Fixed Income Flashcards
Spot and Forward Rates
Spot Rate: Yield of a zero coupon bond today
Forward Rate: Annualized interest rate of loan to be initiated in the future
- f(j,k) is notion, where j = time until the loan is initiated and k = length of loan once initiated
- f(j,k) = ((1+S_(j+k))^(j+k) / (1 + S_j)^j)^1/k -1, can plug in j = 2 and k = 3 to make cleaner
- f(2,3) = ((1+S_5)^5)/ (1+S_2)^2)^1/3 -1, basically taking the total return from 5 year spot rate divided by total return of 2 year spot rate and than annualizing by taking the return to 1/k
- (1 + Spot rate in 5 years)^5 = (1 + Spot rate in 2 years)^2 + (1 + Forward in 2 years for 3 years)^3
Managers make money by predicting that forward rates are not perfect indictors of future spot rates. If a manager believe that the future spot rates will actually be lower than the forward rates today they should buy a bond today because they will be locking in the higher yield today and in the future they believe that yield will actually be lower
Swap Rate Curve, Rolling Down, Factors Driving Yield Curve (Effective & Key Rate Duration)
Rolling Down Yield Curve: If yield curve is upward sloping than an investor can create excess return by buying long term bonds as they can lock in a high yield than short term bonds and as the bonds mature you will be still having higher yields than short term bonds. This trade fails though when the yield curve inverts and is no long downward sloping then you are no longer getting higher yield but you have more risk
Swap Rate Curve: Benchmark for interest rates, rates represent the fixed leg of an interest rate swap. Preferred over govt yield curve since it is reflecting the credit risk of banks rather than the govt, its not regulated by govt, and there are more maturities available for observation
Effective Duration: Bond price sensitivity to parallel shift in all rates
Key Rate Duration: Bond price sensitivity to a single rate change while other rates are all held the same. (Think of it as the bonds sensitivity to the one Key Rate)
Bond prices will also be sensitive to 3 shape movements: Parallel, Steepness, and Curvature
Swap Spreads (Z, TED, & LIBOR-OIS) & Interest Rate Theories (Pure, Local, Liquidity, Segmented, Preferred Habitat)
Swap Spread = Swap Rate - Treasury Bond Yield, represents the credit risk taken by buying the bond vs buying a govt bond
Z-Spread: Spread added to each spot rate to make the PV(Bond CF) = Bonds Market Price, which implies that the interest rates will not change at all and that there is Zero Volatility
TED Spread = Three Month LIBOR - Three Month Treasury, measures the entire economies credit risk
LIBOR-OIS: Measures economy risk (LIBOR) vs bank risk (Overnight Indexed Spread) to give an idea of the health of banks. If spread is high than there is a fear that banks will not be able to pay down their short term debt obligations
Interest Rate Theories:
Pure Expectations: Forward rates are perfect predictors of future spot rate, no risk at all
Local Expectations: No risk in the short term (bonds should all return Risk Free Rate), but in the long term there will be a risk premium needed
Liquidity Preference: Investors prefer liquidity and will require a premium for longer maturity bonds relative to how long the maturity is
Segmented Markets: Yield curve is divided into three markets, short/medium/long and investors will only ever pick one market to be apart of.
Preferred Habitat: Investors will prefer to stay in one market however if the return is adequate enough in another segmented market they will switch into that market
Economic Bond Factors & Investor Actions
2/3 of bond yields in short/medium term are driven by inflation with the other 1/3 being driven by Monetary Policy and GDP growth.
Bond Risk Premium: Also known as Term Premium or Duration Premium, govt bonds that are of longer maturities will require a premium vs govt bonds that are shorter maturity
Monetary Policy Influences on Yield Curve:
Bearish Flattening: Economic boom, fed wants to fight inflation and cool off economy so they will raise rates in the short term, flattening out the yield curve
Bullish Steepening: Economic recession, feed wants to kickstart economy so they will lower rates in the short term, making the yield curve more steep
Bullish Flattening: Market turmoil, investors will go to safe investments like long term govt bonds which will flatten the long term yield curve and make them less attractive
If investors thinks rates are going to go up they should reduce the duration (interest rate risk) in their portfolio, if steepening is expected an investor should buy short term bonds and sell long term bonds because they will be able to get better rates once the short terms mature
Bullet Portfolio: Investing in bonds of one maturity
Barbell Portfolio: Investing in bonds of both short term and long term maturity
Investors who expect a bullish flattening may go from a bullet to a barbell portfolio
Arbitrage Opportunities (Value Additivity and Dominance) & Term Structure Models
Two types of Arbitrage Opportunities:
Value Additivity: Parts are not equal to final product, therefore can either strip down the final product into parts and sell or reconstruct parts into final product if they are cheaper
Dominance: Two identical assets are not equal to each other therefore buy one and sell other
Term Structure Models: Explain how interest rates move and why
Equilibrium Models: Assume the economy has long run interest rate (b) and short term interest rate (r), overtime r will always converge to b, just depends on how quick it converges which is just a factor (a). Therefore ∆r = a(b-r)*∆t + error term. CIR model and Vasicek both explain same thing, only difference is Vasicek assumes interest rate volatility is independent
Arbitrage Free Model: Calibrate using market prices to get drift term which is used to ensure no arbitrage opportunity exists, Ho-Lee and KWF are same except KWF assumes model is lognormal.
Gauss + Model: Uses short (random) / medium (mean-reverting) / long (macroeconomic) rates to predict and explain interest rate movements
Binomial Interest Rate Tree
Zero Coupon Bonds can use yield curve but option embedded bonds can’t due to option value not being tied to the yield curve entirely, instead use binomial interest rate tree.
Binomial Interest Rate Tree: Values using backward induction as we will know the ending value of the bond (interest + principal) as well as the rates, therefore we can discount back each period with the rates to get the original value. The tree also assumes a 50/50 chance to go up or down with rates.
First step is to discount back ending value at cash paid out (interest + principal) to get our values, then use these values to discount back to get combining nodes. Remember 50/50 so we need to take the value of top and bottom node and average them together then discount to get the value we are looking for.
There are 2^n-1 total paths in an interest rate tree, n is the number of periods
Monte Carlo: A lot of random paths generated, used for Mortgage Backed Securities as they can’t use binomial tree as they have path dependent cash flows
Callable and Putable Bonds Calculation and Value & Effective Duration
V_Call = V_Straight - V_Callable (Call option has negative value to the investor)
V_Put = V_Putable - V_Straight (Put option has positive value to the investor)
Calcing value of callable or putable is same idea as binomial interest rate tree except that call will put a price cap of 100 on any node in the tree while a put will put a price floor of 100 on any node in the tree since at any point the investor could exercise there options (take note of times when an option becomes available to the bondholder as well)
As volatility goes up both call and put values will go up, so the callable bond value will go down while the putable bond value will go up.
Call option has value when yield curve is flattening because that means rates are going down in future which means today’s bonds are worth more however with a call option in place the max amount those bonds can be worth is 100
Put option is opposite of this, it will have value when yield curve is upward sloped because it means rates will be higher in future which means that the bonds today will be worth less in future but the put option puts a floor on bond value of 100
Effective Duration = (Bond Value if rates go down - Bond Value if rates go up) / 2 * Bond Value_0 * change in rates, basically measures how much a bond price is going to move given a 100 bps movement in rates
Callable and Putable bonds will have less duration than straight bonds as they have protection to one side of the value (either rates going down for callable or rates going up for putable)
Zero coupon bond duration will depend on maturity as the rate that matter is the one when the bond matures since its not paying anything before than and its value will be if the coupon it pays at maturity will be less than or greater than the rate at the time of maturity
Floater coupon bond duration will depend on the time it resets for the same reason because it will reset and will depend on what rates are at that time
Option Adjusted Spread (OAS) & One Sided Duration
Option Adjusted Spread: Constant spread added to option embedded bond to make equal to market price, identical to Z-Spread if there is zero-volatility in interest rates. However when there is volatility introduced the OAS will go up for putable bonds and down for callable bonds. This is because when a node is putable we need to add more option adjusted spread and when a node is callable we need to subtract more option adjusted spread
One Sided Duration: When these options are near-the-money or at-the-money they will have higher one sided duration in a certain direction. For putable bonds they will have a high down duration because if interest rates go up they will just put the bond and value will not go above 100 but if rates go down the value of the bond will increase and that is not capped. Opposite for callable bonds, they will have high up duration because the value can’t go above 100 even if rates drop alot but if rates go up alot the value of the bond can fall.
Low coupon callable/High coupon putable are unlikely to be exercised because bond issuer will want to pay low coupon/bond holder will like getting high coupon. Due to this the key rate for these bonds will be the maturity rate, that will be the rate that drives the value of the bond. (if bond matures in 3 years than the rate in 3 years vs current coupon will drive what the value of the bond is)
High coupon callable/low coupon payable are likely to be exercised because they are unattractive for both sides that have option. Due to this the key rate that will drive the value of the bond will be the rate at the time when the option becomes available for exercising
Convexity, Capped & Floored Floater & Convertible Bonds
Convexity: All bonds will have positive convexity except callable at low rates because no matter how low the rates go the callable bond will never go above 100 in value as it will be called
Capped Floater prevents a resetting coupon to go abode a certain max
Value of capped floater = Value of straight floater - value of cap (negative value to investor)
Floored Floater prevents a resetting coupon to go below a certain minimum
Value of floored floater = Value of straight floater + value of floor (positive value to investor)
Convertible bonds can be converted into share of stock in exchange for the bond. This gives the investment downside protection because you will get stock if the stock price goes up but if it stays the same or goes down you will keep the bond. Downside to convertible bond is when the stock does go up you will get less of a return vs just buying the stock because you needed to pay a premium for the bond. Convertible bond gives downside protection at the cost of some part of potential upside.
Bond Value = Straight Bond Value + Value of Convertible Option - Value of Callable Option + Value of Putable Option
Converting and Putting are good for investor, having bonds called is bad
Credit Risk Measures (Expected Exposure, Recovery Rate, Loss Given Default (LGD), Probability of Default, CVA), Credit Score/Rating, Credit Migration & Structural Models
Credit Risk Measures:
Expected Exposure is total amount at risk if default occurs
Recovery Rate is the % that you are able to recover if default occurs
Loss Given Default (LGD) is % you didn’t recovery * exposure
Probability of default is the chance that you will default in any given year, this probability will most likely change over the life of a bond as economic conditions change
Probability of survival is the chance that the bond survives its entire life, if 98% chance to survive each year for 5 years than do (.98)^5
Credit Valuation Adjustment (CVA): The sum of the PV of the expected losses each year (find your Loss Given Default and take the % chance that happens to get your expected loss then take PV of that). Risky bond value = risk free bond value - CVA, the CVA is the expected amount you are possibly going to lose
The higher the expected loss on a bond the more credit risk there is
Credit Ratings are for firms while Credit Scores are for small business/individuals, higher score = better
Credit Migration, when a bond is up or downgraded a portfolio manager will want to know how much they can expect to gain or lose.
△%Price = -(modified duration of bond) * △spread, duration is amount it would move based on 100 bps movement, price will go up if spread goes down, price will go down if spread goes up
Structural Models put option value on a balance sheet.
Stock is viewed at as a call option because if the Assets > Liabs you will get the value of Asset-Liabs however if that is the not the case you simply have 0 value, not on the hook. Debt is viewed as a put option because you will get the minimum of the value of your debt or the value of the assets in the firm if assets < liabs. You have the option to put your debt and force the firm to sell their assets to pay you.
In theory this is good however in practice there are a lot of other variables involved and debt can’t just force asset sales most of the time. Also don’t have a super in depth look at financials.
Reduced Form Models, Credit Spread Term Structure, and Securitized Debt
Reduced Form Models: Explains when a default is going to occur rather than why. Uses a regression model with a lot of variables to predict when a firm is nearing default as the state of the economy changes. Default is usually viewed as a surprise however it is actually very predictable with these models
Credit Spread is the difference between the YTM of a risk free bond and the YTM of a risky bond, the two curves will be plots one on top of the other. If the gap widens that means that the risky bond is now more risky.
Shape of a credit spread will depend on the following:
Quality of Bond (if bond is good quality than the spread should be narrow)
Overall Economic Conditions (if economy is doing good than spreads should narrow)
Supply/Demand of Bonds (if there is more demand than supply of debt than spreads will narrow)
Equity Market Volatility (if there is high volatility than credit spreads will widen as bonds become more risky)
Securitized Debt: Debt that is backed by a specific asset or collateral usually for a specific purchase through a Special Purpose Entity. Value of this debt depends on how much collateral is backing it, the quality of the debt issuer, and the structure of payments for the debt (who gets paid first). Covered Bond is a type of securitized debt, it is a bond that is back by both an asset and the issuer of the bond so recourse is available
Credit Default Swaps (CDS)
Credit Default Swaps: One side buys protection from a credit event (bond defaulting) on their debt for a premium while the seller insures the buyer that they will be made whole if the bond defaults. Seller of the CDS is long credit risk while Buyer is short credit risk
Payoff is the cheapest to deliver bond that is similar seniority to the bond that is insured
Index CDS: CDS on an index where a seller will insure the entire index, if one of the members of the index has a credit event than a payment is made and the principal is reduced
CDS Pricing: Expected Loss = Hazard Rate * Loss Given Default, Hazard rate is default chance given that bond has not defaulted in the previous periods
Upfront Payment by buyer = PV(protection leg) - PV(premium leg) or
Upfront Premium = (CDS spread - CDS coupon) * CDS duration
After swap is initiated the potential profit that the buyer will receive will be △spread * CDS duration, value of the swap will change by this amount since its tied to spreads
CDS can be used to speculate (naked CDS), long/short two different reference positions or Curve Trade. In a curve trade you will either believe the spread will widen or narrow in the future, if you believe the spread will widen you will long short term and short long term of the same bond since the spread will be greater in the future. If you believe it will narrow you will short short term and long long term since the spread will be lower in the future
