The Yield Curve

on Saturday, October 1, 2011

What can rental cars teach us about yield curves? What are credit default swaps? Is borrowing a car fundamentally different from borrowing a dollar?
The longer one borrows a car, the lower the daily rate, but the longer one borrows money, the higher the daily rate becomes. Why? Another way to ask this question might be: "Why do yield curves slope upward for dollars and downward for cars?" Additionally, why does the shape of the curve change? What affects the rates that lenders charge to borrow cars and cash?
At first the question seems strange; we don't normally think of cash and cars as assets that are both being leased. We tend to avoid thinking of dollars as an asset that is being rented out, just like a rental car. We even have different names for the same things when speaking about dollars. Dollars do not depreciate, they are subject to inflation. Dollars are not rented to customers, they are loaned. One can lease a new car, but one borrows money. One pays a fee to rent a car, and pays interest to lease dollars. One signs a rental contract to borrow a car from Hertz, but signs a mortgage to borrow dollars from Chase.
The situation is further complicated when considering financial investments and bonds. If a company issues a bond it is borrowing dollars from an investor; it is renting dollars from an investor. Buying a bond is the same as loaning dollars; the same as renting dollars to the company that created the bond. Buying a bond is considered an investment, but buying a bond is the same as loaning money, the same as renting out dollars for a fee. In this sense a rental-car agency is no different than an investor -- it is renting out cars for a fee. Because both cars and dollars are assets, Hertz is "investing" in rental-car contracts, just as you might "invest" in bonds. A bond is a loan contract, and a loan contract is a rental agreement.
When a rental agency rents a car to its customer, the customer issues a bond (rental contract) -- the customer agrees to pay a fee for the right to use the car. The same thing occurs when a company issues a bond -- it agrees to pay a fee (interest) for the right to use an investor's dollars. Both borrowers agree to return the borrowed asset (dollars or a car) at a certain time and place, and pay a fee for the privilege of its use. The paperwork for this agreement is the same.

Back to the original question: why does a 15 year mortgage carry lower rates than the same agreement over 30 years... but borrowing a car becomes cheaper if it is rented for a week instead of a day? At first it seems like the answer is simple -- it costs money to process the car rental (building, desk clerks) and its return (cleaning etc), and the car loses value over time, and it can't make money sitting idle on the lot, so long-term rentals are more cost effective and avoids the paperwork of the turnover. The trouble is that mortgages require all of the same things, and dollars lose value over time as well. 
The are three main differences here. One is that dollars can be immediately loaned to new customers upon their return; they can always find a new investment (person willing to borrow them). Dollars sitting in an account are not idle like cars in a parking lot. The cars are losing value and failing to generate income, but the dollars in the account are generating interest.
Money is so liquid that it can be instantly rented out to myriad borrowers. Cars are illiquid, and there is always a risk that they will sit idle on the lot, which gives rental agencies an incentive to make long-term agreements to avoid idle time. Companies that rent cash (make loans) have zero risk of the money sitting idle, and therefor no incentive to offer a long-term discount on the rental fee. The yield curve flips when any asset becomes so liquid that there is no risk of the asset sitting idle. Until that point, short term rates are higher because of the idle-time risk.
Another main difference is the rate of depreciation; care lose value much faster than dollars, and because of the exponential nature of depreciation (assets lose value faster tomorrow relative to next year), depreciating assets cost more to hold now than in the theoretical future. This is obvious with cars and not so obvious with dollars, but both experience an exponential loss of value over time. 
A new car will lose perhaps half its value in the first three years, but may take another six years to lose half its value again. Eventually, a very old car will be nearly stable, losing only a fraction of a percent of its original value. The same thing happens to dollars, and for this reason if inflation (depreciation of dollars) is expected to be high, short-term rental of the dollars must be more expensive -- because of the nature of exponential depreciation -- dollars are always losing more value today than they will lose tomorrow. A dollar is today worth about 5% of its value 100 years ago, and is expected to again be worth 5% of its current value 100 years in the future, relative to today.
For example, if a car is expected to lose half its value per year, I must charge more than half its value to earn a profit on one-year rentals. But I can charge less per year if you agree to rent it for two years -- at the end of two years a $20,000 car will be worth $5,000; it will have lost $7,500 per year. So I must charge over $10,000 per year on one-year rentals and over $7,500 per year on two-year rentals, and so on. The price per year will be lower if you agree to lease it for a longer term. The same pressure applies to dollars, but because they are expected to lose only 2% of their value per year, I can make a profit by renting dollars (loaning them) at anything over two cents per year, per dollar.

A third difference is that the car rental-fee is paid in an asset other than cars; it is paid in dollars. With the rental of dollars, the fee is paid in the same asset that is leased. Borrowers of dollars pay their fee in dollars, and this turns out to be a key difference, and perhaps a good reason for using different terminology. To judge its equivalent in car rental, one would need to pay for the rental with rental cars.
So why do we see upwardly sloped yield curves if dollars are expected to depreciate in the same way as cars? Shouldn't I be willing to offer discounts on the rental fee if you agree to rent my dollars for a longer term? Yes, but there is another consideration in the business of renting property: default risk.
If I rent out my car, there is virtually no default risk; I will always get my car back. It is insured against theft and destruction. If I rent dollars to you and you default, I have no insurance, and have little hope of having my dollars returned to me. For this reason, the rental of dollars becomes more risky the longer I agree to let you use them. Credit default swaps, invented in the 1990s, are a form of insurance on rented (loaned) dollars. They allow people who rent out dollars to purchase insurance that pays if the rental money is not returned in the same way that Hertz carries insurance that protects them from renters who do not return their vehicles. CDS obligations lower the cost of lending money, making borrowing money cheaper in the same way that auto-insurance lowers the cost of renting a car from Hertz.
If you screw up and crash my car, I am not affected. If you screw up and go bankrupt, I lose my dollars (unless I purchase CDS insurance). Running your life without going broke is like driving a car without running it off a cliff -- both are more likely as the time-frame grows larger. The longer you drive, the more likely you are to crash. The longer you use my dollars, the more likely you are to go broke. However, we must not make the mistake of assuming that the fee is higher because the total risk of default is higher. The risk of default each year is what matters, because the rental fee (interest fee) is paid per year. It is paid based on the amount of time a borrower holds the car or the cash. 
If the borrower is a robot and his credit rating will not change from this moment until 30 years from now, the risk of default is the same for each of those 30 years, all else being equal. But he is not a robot; he is human, and I may be able to see how he drives today, but it is impossible to see how he will be driving 30 years from now. I know a loan may be a good idea for a business in its current state, but how can I know how stable it will be in the distant future? To take that gamble I must be paid a higher fee, and it is because conditions change over time. That man who borrows a car today will not be the same man in 30 years. Short-term loans are safer because it is easier to predict the near future. Tomorrow's weather is easier to predict than next week's.

However, a lender with a default swap does not fear losses in the same way that a rental-car company does not fear losing a car. The price of insurance varies from year to year, and there are myriad insurance companies available. The only risk in lending is that the insurance company will go broke at the same time that a car is stolen (which happened in 2008 with AIG's failure), so the credit rating of borrowers should not affect long and short-term rates differently.
Additionally, the final consideration is commonly known as a liquidity premium. If I am going to invest in something that is difficult to sell, I want to be paid a higher rental fee. If I invest in a rental property I want a higher return than if I invest in treasury bonds, because I can sell the bonds almost instantly, but may have to wait months or years to find a buyer for my rental property. The liquidity premium is the additional fee that is charged by people who rent their assets (invest) for longer periods of time. This consideration is mitigated by the securitization of long-term loans. Because lenders can resell their loan contracts easily, their capital is not tied up for the full term of the loan, and they don't mind issuing long-term loans any more than short-term loans.
The shape of the yield curve (price list lenders and investors set) is determined by four distinct forces:
1. The risk that the asset may sit idle and lose value when it is not leased to a customer. This relates to the liquidity of the asset. Liquid assets carry lower risk of idle time, and liquidity is a measure of how quickly an asset can be traded, sold, or rented. Mitigated by liquid markets.
2. The risk that the asset may not be returned as agreed (default). This depends on both the credit of the borrower and the uncertainty of the future. If war is on the horizon this risk is higher for all borrowers. Mitigated by CDS.
3. The expected depreciation rate of the asset; inflation.
4. The liquidity premium (term premium). People don’t like to own assets that are difficult to sell (trade), preferring to own 100 gold coins instead of a solid block of gold weighing the same amount, because the coins are more liquid; they can be traded more easily. Mitigated by securitization of loans -- similar to selling paper shares that represent a fraction of the gold brick. Because they are smaller, they are more liquid.

5. The difference between the depreciation of the loaned-asset and the asset used as payment. With rental cars, this involves comparing the depreciation of the car with the depreciation of the dollar. 
The liquidity of the asset affects the first risk -- the more liquid the asset, the lower the risk of holding an "idle asset." If it is very liquid, idle time will be near zero; cash can be immediately re-leased (invested at interest) upon its return.
A person in the business of renting out his assets must consider all of five items in setting a pricing schedule. Each lender (investor) will have a different yield curve (pricing schedule).

Long-term leases of depreciating assets should be cheaper (per day). Long-term leases of assets expected to rise in value should be more expensive (per day), even if no change in the value of money or interest rates is expected. Renting a building for five years should paradoxically carry a higher daily fee than renting a car for same amount of time, if default-risk is removed (with insurance).
Lower liquidity of the rented (loaned) asset flattens the curve, moving it towards inversion, because of the risk that the asset may sit idle between rentals. The leasing company has an incentive to avoid idle time by charging a lower daily rate on long-term rental agreements. 
Increased depreciation expectations flatten the curve for rental-car loans, making long-term rates lower relative to short-term rates, because cars exponentially decay to zero value over time. The faster the asset loses value, the cheaper long-term rentals become (relative to short-term rentals). However, if payments are made in the same asset, rates must be higher over longer periods (curve becomes steeper).

To see why, imagine that you and I both own 100 sealed bags of grain, each weighing 100 grams. Each year, mice are expected to pilfer 5% of the grain from each bag, until all of them are nearly empty. If I want to borrow all of your bags (10,000g) for a year, and wish to pay interest payments with my unopened bags, how much would you ask me to pay? Because the bags will be 5% lighter, and will weigh 95g each, you would need to be paid at least 105.26 bags (10,000/95) just to receive the same amount of grain (value) you loaned to me a year ago. The interest rate in whole bags of grain would have to be 5.26% per year, if I wished to repay exactly the amount of grain I borrowed.

Now imagine I want to borrow them for two years. What would you charge then? The bags would after two years contain 90.25g of their original grain (value), and they would have lost 9.75g of their original weight (value). To repay you I would need to give you 10,000g of grain(value), or 110.8 bags, which is 5.26% more than the 105.26 bags I owed from the first year. In the third year, the bags weigh 85.7375g each, and you would have to receive 116.64 bags.

So even if we know the exact inflation rate of our grain-bags, borrowing them for one year will cost 5.26 bags, and borrowing them for three will cost 5.55 bags per year (116.64/3). The cost to borrow for longer periods must be higher because payments are made in the same asset, depreciating at the same rate. Over time, the payments become devalued as well, which is not the case with car rentals, where dollars hold their value relative to the leased asset.
Uncertainty about the future does not make the curve steeper. Long-term rentals of cash seem like they should have higher fees because it is easy to judge the risk of a loan today, but very hard to predict risk of default in the distant future, but because of credit default swaps (insurance), the risk is mitigated. Reduced solvency of insurers affects the slope, not the solvency of borrowers.
The yield curve does not indicate where investors think interest rates will move in the future. It is a reflection of the pricing schedules of firms and individuals that rent out their dollars, and those pricing schedules are based on the five points above. The curve does not represent the expected change in inflation rates, but shows the current opinion regarding the average inflation rate in the future. It is a snapshot of lender opinions regarding how much grain the mice are expected to steal from the bags, on average.
Flat yield curves do not indicate an economic slowdown.  They indicate that people don’t expect any grain to be stolen from the bags in the future; inflation expectations are zero.

Liquidity preference does not affect the yield curve of bonds, because  bonds are also highly-liquid assets. Rental-car contracts are very illiquid. It would be difficult for a rental company to sell a rental contract, midway through a car loan. They must wait for the contract to reach maturity to get their car back. Bond holders don’t have to wait to get their dollars back; they can easily sell the bond in organized bond-markets.

Bonds that are not heavily traded would have a lower price, and thus a higher interest rate across all maturities, because lenders would be trading highly-liquid assets (dollars) for a less-liquid asset (bond). Similarly, a person trading a gold brick for gold coins may accept slightly less total weight in coin, because he values their liquidity; 15.9 oz of gold may trade for a one-pound brick.

High interest rates do not indicate high inflation expectations. If all rates are high and the curve is flat, inflation is expected to be zero. If all rates are high it indicates that there is a high demand for dollars. This is similar to high prices for rental cars -- it does not indicate that the cars are expected to depreciate (inflate) faster than normal, it means that people have a greater need for cars and rentals cost more. The shape of the curve indicates inflation expectations and the general level indicates how badly businesses and individuals want to borrow money.