of illiquid loans to liquid deposits. High measure of loan-to-deposit ratio indicates high liquidity risk. Loans Deposits Ratio $108,379,794 $163,737,665 0.662 • Banking system as a whole had ratio of 2255196 .597 3423510 Liquidity and Reserves • If banks face an unexpected withdrawal of funds and they have no liquid assets they will face the cost of liquidating some assets to meet depositor demand. • Banks hold reserves of liquid assets to meet depositor withdrawals. • Primary Vault Cash Reserves = + Balances at Central Bank Required Reserves & Excess Reserves • In many countries, banks are required to keep some percentage of their deposit liabilities as reserves at the central bank 1. Required Reserves: The minimum amount that depositary institutions are required to hold as deposits at the central bank. 2. Excess Reserves: Reserves held beyond the minimum amount. Reserves in Hong Kong • In Hong Kong, banks reserves at the central bank (the HKMA) are called Clearing Balances. • These are held for the sole purpose of clearing interbank transactions. – For example, if a depositor at Hang Seng bank writes a $100 check to a depositor at Bank of China, the transaction will be cleared when the HKMA deducts $100 from Hang Seng’s Clearing Balance account and credits the BOC’s Clearing Balance account. • Banks in Hong Kong are only required to have enough on hand in their clearing balances to meet their payment obligations. Beyond that there are no required reserves. Secondary Reserves • Having primary reserves directly reduces the liqudity risk. But banks cannot eliminate liquidity risk entirely, since primary reserves pay 0 interest. • Secondary reserves are interest paying government securities held by banks to meet short-term liquidity needs. – In Hong Kong, Hong Kong dollar secondary reserves are Exchange Fund Bills. – U.S. Treasury Securities are US dollar secondary reserves. Managing Liquidity Risk • A bank faces withdrawals of $5 million. Assets Cash - $5 Liabilities Checkable Deposits -$5 • This reduces liquidity. The bank can restore liquidity by managing assets or liabilities. • Liquidity can be restored by converting secondary reserves (market securities) into primary reserves (cash) Assets Liabilities Cash +$5 Securities -$5 Assets Liabilities • The bank can also engage more short-term Cash +$5 Borrowings+$5 liabilities by increasing borrowings from other banks or central bank. Banks manage their interest rate/liquidity tradeoff by keeping more primary reserves. Reserves to Deposit Ratios 7.00% 6.00% 5.00% 4.00% % Primary Secondary 3.00% 2.00% 1.00% Mar-05 Sep-04 Mar-04 Sep-03 Mar-03 Sep-02 Mar-02 Sep-01 Mar-01 Sep-00 Mar-00 Sep-99 Mar-99 Sep-98 0.00% Asset Management • • • • Banks must maximize the returns earned by their assets while minimizing risk of default. Credit Risk: The risk arising from the possibility that the borrower will default. Financial Intermediaries in general and banks in particular exist because of their efficiency in dealing with credit risk. Much of credit risk in financial markets occurs due to asymmetric information and its associated phenomena, adverse selection and moral hazard. Principles for Maximizing Returns while dealing with credit risk Diamonds in the rough – Banks try to find borrowers who will pay high interest rates but who are unlikely to default. – Borrowers who are well known to be good credit risks will have many sources of funds. – Banks need to find information about certain borrowers not publicly available. Strategies for Managing Credit Risk 1. Credit-Risk Analysis – A loan officer manages banks relationship with borrowers and evaluate potential borrowers. • • Loan officers may have some specialization with certain industries or businesses. Loan officers also use credit scoring systems which use statistical data to measure default probabilities and charge interest rate commensurate with risk. Strategies for Managing Credit Risk (cont.) 2. Monitoring – Loan agreements may contain restrictions on borrower behavior or value of assets. Loan officers monitor behavior and may recall loans if covenants are violated. Strategies for Managing Credit Risk (cont.) 3. Collateral – Loans identify physical assets which may be taken by the bank in case of default. 4. Long-term Relationships – Banks often have relationships with certain businesses which reduces information problems. Relationships have value to businesses which they are loathe to jeopardize by engaging in moral hazard behavior. Strategies for Managing Credit Risk (cont.) 5. Diversification – Banks can limit the likelihood of default by reducing exposure to a particular borrower or class of borrower. • Sometimes there is a trade-off between diversification needs and strategies for finding diamonds in the rough, such as specialization or long-term relationships which may tend to reduce Liability Management • Before 1960’s, banks relied on retail deposits (checking and savings accounts) for liabilities. Liabilities management was easy. • Since then, banks have increasingly looked for flexible, low cost liabilities to fund asset positions. Bank Liabilities 1. Checkable Deposits – Demand Deposits 2. Non-transactions Deposits – 1. Savings Accounts, 2. Money Market Deposit Accounts, 3. Time Deposits, Certificate Deposits. 3. Borrowings – Short-term loans from central bank or other banks. As an international financial center, HK has large borrowings from foreign banks. Core Deposits vs. Managed Liabilities • Bank Liabilities can be divided into two parts. 1. Core Deposits – Demand Deposits, Savings Accounts, Small Time Deposits (Retail Funds) 2. Managed Liabilities – Borrowings from Other Banks, Securities, Large CD’s and Time Deposits (Wholesale Funds) • Retail funds have lower interest costs and are thought to be more stable. Capital Adequacy Management • • Bank capital is the funds invested by the owners of banks in the bank. Three factors affect the decisions of bank owners to invest capital: 1. Bank capital prevents bank failure. 2. Bank capitalization affects returns to shareholders 3. Government regulations affect capitalization (next chapter) Bank Failure • Bank failure occurs when a bank cannot pay its depositors in full. • Riskier and less liquid assets make bank failure more likely. • Banks with high levels of capital can have some negative profits and still avoid failure. • Bank owners need to invest their own funds to offset its own moral hazard issues., How Bank Capital Prevents Bank Failure • Consider two banks with identical balance sheets except that Bank A is well capitalized while bank B is poorly capitalized. Assets Liabilities Assets Liabilities Reserves $10 Deposits $90 Reserves $10 Deposits $96 Loan $90 Capital $10 Loan $90 Capital $4 How Bank Capital Prevents Bank Failure • Bad economic times cause borrowers to default on $5 million in loans. This wipes out the capital of the weakly capitalize bank but leave the highly capitalized bank in business. Assets Liabilities Assets Liabilities Reserves $10 Deposits $90 Reserves $10 Deposits $96 Loan $85 Loan $85 Capital -$1 Capital $5 Capitalization • The owners of banks will protect against bank failure if they invest a high level of capital. • The owners of banks will earn high returns if they invest relatively little compared to the size of assets. • We can compare capital as a share of assets for Bank of East Asia with banking system as a whole. 2003 Authorized Institutions Bank of East Asia Net Worth 717149.3 26164 Total Assets 6490721 198476 Capitalization 11.05% 13.18% Equity Multiplier • Another measure of leverage is the equity multiplier. • This is assets relative to shareholders equity (i.e. net worth less loan capital) EM ASSETS EQUITY CAPITAL Total Assets $198,476,118 9.874 Shareholders Funds $20,101,500 ROA/ROE • A measure of the returns PROFITS DUE TO SHAREHOLDE RS ROA earned on assets is TOTAL ASSETS Return on Assets PROFITS DUE TO SHAREHOLDE RS • Owners of banks are ROE concerned with the payEQUITY CAPITAL off they earn per each dollar originally invested ROE ROA EM in the bank. They care about the Return on Equity • Its easy to see that equity returns are a positive function of ROA and leverage Profits Assets Shareholder's Funds B of EA 2003 1921714 198476118 20101500 ROA ROE 0.009682344 0.095600527 Performance of US Banks COMMERCIAL BANK CONCENTRATION, NUMBERS AND ASSETS, 2000 AND 2004 ($ billions, end of year) By asset size Less than $100 $1 billion $100 Percent of million to Percent of to $10 Percent of Greater than Percent of million total $1 billion total billion total $10 billion total 2000 Number of banks 4,842 Total assets $231.20 Total deposits 194.9 Return on assets 1.01 Return on equity 9.09 2004 Number of banks 3,655 Total assets $189.00 Total deposits 158.2 Return on assets 0.99 Return on equity 8.46 NA=Not applicable. Total banks 58.20% 3,078 37.00% 313 3.80% 82 1.00% 8,315 3.7 $773.00 12.4 $884.10 14.2 $4,350.40 69.7 $6,238.70 4.7 632.5 15.1 621.6 14.9 2,727.60 65.3 4,176.60 NA 1.28 NA 1.29 NA 1.16 NA 1.19 NA 13.56 NA 14.57 NA 14.42 NA 14.07 47.90% 3,530 46.30% 360 4.70% 85 1.10% 7,630 2.2 $953.40 11.3 $973.00 11.6 $6,297.30 74.9 $8,412.80 2.8 770.9 13.8 666.5 11.9 3,997.20 71.5 5,592.80 NA 1.28 NA 1.46 NA 1.3 NA 1.31 NA 12.88 NA 13.48 NA 14.24 NA 13.82 Source: Federal Deposit Insurance Corporation. Market Risk • • • Interest Risk – The risk to an institution's financial condition resulting from adverse movements in interest rates. An asset (or a liability) represents a set of payments that must be made at times in the future. Define PVT as the present value of a future payments made to an asset or a set of assets in T periods. Duration Measure of Interest Rate Risk • Define market value, MV, of an T asset or a set of assets as the MV PVt sum of present values derived t 1 from payments made in each future period. • Define the duration of an asset as d d • The % change of the market value of an asset to a change in the interest rate is approximately proportional to the duration of an asset. T PVt t MV t 1 MV i d MV 1 i Example • Two period coupon bond with face value of 100, coupon of 10 and yield to maturity of 10%. • Yield Rises to 11% MV 10 110 100 2 1.1 (1.1) d 1 10 110 21 2 110 121 11 10 110 MV 98.2875 .017125 2 1.11 (1.11) MV i 21 .01 d .01736 1 i 11 1.1 MV Example • 5 period fixed payment loan of 100. Yield to maturity of 10% 100 100 100 100 100 2 2 3 3 4 4 5 5 1.1 1.1 1.1 1.1 2.8101 d 1.1 379.1 MV .01 2.8101 .0255 MV 1.10 1 1 100 100 100 100 100 1.15 MV 2 3 4 5 100 379.1 1.1 1.1 1.1 1.1 1.1 .1 Measuring Interest Exposure • Calculate the duration of A d i A A 1 i a banks assets, dA. Calculate the duration of L i d L a banks liabilities, dL. L 1 i • An increase in the interest rate will have the following effect on assets and liabilities. • Calculate the GAP as a L function of duration of GAP d A d L A assets and liabilities. An increase in interest rates changes the value of a banks assets and liabilities. NW A L A L L A A A A L A i i L L i d A dL d A d L 1 i 1 i A A 1 i NW NW NW i GAP A NW A 1 i NW i EM GAP NW 1 i Managing Interest Rate Risk • A bank which has a large stock of assets which will pay a fixed interest rate may face losses if market interest rates rise. • Since deposits must be redeemed at any time, the bank must offer market interest rates. If market interest rates rise, loan spreads will be cut. • Banks may use asset and liability management to match the sensitivity of assets and liabilities to interest rates. Responses to a perceived risk of a rise in interest rates. • If interest rates are expected to rise, banks would like to reduce their long-term fixed interest rate assets and lock in current low interest rates by increasing their stock of long-term liabilities – Asset Management: Increase holdings of short-term loans or securities. – Liabilities Management: Sell long-term certificates of deposit. Floating Rate Loans • Fixed payment loans have a constant payment based on a fixed interest rate. • Floating rate loan payments are based on an interest rate that changes as some benchmark interest rate changes • A HK$ floating rate loan will be quoted as a spread over Hong Kong InterBank Offered Rate (HIBOR), the rate at which banks lend aggregate balances to each other. • A US$ floating rate Euroloan will be typically quoted as a spread over London InterBank Offered Rate (LIBOR). Interest Rate Swaps • Problem: Hang Seng can make fixed rate loans but does not want to bear interest rate risk. • Hang Seng may find a firm willing to swap interest payments for a floating interest rate payment. • “Plain Vanilla” Interest Swap – Two parties agree on a notional amount of principal (which does not change hands). – One party will pay the counterparty a fixed interest in every period. – The counterparty will pay the first a floating interest rate as a markup over LIBOR. – Only the net difference in interest is actually paid. Loan Securitization • Banks make loans in a certain class, bundle the loans into a portfolio, sell securities, and dedicate the principal and interest payments on the loans to making coupon and face value payments on the securities. – Banks profits come as fee for setting up loan back securities. • Banks reduce the maturity mismatch between assets and liabilities by raising funds this way instead of deposits reducing interest rate and liquidity risk. – Primarily mortgage loans are securitized but also securitization of credit card receivables, auto loans and even leasing payments by rental companies. Growth in Asset Backed Securities Asset Back Securities USA 1800 1600 1400 1200 Billions US$ 1000 800 600 400 200 0 Mortgages Consumer loans 1985 1995 Business loans 2004 Trade receivables Market Risk • Market Risk : Risk to banks liquidity and/or assets from movements in asset prices 1. Exchange Rate Risk – The potential fluctuations in the value of assets or liabilities denominated in foreign currencies due to fluctuations in the exchange rate. 2. Collateral Risk - The risk to the value of assets used as collateral for loans. Exchange Rate Risk • International banks often issue liabilities and assets in a variety of different currencies. If there is an imbalance, a change in the exchange rate can affect firms balance sheets. • Example 1 : In Hong Kong, banks accept large foreign currency deposits but less demand for US$ loans in Hong Kong. • Example 2: In Argentina, fixed exchange rate with the US dollar (EX = 1) has led Argentine banks to engage in large scale off-shore borrowing denominated in dollars and on-shore borrowing in pesos. What happens to Argentine banks if the peso is devalued? Licensed Banks Balance Sheets in 2001: HKMA Total Assets HK$ Assets -Loans to Customers -Advances Due from Banks -Securites and Other Foreign Currency Assets -Loans to Customers -Advances Due from Banks -Securites and Other 6241252 2662668 1502813 120477 1039378 3578585 606001 2160909 811675 Total Liabilities and Capital HK$ Liabilities -Deposits -Due to Banks Abroad -Other Foreign Currency Liabilities -Deposits -Due to Banks Abroad -Other 6241252 2849549 1830474 171007 848068 3391704 1605443 1349723 436537 • Balance Sheets Prior to Devaluation Assets Loans P$125 Liabilities Borrowings US$100 (= P$100) Net Worth P$25 • Balance Sheets After Devaluation (EX = .8) Assets Loans P$125 Liabilities Borrowings US$100 (= P$125) Net Worth P$0 Managing Exchange Rate Risk • Asset and Liability Management: Hong Kong Banks Borrow and Lend to Foreign Banks to Match Currencies of Liabilities and Assets. • Exchange Rate Swaps • Exchange Rates Futures and Forward Contracts. Plain Vanilla Exchange Rate Swaps • • One bank has two many dollar loans and another has two many yen loans. Two banks swap loans. 1. Principal is exchanged at the current exchange rate. 2. Interest on the loans is paid at subsequent periods. 3. At end of loans, the party’s buy back the principal at final maturity date exchange rate. Exchange Rate Futures • A futures contract is an agreement that specifies the delivery of commodity, currency at a prespecified date and price. • A bank with a foreign currency loan may engage in a futures transactions that will entitle them to sell the principal for domestic currency at a prespecified price shifting exchange risk to their counterparty. • Participants in futures markets are hedgers if they are getting rid of risk to speculators who will take it at a price. Non Interest Income is growing as a share of operating income Non Interest Income as % of Income 45.00% 40.00% 35.00% 30.00% 25.00% % 20.00% 15.00% 10.00% 5.00% 0.00% 1991 1996 USA Japan 2001 New Sources of Bank Profits • Before 1970’s, banks had almost complete dominance of external finance markets. Over time, in all markets, securities markets make up a larger share of finance. • Banks have innovated to find new sources of profits taking advantage of their competitive strength in analyzing and monitoring borrowers. Off-Balance Sheet Lending • Standby Letters of Credit: Commercial paper is short-term uncollateralized debt. An SLC is a promise by a bank to lend money to CP issuers when they must repay their CP. This service earns fees even if no loans are made. • Loan Commitments – A more general promise to make loans in exchange for a fee. • Loan Sales – Banks sell loans to 3rd parties willing to bear the credit risk. • Fees for information Services. Credit Derivatives • Banks can reduce their exposure to credit risk through credit derivatives. • Credit Swaps – Bank pays a protection fee, counterparty pays a contingent amount if a “credit event” occurs. • Total Return Swap – Bank swaps total return of a given asset for specified floating interest payments Notional Amounts of Credit Derivatives Credit Derivatives $14,000.00 $12,000.00 $10,000.00 $8,000.00 $6,000.00 $4,000.00 $2,000.00 $0.00 1H01 2H01 1H02 2H02 1H03 2H03 Source: Goldman Sachs, In Trillions of US $ 1H04 2H04 1H05 Credit Risk Models • Banks construct statistical models of their portfolios of assets to calculate the probability that the value of overall portfolio will drop by a certain amount. • Value at Risk models. – Input: classifications of quantities of different types of assets in the portfolio (cash, loans to borrowers by credit scores, securities, …) – Output: Minimum value of portfolio at different horizons at given probabilities (i.e. your portfolio has a 95% chance of being above $1.23769 billion next year.