New material covered in class from Weeks 1 through 5 and in the Week 6 in-class problem review
New material introduced in Week 6, while useful for understand the previous material, is not explicitly covered on the mid-term
The main focus is what is covered in class and a basic understanding of the world’s current financial situation (exchange rates, yields, the Fed, etc.)
Although you should try doing Week 5 problems yourself before looking at the answers, the answers are already posted here
It contains within it a forecast of future interest rates and provides the prices of forward rate agreements (FRAs) and interest rate swaps
The bond yield curve that is typically given in newspapers is not the one used by professionals because it is complicated by the coupon payments of the bonds
Professionals use the pure yield curve based on zero-coupon securities
It is a reminder that despite the fact that most of financial analysis assumes all cash flows from an investment can be discounted using the same interest rate, doing so it sometimes unwise
Estimates of forward interest rates are there for the taking
An example of the basic idea: If you know the yield out 2 years ( y 2 ) and the yield out one year ( y 1 ), you can reverse engineering the what the one-year yield should be one year from now (this is an implied forward rate )
The same idea works between any two points in time
BKM introduce the idea of a short interest rate (itself a confusing term)
As BKM note repeatedly, these rates do not really exist in reality, they are purely theoretical constructs that act just like the forward rates that can be implied from the pure yield curve
If they help you understand things, that is fine; otherwise, feel free to ignore them
$1,000 invested in the 2-year 8.995% zero-coupon bond gives: $1,000(1+ y 2 ) 2 = $1,000(1.08995) 2 = $1,187.99
$1,000 invested ion the 1-year 8.000% zero-coupon bonds gives: $1,000(1+ y 1 ) 1 = $1,000(1.08) = $1,080.00
Turning the $1,080 into $1,187.99 requires a one-year forward yield of: f 2 = $1,000(1+ y 2 ) 2 / $1,000(1+ y 1 ) 1 – 1 = $1,187.99/$1,080.00 – 1 = 10.00%
17.
The General Formula for Computing Implied 1-year Forward Rates
18.
Theories of the Term Structure of Interest Rates
(Unbiased) Expectations Hypothesis : At a given point in time, the yield curve reflects the market’s current expectations of future short-term rates
Liquidity Premium Theory : Investors will only hold long-term maturities if they are offered a premium to compensate for future uncertainty in a security’s value
Market Segmentation Theory : Investors have specific maturity preferences and will generally demand a higher maturity premium
19.
The (Unbiased) Expectations Hypothesis and Implied Forward Rates
Under this hypothesis, the forward rates implied from the yield curve are an unbiased (accurate on average) estimate of what interest rates will be in the future
Even without this hypothesis, the implied forward rates from the curves should be very close to the actual forward rates that prevail in the market
If they were not, profitable arbitrage could occur
In the example, the implied 1-year forward rate 1 year from now, f 2 , was 10%
If we were a financial institution , we would like to have a way to “lock in” a 10% one-year fixed rate on money we borrow for 1 year beginning 1 year from today (Note: This is a liability to the bank)
If the agreement involves a loan of $1,000,000, we would pay $1,100,000 a year later
21.
What if a Year from Now (When the Loan Begins) the Interest Rate Rises to 11%
Then, instead of paying $1,100,000 back on the loan, we would have to pay $1,110,000 back, which is $10,000 more
We could, however, be “made whole” (get back to the original 10% rate) if someone gave us $9,009, which is $10,000/1.11
At the end of the year, this $9,009 (at 11% interest) would become $10,000—making up for the extra $10,000 in interest we would need to pay out
22.
What if a Year from Now (When the Loan Begins) the Interest Rate Drops to 9%
Then, instead of paying $1,100,000 back on the loan, we would have to pay $1,090,000 back, which is $10,000 less
Now to get back to the 10% rate we would pay $9,174, which is $10,000/1.09
At the end of the year, this $9,174 (at 9% interest) would become $10,000—offsetting the extra $10,000 in interest we made when rates dropped
Very similar because most FRAs are quoted in terms of Eurodollars
Like FRAs, Eurodollar futures are settled in cash rather than through the actual physical delivery of a 3-month Eurodollar
Because it is a futures contract, cash flows related to posting and maintaining margin take place prior to the “delivery” date
25.
The PV of Fixed-Rate Securities Vary With Interest Rates
Obviously this is the case for a bond that pays a fixed coupon rate—rates go up and its PV goes down and vice versa
Somewhat less obviously this is the case for a bank that enters into an FRA that commits it to borrow money at a fixed rate
When rates go up, this commitment becomes less of a liability to the bank (someone else would be happy to pay the bank to be committed to borrow at 10%, for example, when the going rate is 11%
Suppose that the bank agrees to borrow $1,000,000 for a year one year from today; however, it will not commit to a rate now. Instead, it will pay whatever the prevailing rate is in a year
What would someone be willing to pay (or need to pay) to relieve the bank of this liability?
27.
Duration : A Measure of Interest Rate Sensitivity
The duration of an investment is a quantitative indication of its interest rate sensitivity that is measured in years
Higher duration means a greater sensitivity (in percentage terms) to changes in interest rates
Doubling duration means doubling the interest rate sensitivity
A bond with a duration of 6 years will drop twice as much (as a percent of its price) as a bond with a duration of 3 years when interest rates increase by 1 basis point (0.01%)
The higher the coupon payment, the lower is a bond’s duration
Duration and Yield to Maturity
Duration decreases as yield to maturity increases
Duration and Maturity
Duration increases with the maturity of a bond but at a decreasing rate
33.
Relationship Between Maturity and Duration on a Coupon Bond
34.
Duration, Zero-Coupon Bonds, and Bond Portfolios
For a zero-coupon bond, duration equals maturity
The duration of a portfolio of bonds is the value-weighted average of the durations of the individual bonds
A portfolio with $1,000 in a bond with a duration of 2 years and $2,000 in a bond with a duration of 4 years has a duration of [2(1,000)+4(2,000)]/(3,000) = 3.33 years
The fixed side of the swap has the standard positive duration of a fixed coupon bond—for short-term bonds this duration is a bit less than the bond’s maturity as we saw earlier
The floating side of the swap has only a slightly positive duration because the interest rate is reset frequently—if the interest rate were continuously reset, the floating side would have zero duration
The net duration is then almost that of the fixed-rate side
46.
Swaps from the Financial Institution’s Perspective
Interest rate swaps affect the duration of the financial institution’s liabilities
The buyer of a swap (the one who pays the fixed rate) increases the duration of the liabilities, which decreases the value of the duration gap
The seller of a swap (the one who pays the floating rate) decreases the duration of the liabilities, which increases the value of the duration gap
47.
An Important Point: The Assets Themselves Are Not Swapped
An interest rate swap is equivalent to a series (or bundle) of FRAs
Both parties to the swap continue to own the “swapped” assets
Only the cash flows are swapped
Each period during the swap, the difference in cash flows between the two assets is computed and the difference is paid
48.
Swaps Are the Prime Determinant of U.S. and Most Global Interest Rates
Swaps are the easiest way for large financial institutions to adjust their duration
Large swap transactions by influential market participants affect the entire yield curve
Hence, the influence of swaps on interest rates is similar to what we will see is the effect of S&P Index futures on the stock market
A large firm pays a fixed interest rate to its bondholders, while a smaller firm pays a floating interest rate to its bondholders.
The two firms could engage in a swap transaction which results in the larger firm paying floating interest rates to the smaller firm, and the smaller firm paying fixed interest rates to the larger firm.
55.
Diagram of a Plain Vanilla Swap Big Firm Smaller Firm Bondholders Bondholders LIBOR – 50 bp 9.55% 8.05% LIBOR +100 bp
Idea: Strip the credit risk off of corporate bonds
Pioneered by J.P. Morgan (before the Chase merger)
An important part of the process where banks got loans off their balance sheets in the late 1990s (a result of high loan reserve requirements)
68.
One Last Piece of the Interest Rate Puzzle to be Covered Next Time
The long-term debt market as seen by financial institutions in the U.S. is dominated not by corporate borrowing, but by mortgages and other forms of securitized debt
The origination of mortgages is what drives the swaps market (and, in turn, the other debt markets)
According to the Fed , in the second quarter of 2005, annualized home mortgage borrowing ($892.4B) was almost a third greater than all business borrowing ($672.0B)
Interest rate swaps are traded on the New York Stock Exchange.
If interest rates rise sharply after an interest rate swap has been initiated, the buyer of the swap will be the party to receive the difference checks for the swap.
In an interest rate swap, the duration of the fixed-rate side of the swap is higher than the duration of the floating-rate side.
Personal Credit Corporation (PCC) is a financial institution that specializes in making personal loans. Its current balance sheet appears as follows (all dollars are in millions and all durations are in years):
Assets Cash (duration=0) $ 400 Personal loans (duration= 1.4) $2,600
73.
A Duration Gap and Interest Rate Problem (continued)
All of PCC’s notes mature in 3 years. The floating-rate notes pay exactly LIBOR and are reset annually. LIBOR rates are currently the following:
1-year LIBOR: 4.32%
2-year LIBOR: 4.45%
3-year LIBOR: 4.51%
74.
A Duration Gap and Interest Rate Problem (continued)
What is the implied 1-year forward rate for LIBOR for funds invested 1 year from now?
Using the appropriate LIBOR rate for discounting, if PCC’s zero-coupon notes maturing in 3 years and have a face value of $1,500 (all numbers are in millions), what is its PV to the nearest million?
What is the duration of PCC’s zero-coupon notes?
75.
A Duration Gap and Interest Rate Problem (continued)
Based on your answers in 2. and 3., what is PCC’s duration gap?
Should PCC swap some of its floating-rate notes for fixed-rate notes in order to reduce its duration gap?
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