Accounting Fundamentals for Financial Institutions Midterm
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BondsBondPricingInterestRateSensitivity.pptxAsset and Liability Management
FIN 6102
Ferriter – Spring 2018
Debt Securities
A Debt Security is a claim on a specified periodic stream of cashflow. Debt Securities are often called fixed-income securities because they promise either a fixed stream of income or one that is determined by a formula.
A typical bond requires semi-annual payments for the life of the bond. These are called coupon payments. The interest rate that determines the coupon payment is called the coupon rate.
When a bond matures, the issuers repays the debt by paying the bond’s par value (also known as the face value).
If no face value is given, assume $1000 for the face value
Zero-Coupon Bonds
Bonds are usually issued with a high enough coupon rate to induce investors to purchase the bond. However there are zero-coupon bonds, where the buyer only receives the face value at the maturity date but receive no coupons payments.
When these bonds are issued they are priced considerably lower than the par value. They may be issued by federal, state or local governments or by corporations. Then there are the tax exemptions. If issued by a government entity, the interest generated by a zero-coupon bond is often exempt from federal income tax, and often from state and local income taxes too.
Accrued Interest and Quoted Prices
The bond prices quoted in financial papers are not the actual prices that an investors pays for the bond. This is because the quoted price doesn’t include the interest that accrues between coupon payments.
Accrued Interest = Annual Coupon Payment/2 *Days since last coupon/Days between payment
For example, a bond with a coupon rate of 8%. The annual coupon payment is $80 and the semi-annual coupon is $40. 30 days have passed since the last coupon payment. What is the the accrued interest?
$40 x (30/182) = $6.59
This value would be added to the quoted price when the bond is sold.
Call Provision and Callable Bonds
Some corporate bonds are issued with a call provision. A call provision allows the issuers to repurchase a bond at a specified call price before the maturity date. Why might a company issue a callable bond?
When corporate bonds are issued in a high interest rate environment, they will most likely issue bonds with a high coupon rate. Over the course of the bonds life, interest rates might fall. A corporation might take advantage of the call provision to retire the bond early and issue a new bond at a lower coupon rate
Callable Bonds typically come with a period of call protection, an initial period of time when the bond cannot be called.
The option to call a bond is useful to the issuer, but what is beneficial to an issuer is detrimental to bond holders. To compensate for this, callable bonds are often issued with higher coupons and promised yield to maturity than non callable bonds.
Convertible bonds
Convertible bonds give bondholders an option to exchange each bond for a specified number of shares. The conversion ratio is the number of shares for which each bond may be exchanged. For example, a convertible bond is issued at a par value of $1,000 and is convertible in 40 shares of stock. If the current stock price is $20 it is not profitable to convert. As $20*40 = $800 is less than the par value of the bond. If the stock price increases to $30, the conversion is profitable ($30*40 = $1,200)
The market conversion value is the current value of shares for which the bonds may be exchanged.
Convertible bond holders benefit from price appreciation of the company’s stock. Because of this benefit, convertible bonds tend to have lower coupon rates and lower yields to maturity.
Floating Rate Bonds
Floating rate bonds make interest payments based on some measure of current market rates. For example, the might might be adjusted annually to the current T-bill rate plus 2%. If the One year T-Bill rate is 4%, then the coupon rate over the next year will be 6%.
The major risk with floating-rate bonds is that while the spread between the market interest rate and the floating coupon rate is fixed, the adjustment is not connected to changes in the issuing firm’s financial position. For example, if a company runs into financial distress, investors might demand a greater yield premium than is offered by the security. In this case the price of the bond will fall.
Bond Pricing
Bond Value = Present Value of Coupons + Present Value of par value
For example consider an 8% coupon, 30-year maturity bond with a par value of $1,000, paying 60 semiannual coupon payments of $40 each. Suppose that the market interest rate is 8% annually.
The present value of all the coupon payments is $904.94 and the present value of the par value is $95.06. When the market interest rate is equal to the coupon rate, the par value and equals the bond price.
If the market interest rate were to rise to 10%. The bond price would fall by $189.29 to $810.71
Yield to Maturity
As you may have guessed, the coupon rate and the market interest rate are rarely equal. As such bonds usually do not sell for par value. Therefore we need to measure or rate of return that accounts for both current income (coupon payments and the return of principle) and the bond price increase or decrease over the lifetime of the bond.
The yield to maturity is the interest rate that makes the present value of a bond’s payments equal to its current price. This interest rate is interpreted as the average rate of return of a bond if held to maturity.
Example: An 8% coupon, 30-year bond is selling at $1,276.76. What is the yield to maturity?
In this example, the semiannual yield to maturity is .03. However, yield to maturity is generally quoted as an annual figure. Therefore we must annualize the yield. 1.03^2 =1.0609.
In this case the yield to maturity in 6.09%
YTM vs Current Yield
A bond’s yield to maturity (YTM) is the internal rate of return on the investment in the bond. YTM differs from the current yield of a bond. The current yield of a bond is defined as the annual coupon payment divided by the current price of a bond.
For example, take the 8% coupon bond selling for $1,276.76. The current yield would be 80/ $1,276.76 or .0627 (6.27%). Recall that the YTM on this bond was 6.09%. This bond is considered to be selling at a premium. In this case the coupon rate is higher than the current yield which is higher than the YTM.
The reason for this is that the Coupon rate is divided par value, the current yield is divided by the current price. The current yield is higher than the YTM because the YTM accounts for the capital loss as the bond will eventual repay only $1,000 at maturity.
As a general rule, for premium bonds the Coupon rate > Current Yield > YTM. For discount bonds the relationship is reversed.
Yield to Call
Yield to maturity assumes that the bond will be held to maturity. However, a bond maybe be retired prior to that time. This is especially true if the bond has a call provision. For example, a $1,000, 30 year bond with a coupon payment of 8% has a callable provision at 110% of par value and has call protection for 10 years. If the bond currently sells for $1,150 what is the yield to call?
In this example, the Yield to Call (YTC) is equal to 6.64% (3.32 x 2) and the YTM is 6.82%
| Yield to Call | Yield to Maturity | |
| Coupon Payment | 40 | 40 |
| Number of Periods | 20 | 60 |
| Final Payment | $1,100 | $1,000 |
| Price | $1,150 | $1,150 |
Zero-Coupon Bonds
US Treasury Bill are common for of short term zero-coupon bonds. Since these have no coupon the return on these bonds is completely from price appreciation. However, Long-Term Zero coupon bonds are usually created by separating a coupon payment stream from the principal repayment. An investor can request that the US Treasury split or strip the coupon payment from the principal repayment. In this case each component is assigned a CUSIP number, the CUSIP allows the each security to trade on the FEDWIRE system.
The process is governed by the US Treasury Program called STRIPS (Separate Trading of Registered Interest and Principal of Securities)
The primary purpose of strips is to appeal to different investor types. Much of the reason is related to cashflow matching.
Term Structure of Interest Rate
The term structure of interest rates refers to the the process of discounting cash flows of different maturities.
Most often investors will plot the YTM against Maturity. This is called the yield curve. There are three common “types” of yield curve; rising, flat and inverted. A rising yield curve is the most common in normal economic times, and it suggests that interest rates will rise in the future. The yield curve is derived from plotting zero-coupon bonds.
| Maturity | YTM | Price |
| 1 | 5% | $952.38 |
| 2 | 6% | $890 |
| 3 | 7% | $816.30 |
| 4 | 8% | $735.03 |
Yield Curve and Future Interest Rates
In order to derive the Yield curve we’ll be making some assumptions. First, we assume that there is no possibility of arbitrage. Under this scenario, yields from must be identical.
To see what this means, consider the following. There are two strategies, a choice to buy and hold a two year zero-coupon bond and to buy a one year zero coupon bond and reinvest in another one year zero coupon.
For the two year bond, assume a 6% market rate for a holding period of years. In this case the bond price would be $890. For the second strategy, assume the interest rate for a one year holding period is 5%. What will the one year rate be in one year
We can set this up as the following:
$890 x 1.062 = $890 x 1.05 x (1+r)
Or 1.062/1.05 = 1+r = 1.0701 or 7.01%
Finding Future Short Rates
Now let’s compare a three year strategy. One will be to purchase a 3 year zero coupon bond with a YTM of 7%. This bond would be priced at $816.30
The alternative strategy is to buy a 2 year zero and reinvest in a 1 year zero.
We can structure this as follows
1.073 = 1.062 x 1+r
1.073/1.062 = 1+r
This equals 1.09025 or 9.025%
Additionally, 1.07 = (1.05 x 1.0601 x 1.09025)1/3
Interest Rate Types
One thing you might have noticed is that there are a lot of interest rates when doing these calculations. In order to differentiate between them, investors coined two terms to describe them.
The spot rate refers to the yield to maturity for a zero coupon bond that prevails today.
The forward rate (also called the forward yield) is the theoretical, expected yield on a bond several months or years from now. It is common to denote a forward rate as xRy this can be read as the x forward rate y years from today.
Theories of the Term Structure
The expectations hypothesis is the theory that the forward rate equals the market consensus expectations of what future short-term rates will be. This means that there is no liquidity preference.
This can be stated as E(r) = f2 or that the expected rate in period two will be the rate in period two.
Using this hypothesis is what allows use to generate a yield curve, as no other information is required aside from current spot rates.
Theories of Term Structure
Liquidity Preference: Essentially states that investors or buyers of fixed income securities have a preference of either short term or long term securities. Part of the reason for these preferences can be seen as attempts to match cash inflows to cash outflows.
In any event, short term investors would require a premium to invest in longer term securities, or the f2 > E(r2) or the rate in period 2 must be greater than the expected rate in period 2.
Advocates of liquidity preference believe that short term investors dominate the market which pushes the short term interest rate down.
Interest Rate Risk
As we’ve seen bond prices move inversely to changes in the market interest rate. Therefore, bond investors are particularly concerned with the sensitivity of bond prices. We also note that bond prices are convex and therefore decreases in YTM have bigger impacts on price than increases in YTM for the same magnitude. A summary of bond price observations
Bond prices and yields are inversely related.
An increase in a bond’s yield to maturity results in a smaller price change than a decrease
Prices of long-term bonds tend to be more sensitive to changes in interest rate.
Sensitivity to price changes increases at a decreasing rate. A 30 year bond is not 6x more sensitive than a 5 year
Interest rate risk is inversely related to a bond’s coupon rate
The sensitivity of a bond’s price to a change in its yield is inversely related to the YTM at which the bond is currently selling.
(Macaulay’s) Duration
Frederick Macaulay termed the effective maturity concept the duration of a bond. Macaulay’s Duration equals the weighted average of the times to each coupon or principal payment. The weight associated with each payment time should be related to the “importance” of that payment to the value of the bond. Timing of cash flows is designated in years
Wt = PV of CFt/Bond Price
D = Σ T x Wt
Duration Example
| Time until | PV of CF | Column C | ||||
| Payment | Discount rate = | times | ||||
| Period | (Years) | Cashflow | 5% per period | Weight | Column F | |
| A. 8% Coupon Bond | 1 | 0.5 | 40 | $ 38.10 | 0.039496 | 0.0197 |
| 2 | 1 | 40 | $ 36.28 | 0.037615 | 0.0376 | |
| 3 | 1.5 | 40 | $ 34.55 | 0.035824 | 0.0537 | |
| 4 | 2 | 1040 | $ 855.61 | 0.887065 | 1.7741 | |
| Sum: | $ 964.54 | 1.8852 | ||||
| B Zero-Coupon | 1 | 0.5 | 0 | $ - | 0 | 0.0000 |
| 2 | 1 | 0 | $ - | 0 | 0.0000 | |
| 3 | 1.5 | 0 | $ - | 0 | 0.0000 | |
| 4 | 2 | 1000 | $ 822.70 | 1 | 2.0000 | |
| $ 822.70 | 2.0000 |
Why is duration important
There are three primary reasons why duration is important.
First it is a simple summary statistic of the effective average maturity of the portfolio
It is an essential tool in immunizing portfolios from interest rate risk
Duration measures interest rate sensitivity of a portfolio
Modified Duration
Reason for – sign The price-yield relationship is negatively correlated; when prices go down, the implied yield goes up. The minus sign allows the modified duration to be positive for a normal bond.
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Modified Duration Example
Consider the 2-year maturity, 8% coupon bond, selling at a price of $964.54 for a YTM of 10%. The semi-annual duration of this bond is 1.8852 years. The annual duration is 3.7704. Therefore modified duration is 3.7704/1.05 = 3.591.
No suppose that the semiannual interest rate increases to 5.01%
-3.591x .01% = -.03591%
This would be interpreted as a .01% increase in interest rates would cause the price of a bond to fall by .03591%
Rules for Duration
The duration of a zero-coupon bond is its time to maturity
A coupon bond can have a duration less than one.
Holding maturity constant, a bond’s duration is lower when the coupon rate is higher
A bond’s duration increases with its time to maturity.
The duration for a coupon bond is higher when the bond’s yield to maturity is lower
Duration of a perpetuity (1+r)/r
Convexity
Convexity Formula
Duration with Convexity Example
Immunization
Immunization techniques refer to strategies used by investors to shield their overall financial status from interest rate risk.
Many banks and thrifts have naturally occurring mismatches between their liabilities and assets. Much of there liabilities are deposits that are short term and have low duration. Bank assets are primarily consumer and commercials loads and have higher duration. This means that banks are sensitive to changes in interest rate as they can directly effect new worth.
Immunization Example
Consider an insurance company offering a Guaranteed Investment Contract for $10,000 and guarantees and interest rate of 8%. If the GIC has a maturity of 5 years. The future value of the liability is $14,693.28.
If the company chooses to fund the liability with a 10,000 8% coupon bond. Then as long as interest rates remain at 8% the liability will be exactly matched.
But what happens when interest rate rise or fall?
Immunization Example
| Payment Number | Years Remaining until Obligation | Accumulated Value of Invested Payment | ||
| A. Rates Remain at 8% | ||||
| 1 | 4 | 800 x (1.08)^4 | = | 1088.391 |
| 2 | 3 | 801 x (1.08)^3 | = | 1007.77 |
| 3 | 2 | 802 x (1.08)^2 | = | 933.12 |
| 4 | 1 | 803 x (1.08)^1 | = | 864 |
| 5 | 0 | 804 x (1.08)^0 | = | 800 |
| Sale of Bond | 0 | 10800 | = | 10000 |
| 14693.28 | ||||
| A. Rates Fall to 7% | ||||
| 1 | 4 | 800 x (1.07)^4 | = | 1048.637 |
| 2 | 3 | 801 x (1.07)^3 | = | 980.0344 |
| 3 | 2 | 802 x (1.07)^2 | = | 915.92 |
| 4 | 1 | 803 x (1.07)^1 | = | 856 |
| 5 | 0 | 804 x (1.07)^0 | = | 800 |
| Sale of Bond | 0 | 10800 | = | 10093.46 |
| 14694.05 | ||||
| A. Rates Fall to 7% | ||||
| 1 | 4 | 800 x (1.09)^4 | = | 1129.265 |
| 2 | 3 | 801 x (1.09)^3 | = | 1036.023 |
| 3 | 2 | 802 x (1.09)^2 | = | 950.48 |
| 4 | 1 | 803 x (1.09)^1 | = | 872 |
| 5 | 0 | 804 x (1.09)^0 | = | 800 |
| Sale of Bond | 0 | 10800 | = | 9908.257 |
| 14696.03 |
Immunization
In this example because we have successfully matched the duration of our asset with our liability we can be considered immune from interest rate risk. However, immunization investors have a risk trade off between price risk and reinvestment risk.
Price risk is essentially the capital loss or gain that occurs because of a change in interest rate. Now a key consideration with immunization is the need to rebalance. As interest rate change a portfolio manager must rebalance as the duration will have changed.
Additionally, even if interest rates stay the same durations will change solely because of the passage of time.
Constructing and Immunized Portfolio
A bank must make a payout of $19,487 in seven years. The current market interest rate is 10%, so the PV of the payout is $10,000. The portfolio managers wants to fund the obligation with a 3 year zero coupon bond and a perpetuity paying 10%. How can the manager immunize the portfolio?
Calculate the duration of the liability. In this case because it is a single payment obligation, the duration is 7 years
Calculate the duration of the asset portfolio. The portfolio duration will be the weighted average of each assets duration. In this case the zero coupon bond has a duration of 3 years and the perpetuity a duration of 11 years.
Asset Duration = w * 3 + (1-w)*11
Find the asset mix that sets the duration of assets equal to 7 years
w * 3 + (1-w)*11 = 7 in this case w = .5
Fully fund the obligation. In this case it means that 5,000 should be invested in the zero coupon bond and 5,000 should be invested in the perpetuity.
Duration Gap Analysis
Duration Gap:
From the balance sheet, A = L+E, which means E = A-L. Therefore, DE = DA-DL.
In the same manner used to determine the change in bond prices, we can find the change in value of equity using duration.
DE = -[DA - DLk]A(DR/(1+R))
DLk is total liabilities / (Total Liabilies + Equity) or the proportion of assets funded by liabilities
Suppose a manager has the following situation:
Duration of Assets = 5 years and Duration of Liabilities = 3 years
The current interest rate is 10% and it is expected to rise to 11% what is the impact on the net worth of the company
Duration Gap Analysis
| Assets | Liabilities and Equity |
| Assets 100 | Liabilities 90 |
| Equity 10 |
First let’s calculate the potential impact on net worth.
DE = -[DA - DLk] x A x (DR/(1+R))
-[5-3*.9] x 100 x (.01/1.10)
or -2.09 decrease in equity
What about Asset and Liability Accounts?
DA = -5(.01/1.10) = -.04545 = -4.545% = 95.45
DL = -3(.01/1.10) = -.02727 = -2.727% = 87.54
New Balance Sheet
| Assets | Liabilities and Equity |
| Assets 95.45 | Liabilities 87.54 |
| Equity 7.91 |
Duration Gap Example
Duration and Repricing Gap Example
