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Discounting and Risk

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Discount rate Discount rate is the main tool governments and central banks use to fine tune economic activities. It is the cost of borrowing. For simplicity, we don’t distinguish discount rate and interest rate.

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Risk Factors We will examine some risk factors that affect discount rate. –Salvage ratio –Ratio of self funding –Uncertainty

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Salvage ratio and discount rate Banks charge interest rate based on risk of loans. For simplicity, assume the bank’s borrowing rate is zero. A business applies for 1 million loan for a project and plans to repay the loan in one year. A loan officer estimates the payoff from the project will be 2 million with 85% probability and 0.8 million with 15% probability. If a loan defaults, on average, a bank can get 60% of the salvage value. If the bank requires 2% return on its loans, what would be the loan rate?

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Solution If the project payment is 0.8 million, the company will declare bankruptcy and the bank will receive 0.8*0.6 = 0.48 million If the project payment is 2 million and the loan rate is x, the bank will receive 1+x million

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Overall, the bank is expected to receive (1+x)*0.85+0.48*0.15 = 1.02 2% is the bank’s required rate of return on its loans Solving the equation to get X = 11.53% Discount rate is a reflection of downward risk

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Discussion If the salvage ratio is increased to 80%, the interest rate will decline to 8.71%. If the credit and judicial systems can increase the ratio of salvage value, the discount rate will decline.

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Bonds and Stocks Bond owners have higher priority in claiming assets over stock owners. Bonds have higher salvage values than stocks. So cash flows from bonds are discounted at lower rates than cash flows from stocks.

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Ratio of self funding and discount rate A business plans for a project, which will require 1 million initial investment. The business will supply 0.1 million funding itself. It will apply for 0.9 million loan from a bank and plans to repay the loan in one year. A loan officer estimates the payoff from the project will be 1.3 million with 85% probability and 0.7 million with 15% probability. If a loan defaults, on average, a bank can get 60% of the salvage value. Assume the risk free rate to be zero. If the bank requires 2% return on its loans, what would be the loan rate? If the business will supply 0.2, 0.3 million funding itself and apply a loan for the remaining amount of capital, what will be the loan rates? What conclusion you can draw?

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Solution The amount of loan 0.9 million The required payback 0.9*1.02= 0.918 Assume the loan rate is x 0.9*(1+x)*0.85+0.7*0.6*0.15=0.918 Solving the equation to obtain x X = 11.76%

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When the part of self funding are 0.2 million or 0.3 million, the required loan rates are 10.74% and 9.41%. This shows that the higher the percentage of self funding, the lower the discount rate. When you put in your own money, others trust you more.

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Discussion How much trading in banks is funded by traders themselves or by banks? Should banks enjoy such low rate in their own trading activities? It is often said the causes of financial crisis are “complex”. Are they really that complex?

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Uncertainty and discounting A company has a choice to select one of the two projects. The first project requires an initial spending of 10 million dollars. For the next ten years, the project will generate 3 million dollar profit each year. The second project requires an initial spending of 20 million dollars. The project will generate 3 million dollar profit the first year. The profit from the project will increase 10% from each previous year. The project will last ten years. The criterion of selection is NPV of a project. If the discount rate is 3%, which project you will choose? If projects are forced to close down after three years due to unexpected circumstances, what would be the realized value for both projects, discounted at 3%? If the discount rate is 10%, which project you will choose? With the benefit of hindsight, what discount rate one should use to value two potential projects? How should the choice of discount rate be related to uncertainty and our capacity in information processing?

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Solution When discounted at 3% –NPV of project 1 = 15.59 million –NPV of project 2 = 19.86 million –Choose project 2 If projects have to close down after three years –Realized value of project 1 = -1.51 million –Realized value of project 2 = -10.65 million

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Solution (Continued) When discounted at 10% –NPV of project 1 = 8.43 million –NPV of project 2 = 7.27 million –Choose project 1 Choosing project 1 avoid possible heavy losses due to uncertainty.

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Discussion Some people claimed that since future is unpredictable, one should not be held responsible for losses due to uncertainty. However, if we acknowledge future is highly uncertain, one should discount future at a higher rate. This will help you reduce the amount of losses due to uncertainty. Policymakers often attribute bad economic performance to uncertainty but at the same time keep discount rate low. Are they being consistent?

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Discounting, long term forecasting and fraud Low discount rate makes expected earning in distant future more valuable. This encourages long term forecasting. Since we are less capable to forecast distant future, aggressive optimism and downright fraud often flourish in low discount rate environment.

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Interest rate and housing price Suppose a house is bought for 400,000 dollars. The required down payment is 5% of the house price. The rest of the money is borrowed through a 30-year mortgage with monthly payments. What is the amount of down payment and what is the amount of borrowing? The annual percentage rate on the mortgage loan is 6%. Calculate the monthly payment. Now suppose the government tries to making housing more affordable. It reduces the interest rate to 3%. What is the new monthly payment on the mortgage? Does the government policy improve the affordability of housing market over the short term? If the housing supply doesn’t increase, those who can afford the monthly payment of the original 400,000 dollar house will likely to buy the same kind of house. What will be the new price of the house which was originally sold for 400,000 dollars, if monthly payment is kept at the same level when interest rate was at 6% and down payment is 5% of the housing price? What is the new down payment? Over the long term, will lowering the interest rate alone improve or deteriorate the affordability of housing market?

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Solution Down payment 400000*5% = 20000 Amount remaining 400000 – 20000 = 380000 Number of payment 30*12 = 360 Monthly interest 6%/12 = 0.0005

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Monthly payment 2278.292 Formula for calculation

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When the interest rate is reduced to 3%, redo the calculation to obtain the new monthly payment: 1602.095 Over the short term, the low interest rate policy improves the affordability of housing. To maintain the original monthly payment of 2278.292 with the 3% interest rate, the present value becomes 540386.7, which represent 95% of the housing value, which is 568828.1.With this housing price, the new down payment will be 28441.4 Over the long term, low interest rte policy will deteriorate the affordability of housing market

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adjustable rates mortgage A house is bought for 500,000 dollars. The mortgage rate is 3% per annum. The buyer chooses the adjustable rates mortgage that last for 25 years. In the first 5 years, the buyer only needs to pay the interest part. In the next 20 years, the buyer will pay back interest and principle with an equal monthly payment. What is the monthly payment for the first 5 years? What is the monthly payment for the next 20 years? Why people often default on adjustable rate mortgages? Suppose the housing price appreciates 15% over one year. If the capital investment is measured as the first 12 month’s mortgage payment, what is the rate of return from this investment?

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Discussion Many ARMs were initiated after 2001. Housing market peaked at 2006. Defaults occurred in a large scale from 2007 to 2008. Any relations?

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Short term benefits and long term harm To a drug addict, a shot in the arm will stimulate him over a short period. But it will also put him on long term dependency. However, this type of policies and activities are very common in our society.

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Low discount rate policy after 2000 After the burst of internet bubble, discount rate was lower to stimulate economy. Housing prices rose rapidly. Commodity prices rose rapidly. Financial crisis occurred in 2007, 2008.

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Low discount rate after the financial crisis During the financial crisis, central banks dropped the discount rate sharply to stimulate economy. But easy money flows to commodity speculation, driving oil prices from 20s to near 100 over a short period of time. High commodity prices increase the overall cost of economic activities, stifling recovery.

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Women’s participation in labor market and labor shortage When there is a labor shortage, women are encouraged to join the labor market. This relives labor shortage over short term. But the increased labor participation of women reduces fertility rate, which will intensify labor shortage in the future.

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Senior care: Short term and long term Increased input to senior care improves its quality. But it reduces resources for next generation. With less resource to next generation, less next generation is produced. With less future workforce, the quality of senior care is threatened over long term.

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Social safety net and savings rate We all try to avoid uncertainty. Social safety nets are established to countervail negative shocks. But with extensive social safety nets, there is little incentive to save. In most countries with highly developed social safety nets, saving rates are near zero or negative for most people. The reduction of uncertainty at individual level increases uncertainty at social level. In many rich countries, governments accumulate large amount of debts.

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Pain Killer Many athletes use Advil as pain killer. Advil, as an anti-inflammation drug, reduce the inflammation, and hence the healing, of athletes involved in heavy exercises, which damage muscle tissues. Over short term, athletes are less painful and can endure more strenuous exercises, which helps improve their performances. Over long term, tissue damage accumulate.

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Concluding remarks The choice of discount rate has long term consequences. Many central banks fine tune discount rates to improve short term economic performances. However, short term monetary policies have long term consequences, which often do not concern policy makers and general public as much. They don’t fully understand the precise relations among the various factors in social and economic activities.

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Risk free rate There are two types of projects. Each project require an initial investment of 1 million dollar. For the first type of projects, there is a 65% chance that the project will generate 1.3 million dollar payoff after one year and there is a 35% chance that the project will generate 0.7 million dollar payoff after one year. For the second type of projects, there is a 85% chance that the project will generate 1.3 million dollar payoff after one year and there is a 15% chance that the project will generate 0.7 million dollar payoff after one year. (Continued on next page.)

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Each prospective project operator may apply for a loan from the bank. As a rule, the bank will require the project operator to supply 30% funding and provide 70% loan. From past statistics, the bank knows that 60% of the projects are of type 1 and 40% of the projects are of type 2. But the bank cannot distinguish between type 1 and 2 projects without additional cost. (Continued on next page.)

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The bank require 2% return on its loans. If the risk free rate is 4%, what is the loan rate the bank would offer to prospective project operators? The prospective project operators will accept the loan only if the expected payoff is positive. We assume the interest rate the prospective operator will earn is the risk free rate if he decided not to start the project. Will the prospective operators of projects of type 1 and 2 accept the loans? (Continued on next page.)

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If the risk free rate is 6%, what is the loan rate the bank would offer to prospective project operators? Will the prospective operators of projects of type 1 and 2 accept the loans? If only prospective operators of projects of type 2 will apply for loans, what will the bank charge for its loans? Is this rate higher or lower than when the risk free rate is 4%? What conclusion you can draw?

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Solution First we assume the risk free rate is 4%. Assume the interest rate bank will charge is x. For type 1 projects, the payoff is (1.3-0.7*(1+x))*0.65-0.3*1.04 For type 2 projects, the payoff is (1.3-0.7*(1+x))*0.85-0.3*1.04

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For the bank, the expected payoff is (0.7*(1+x)*0.65+0.7*0.8*0.35)*0.6+ (0.7*(1+x)*0.85+0.7*0.8*0.15)*0.4 = 0.7*(1+0.04+0.02) Solving for x to get X = 15.74%

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Plugging into the equations for expected returns for projects of type 1 and 2, we find Type 1: 0.0064 Type 2: 0.1044 Both operators will accept the loans.

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Then we assume the risk free rate is 6%. Following the same procedure, we find X = 18.36% The expected returns for projects of type 1 and 2 are Type 1: -0.0115 Type 2: 0.0828 Operators of project type 1 will not accept the loans.

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As a result, only projects of type 2 will actually start. This shows that higher risk free interest rate reduce information cost for financial systems. Is high interest rate bad for economy? When resource is abundant, waste of resources will not be reflected in human society. When resource is scarce, waste of resources will accelerate the decline of human society.

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