Interest Rates, Discount Rates, Nelson-Siegel and Term Structure Flashcards
What is a bond?
A bond is a loan where an investor lends money to an entity in exchange for periodic interest payments and the return of principal at maturity.
What is the face value or par value of a bond?
The amount the bondholder receives at maturity, typically $100.
What is a coupon rate?
The percentage of the face value paid to bondholders annually as interest.
What is the maturity date of a bond?
The date when the bond’s face value is repaid to the investor.
How are bonds different from stocks?
Bonds are fixed income, lower risk and are a debt investment, while stocks are variable returns, ownership in a company and thus higher risk.
What is the Time Value of Money?
The principle that money today is worth more than the same amount in the future due to its earning potential.
What is a Discount Factor Z(t,T)?
A number that tells us how much $1 in the future is worth today.
How do you calculate the discount factor?
Z(t,T) = 1 / (1 + r)^(T-t) in discrete compounding.
Z(t,T) = e^-r(T-t) in continuous compounding.
If the discount factor Z(0,3) = 0.8638, what does it mean?
It means $1 received in 3 years is worth $0.8638 today.
What is a zero-coupon bond?
A bond that pays no coupons, only a lump sum at maturity.
How do you calculate the price of a zero-coupon bond?
Pz(t,T) = 100 * Z(t,T)
What is a coupon bond?
A bond that pays periodic interest and principal at maturity.
How do you price a coupon bond using discount factors?
P(t,T) = All Coupons * Their Discount Factors + Principal * Discount Factor
What is arbitrage?
A risk-free profit from exploiting price differences between markets.
What is the Law of One Price?
Two securities with the same cash flows must have the same price.
How does no-arbitrage relate to bond pricing?
Bond prices must be consistent with discount factors; otherwise, traders could exploit price differences.
Why do we need bootstrapping?
Because not all maturities have zero-coupon bonds, so we must extract discount factors from coupon bonds.
What is the first step in bootstrapping?
Solve for Z(t,T) using the shortest-maturity bond.
How do you compute Z(t,T1) if it pays coupons?
Z(t,T1) = P(t,T1) / (100 + C)
How do you compute Z(t,T2) if it pays coupons?
Z(t,T2) = P(t,T2) - C*Z(t,T1) / 100 + C.
Rewrite the equation for P(t,T2) = C * Z(t,T1) + (100 + C) * Z(0,2)
What is interpolation?
A method to estimate missing discount factors between known maturities.
What is the Nelson-Siegel model?
A formula that fits the yield curve using four parameters (Theta0, Theta1, Theta2, and Lambda)
What is the yield curve?
A graph showing the relationship between interest rates and maturities.
What are the three shapes of the yield curve?
Normal (upward sloping) = long-term rates higher than short-term.
Flat = long-term and short-term rates are similar.
Inverted = long-term rates are lower than short-term (signals recession)
How do you extract the yield from discount factors?
r(T) = - ln(Z(0,T) / T
What is the difference between clean price and dirty price?
Clean price is without accrued interest.
Dirty price is with accrued interest.
How do you calculate the accrued interest?
Accrued interest = Coupon Payment * (Days since last coupon) / (Days in period)
What is Yield to Maturity (YTM)?
The average annual return an investor earns if they hold a bond until maturity.
If a bond’s coupon rate is 5% and its YTM is 7%, is it trading at a premium or discount?
Discount because YTM > coupon rate
What are the key risks when investing in bonds?
- Interest rate risk - bond prices fall when interest rates rise.
- Credit risk - the issuer may default on payments.
- Liquidity risk - some bonds are harder to sell quickly.
- Inflation risk - inflation erodes purchasing power.
- Reinvestment risk - coupons may have to be reinvested at lower rates.
What is a callable bond?
A bond that allows the issuer to repay early before maturity.
What is a putable bond?
A bond that allows the bondholder to sell it back early before maturity.
Why do zero-coupon bonds have higher price sensitivity to interest rates?
Since they pay only one payment at maturity, their price is entirely dependent on discounting, making them more sensitive to interest rate changes.
If a bond pays semi-annual coupons, how does this affect its discounting?
Z(t,T) = 1 / (1 + r/2) ^ 2T
What happens if bond prices violate the no-arbitrage condition?
Traders exploit mispricings by buying the underpriced bond and short-selling the overpriced bond, forcing the prices back int equilibrium.
What is the role of government bonds in determining discount factors?
Since government bonds are risk-free, their prices directly reflect discount factors without credit risk distortions.
Why does bootstrapping start with the shortest-maturity bond?
The shortest-maturity bond requires only one unknown discount factor, making it solvable immediately.
Why do we subtract known present values in the bootstrapping formula?
To isolate the unknown discount factor by removing the contributions of already known discount factors.
What is spline interpolation?
A method that fits a smooth, flexible curve through known data points, ensuring no sharp changes in the yield curve.
What does an inverted yield curve indicate?
It signals a potential recession because long-term rates are lower than short-term rates, suggesting that investors expect lower future interest rates.
What causes a steep yield curve?
Expectations of higher future interest rates, usually in times of economic expansion.
What is the forward rate formula?
f(T1,T2) = Z(0,T1) / Z(0,T2) - 1
Why do bond traders quote clean prices instead of dirty prices?
To prevent prices from jumping every time a coupon is paid, making it easier to compare bonds.
What happens to the dirty price right before and after a coupon payment?
Before the coupon payment, the dirty price gradually rises as interest accrues. After the coupon payment, the dirty price drops by the coupon amount.
What happens to YTM if bond prices fall?
YTM increases because the investor gets the same cash flows for a lower price.
If YTM = coupon rate, how does the bond trade?
At par
How is YTM different from current yield?
Current yield is annual coupon / bond price, which ignores future reinvestment)
YTM accounts for all future cash flows and reinvestments.
What is the relationship between interest rates and bond prices?
Interest rates and bond prices have an inverse relationship. When interest rates rise, bond prices fall, and when interest rates fall, bond prices rise.
Why do bond prices fall when interest rates rise?
New bonds are issued at higher rates, so investors prefer them over existing bonds. Existing bonds with lower coupon payments become less attractive, so their price drops to remain competitive.
Why do bond prices rise when interest rates fall?
New bonds are issued at lower rates, making older bonds with higher coupons more valuable. Investors are willing to pay more for existing bonds with better yields.
For a given interest rate, more frequent accrual of interest yields a higher final payoff. However, for a given final payoff, more frequent accrual of interest yields a lower interest rate.
