Fixed income portfolio management Flashcards
So we know that when government issues bonds, they are riskless in the sense that you will get coupon payment and the risk of default of this is 0, so where is the risk coming from?
The risk to the portfolio is interest rate risk: the effect that interest rate movements can have on the prices of bonds .
What is yield to maturity of a bond and how do you calculate it?
The yield to maturity (YTM) of a bond is the total return an investor can expect if they buy a bond at its current price and hold it until it matures, assuming all coupon payments are reinvested at the same rate. It represents the average annual rate of return the investor will earn.
What is the relationship between price and yield?
The relationship between bond yield and price is an inverse one. This means that when the price of a bond goes up, its yield goes down, and when the price goes down, its yield goes up. numerous reasons explain this but mainly due to These cash flows are fixed and do not change over the life of the bond in the numerator.
The yields of zero coupon bonds equal what and why?
Yes, it’s true that the yields of zero-coupon bonds are equal to spot rates. This is because zero-coupon bonds do not have periodic coupon payments, so their yield is based solely on the difference between the purchase price and the face value of the bond, which is equivalent to the spot rate.
What is the yield curve( also known as the term structure of interest rates.? and draw the diagram.
A yield curve is a graphical representation illustrating the relationship between interest rates (or yields) and time to maturity for a group of fixed-income securities, typically government bonds. These bonds have comparable credit quality but varying maturity dates. The yield curve’s shape is influenced by market expectations and prevailing economic conditions.
Explain the different shapes of the yield curve?
1) Normal (upward-sloping): Short-term rates are lower than long-term rates, indicating expectations of economic growth and inflation, leading to higher future interest rates.
2) Inverted (downward-sloping): Short-term rates are higher than long-term rates, suggesting a pessimistic outlook with declining future interest rates due to slower growth or deflationary pressures.
3) Flat or humped: Short-term and long-term rates are roughly equal, or there’s a hump in the middle, signaling a transition between economic cycles or uncertainty about future interest rate movements.
What is the forward interest rate?
What is the spot rate?
the forward rate is an implied future interest rate derived from the current yield curve. It represents the market’s expectation of the interest rate for a specific period in the future.
Is the interest rate that applies to a particular loan or financial instrument at a specific point in time, often referred to as the spot date.
How can we calculate the interest rate at time t ( hint it includes forward rate)?
rs is the spot rate for the time horizon ‘s’
rt is the spot rate for the time horizon ‘t’
fs,t is the forward rate for the period agreed at s for time t.
‘s’ and ‘t’ are two different time horizons, with ‘t’ being greater than ‘s’
What are 2 main reasons for forward rates? ( WRITE THE FORWARD RATE EQUATION.
Future interest rate expectations: Forward rates offer insights into market expectations of economic conditions, such as growth, inflation, and monetary policy, based on the current yield curve.
Risk management: Forward rates enable hedging against interest rate and currency fluctuations, safeguarding businesses and investors from unexpected changes in borrowing or lending costs or exchange rates.
fs,t = ((1 + rt)^t / (1 + rs)^s)^(1/(t-s)) - 1
We know that if the equation for interest rate at time t is violated we can make aribtirage profits, so lets say ?
Explain how we can make arbitrage here?
1) You borrow £1 at the rate rt for t years ( you have to payback (1+rt)^t))
2) This £1 you invest at the spot rate rs for s years. So at the end of rs you get £(1+rs)^s.
3) With this money you get £(1+rs)^s, you invest this in the forward rate fs,t negotiated at date 0 for subsequent t-s years
4) So at the end of year t you have the left side and you need to pay back the right side.
Now lets say this is the inequality what is the arbitrage strategy? (1 + rs)^s × (1 + fs,t)^(t-s) < (1 + rt)^t
1) Borrow £1 at spot rate (rs) for time horizon ‘s’.
2) Invest the borrowed amount in a bond or loan with a longer time horizon ‘t’ at the forward rate (fs,t) agreed upon at time ‘s’.
3) Simultaneously, enter into a short position in a bond or loan with the same time horizon ‘t’ at the prevailing spot rate (rt).
4) This means that the cost of borrowing at the short-term rate and reinvesting at the forward rate is less than the return from the short position at the longer-term spot rate. This difference in values creates a risk-free profit opportunity for the investor.
There are 3 hypothesis that determine the shape of the term strcuture of interest rates ( yield curve)?
Expectations hypothesis
Liquidity premium hypthothesis
Segmentation hypothesis
What is the expectation hypthoesis?
Expectations Hypothesis: Forward rates are unbiased predictors of future spot rates: fs,t=E[rs,t].
Thus, if you invest money in the spot rate r2 for 2 years, you will get the same amount of money if you invest at the rate of r1 for 1 year and the amount of money you get at date 1, you invest in the forward rate
With this explain how an upward sloping term structure means that r2 > r1 and therefore f 1,2 > r1?
According to the Expectations Hypothesis, an upward-sloping term structure implies that r2 > r1, meaning the two-year spot rate is higher than the one-year spot rate. This is because the market expects short-term rates to rise in the future.
For the equation to hold, if r2 > r1, it must be the case that f1,2 > r1. In other words, the one-year forward rate one year from now (f1,2) must be greater than the current one-year spot rate (r1). This is consistent with the Expectations Hypothesis, as the higher forward rate (f1,2) reflects the market’s expectation of an increase in short-term interest rates in the future.
If we have a downward sloping term structure, using the expectations hypothesis, explain how r2< r1 and f1,2 < r1?
A downward-sloping term structure implies that r2 < r1, meaning the two-year spot rate is lower than the one-year spot rate. This is because the market expects short-term rates to fall in the future.
For the equation to hold, if r2 < r1, it must be the case that f1,2 < r1. In other words, the one-year forward rate one year from now (f1,2) must be less than the current one-year spot rate (r1). This is consistent with the Expectations Hypothesis, as the lower forward rate (f1,2) reflects the market’s expectation of a decrease in short-term interest rates in the future. The economic conditions we expect this is in a recession.
Using the expectations hypothesis explain a flat term structure means that r2 = r1 and f1,2?
A flat term structure implies that r2 = r1, meaning the two-year spot rate is equal to the one-year spot rate. This is because the market expects short-term rates to stay unchanged in the future.
For the equation to hold, if r2 = r1, it must be the case that f1,2 = r1. In other words, the one-year forward rate one year from now (f1,2) must be equal to the current one-year spot rate (r1). This is consistent with the Expectations Hypothesis, as the equal forward rate (f1,2) reflects the market’s expectation of stable short-term interest rates in the future.
What is the duration ( maculay duration) of a bond? does duration increase of decrease as time to maturity increases?
What term structure does it assume?
Which bond is more sensitive to changes in interest rates?
The duration of a bond is the elasticity of its price with respect to change in yields/interest rates). In general, the duration of a bond increases as its time to maturity increases. This is because longer-term bonds have more cash flows that are further into the future, and these cash flows are more sensitive to changes in interest rates than cash flows that are closer to the present.
It assumes that yield-to-maturity of the bond remains constant as interest rates change. ( so term structure flat) ( is that the bond’s total return, if held until maturity, will not change even when market interest rates fluctuate. In reality, this assumption may not hold true, as yield-to-maturity typically changes with market interest rates.)
or can be defined as Duration is a measure of the weighted average time until a bond’s cash flows are received, accounting for both coupon payments and principal repayment.
The red bond is more sensitive. ( more steep)
What is macaulay duration ( this is duration) ? what does it mean with maculay duration is high?
the weighted average maturity of the cash flows from a Bond in years ( a bond with a higher amount of years for this will be more sensitive to interest changes. )
calculate macaulay duration
How do we actually calculate change in price?
f
Immunisation strategy question, what is the net worth when interest rate is 5% and 4%?
So this change in interest rate means the company runs out of money and goes bankrupt. So we have to conduct an immunisation strategy to help the company. Remember its the liabilities that are sensitive to interest rates not the assets. We need the assets to go up to match the increase in liabilities.
So the immunisation strategy should be investing the assets in bonds, invests £30 mln and buys x3 units of 3-year zeros at price P3 and x10 units of 10-year zeros at price P10.
3-year and 10-year bonds with face value £100.
Work out the prices of each bonds and work out the modified duration of the steam of liabilities?
Here simply do Mac duration / (1+r)
Now that we know that the modified duration is 6.13 years, now lets take a step back to work out change in PV of liabilities ( this is general, so can be used for change in price of a bond, so learn it!!! what is the formula?
Change in PV of liabilities can be approximated to minus modified duration of liabilities x PV of liabilities x change in interest rate. ( basically telling us how PV of liabilities change in response in interest rate .
So now write the budget constraint and approximate change in asset value? What do we want to achieve in this immunisation strategy?
we want our assets to have the same sensitivity of interest rates as our liabilities. ( so when value of assets go up so does liabilities so it offsets equality.
So now what is our immunisation problem ( similar to economics) ?
we want to find the amount of each bond that maximises the sensitivity of assets and liabilities being the same.
To calculate D3 and D10 REMEMBER the Macaulay duration of a zero coupon bond is its time to majority. So, the duration of a 3-year zero-coupon bond is 3 years, and the duration of a 10-year zero-coupon bond is 10 years. now to find Dmod = Dmax/1+r.
Plugging in all the numbers find out find out x3 and x10 and find out the net worth of this strategy?
What is important to remember about immunisation strategy?
Its a dynamic strategy, so it will need to be revised year on year as term structure and duration changes.
How do we work out convexity?
To estimate the bond price change using both duration and convexity, you can use the following formula?
-D* remember is modified duration.
the rate k is such that it costs nothing to enter into the swap contract, k cannot be changed.
What is the duration of a perpetuity?
What is duration of a portfolio?
Dperp = 1 + y/y
How to calculate price of a bond with an annuity coupon payments?
f
