4 Yield Curve

Today

Reading, HW2 Tips, Review, Yield Curve, Credit Spreads, Etc.

Reading:

Sometimes, It's bonds for the long run.

References "Stocks for the long run" by Jeremy Siegel

He says that "Stocks are volatile" it's a way financial people use to describe risk, typically tends to move a long.

Stocks may be volatile in the short term, but are a strong, solid investment in the long term.

If you can be a long term investor, you can have substantial exposure to stocks.

The chart in the article. It's saying that total real returns with income reinvested.

Real Returns: in excess of inflation.

when you look at the chart, the red line is bonds, black line is stocks. For well over 100 years, the red line is over the black line, but in 2007 it touched again.

Why would stocks outperform bonds? The whole idea of a risk premium for stocks so the market adjusts.

Stocks are an asset class, crypto, bonds, real estate, etc.

Are there asset classes that are even riskier than publicly traded equity? Private equity, crypto, etc.

within private equity, venture capital: extremely risky, extremely volatile.

The second article is from the current press, which connects to what is going on right now. The chart, is showing the spreads on CCC and lower bonds. So the spread over treasuries. The source is ICE, InterContinental Exchange, They own NYSE, etc. There was a spike in April 2025 from the liberation day terrorists.

A CCC company would have to offer on the market was like 15% where we add the relevant duration note to the spread. The 10yr note has gone up from 4%-4.8% and the treasury basis spread is like 10% so we can add them together.

Let's go over some vocabulary.

Credit: in finance it means the provision of funds in exchange for a promise of future repayment. - eg, a credit card is a different from a debit card.

when we get to the accounting section, we will learn about debits and credits in terms of balance sheets.

Credit Markets: certain types of loans are illiquid, but you can trade them as well. Loans can be bought and sold, etc.

If someone trades credits, they aren't talking about treasuries, but more about corporate bonds, or asset backed securities. How does the market seem to adjust to match interest rates. But it's the opposite, the credit bond market is what sets in the interest rates.

HW 2:

Bond Pricing:

• a 1m bond offered at a price of:
• 100 would cost 1m. is said to be offered at par
• 99.5 would cost 995,000 is said to be offered at a discount
• 100.5 would cost 1,005,000 is said to be offered at a premium
• the amount of the bond is referred to as its face value or par value, and is principal amount due at maturity.

Zero Coupon Bonds:

• single cashflow at a single point in time.
• no coupon payments, just one bullet payment at maturity, costing the face value
• trade at a discount
• coupon bonds can be thought of as a portfolio of zero-coupon bonds.

Perpetuity, and annuity proofs. The first problem will ask you to prove a few things with perpetuities and annuities. - remembering or looking up infinite series formulas is not necessary. you can use the following trick - \(X = C + Ca + Ca^2 + \ldots\) - \(aX = Ca + Ca^2 + Ca^3 + \ldots\) - $X - Xa = C

Review: long vs short

• Long: you own the asset and hope its price increase
• short: you borrow and sell an asset and hope its price decreases

Bond Markets are modeled by first looking at interest rates then deriving bond prices.

Yield exhibits mean reversion

the long term average for the 10 year note is approx 6%

the average over the last 20 years is approx 3%

Any bond or fixed income instrument that matures in less than a year is called a money market instrument. Anywhere out to a year, is known at a higher yield. How can money market rates be at 0.5% there is a gap. that gap is known as a historical high. Why would anyone keep their money in a bank and easily open a brokerage and need treasury bills and get 4%

It's just lack of knowledge. If you have relatives who could be getting 4% risk free without their own. They are investing all these deposits, and could be living risk free right now. there is a competition between wall street and main street banks, what wall street people used to say, that the main street people are part of the 363. The banks take for 3% then will lend out the money for 6% and they are onto the golf course by 3pm. SOFR. when we get to swaps and so on, we will talk about it here. We want to know about a secured loan. something that's collateralized. They will lend money, but if you don't get their money back, the bank will get your car back. A secured loan means the lender has something to take if they don't meet their obligation.

yield Curve

 yield
   |                                  __________
   |                         ________/
   |                  ______/
   |             ____/
   |         ___/
   |      __/
   |    _/
   |   /
   +---------------------------------------------- maturity
      3m   1y    2y     5y      10y      20y   30y

           normal (upward sloping) yield curve

that lower curve, it wouldn't have been too much to say if you were an investor, they would say, i think someday, there's gonna be something that happened. What you are betting is that the shape of the yield curve is going to change.

They wanna say: i think the 3 year yield, is going to go up. People call it a butterfly curve plays out. So they are betting on how that yield curve is going to shape it for youl.a

Why do people demand a better yield for the price to go up. I'm going to demand a better yield. If you are interested in a 30 year bond, or something

Long-term yield: the average expected future short term rates. A very simple example, and you're looking at a bond where the short term rate will be 4% forever, you may owgonter the same.

the concept is that when you are looking at longer terms yields it is a conversation over what the market expects plus a short price to compensate over the same amount.

What causes the shape of the yield curve to change, We talked about how central banks set the interest rate.

• Monetary policy.
• central banks directly influence short term overnight rates.
• Expectation of future policy impact long term yields
• central banks sometimes directly influenced long-term rates by purchasing system.
• that is the first time they had quantitative easing, they don't wanna do it right now, they can make 30yr rates to go down, they will buy them until rates go down. The friend.

they are making money easier, they are putting money into the world. that is what is called quantitative easing. the other thing, is how interest rate expectation. long term yields.

If people think short term rates are going to rise. If shorter term rates are expected to fall. this is all about market forces. should i buy a purse, or should i do more kick rocks around. Market participants will purchase more of one bond than the other.

Interest rate expectations

• long term yields reflect expectations of future, short term interest rates
• if short term rates are expected to fall.

How can a yield curve be inverted then?

Signals expectations of lower future interest rates, often due to an anticipated economic slowdown or recession (=> fed will have to lower short term rates to stimulate the economy.)

long-term yield = average expected future short term value

People talk about the yield curve steepness. This is the difference between 2 and 10 yr notes. a yield curve often anticipates a recession.

The most common measure of the yield curve steepness is the 2yr to the 1-10 yr bond. there is also simply flattening and steepening trades. You can understand the buying and selling. You think that the 10 year bond price. So you buy 10yr notes and sell 2 yr notes.

if i wanna bet through that omega, etc.

that's what people call the curve, because. So you can do the spot-rate curve. watch. This one is going to get more subtle. the 1yr forward. the concept is this: when we get to derivatives, is that we are going to agree on a price, etc. but we will dont do the trade until sometime blame it onty sister. Imagine i lend you 25% and you pay me back on the 16th year. I'm not gonna dive into this too much, and there's specific parts of QF where we get into certain rates and yield. The curve will be 2r rates every half year, etc.

The difference between the central bank and finance ministries.

Central Bank implements monetary policy, etc. That is the fed in the United states - monetary policy -> participation in the overnight loan market to influence money supply - meant to be independent from political leadership in the US (although the President gets to appoint the FED chair)

Ministry of finance: manages government finances, collects taxes, issues bonds to fund public spending. Does not impact the money supply. they are just participants. Explicitly controlled by the political leadership. - the treasury in the US

most developed countries keep these functions separate. Why? - political leaders typically have short-term incentives - lower interest rates now => inflation in the future

Central banks implement monetary policy

financial ministry influence fiscal policy

Inflation in Türkiye peaked at 85% in 2022

any market participant can impact interest rates, if they have enough capital.

George Soros would move the market, and would buy buy buy for days on end, and people would try to predict what he would do and he would do something else. Made england run out of money. Scott Bessent tried this and was selling short term bonds to buy long term bonds, and it move the market a little bit and then it went back to what it was.

Duration and Convexity

Let's look at some curves.

Price Yield relationship, all of these have a negative slope. it's a consequence of that eqn down below. it clearly semi annual. It represents 3 versions down below. If it's for a 2 year note, there are 3 coupon payments and one coupon plus maturity.

when you plot the equations, they look a little different.

The price yield relationship for different maturity durations at 5% coupon. they all cross at the 5% yield where they are all served at par. makes sense right.

Convexity quantifies the curvatures of the price yield relationship.

Positive convexity: convex upward. Can convexity ever be negative...? Yes!

for straightforward bonds, they're mostly convex up.

Negative convexity most commonly occurs for MBS/ABS bonds and bonds with embedded options. with interest rates going down, mortgage bonds start to dissolve because people will refinance with a lower interest rate.

there are actually two kinds of duration, the one we talked about was macaulay duration. there's another called modified duration. You can see how they are related in a second.

Macaulay duration

• weighted average time in years until the bonds cash flows are received.
• weights are the PV of each cash flow as a proportion of the bonds total PV.

\(D_m = \sum_{t=1}^{T} t \cdot w_t\) where \(w_t = PV(C_t)/P\) and \(PV(C_t) = C_t/(1+r)^t\) and \(P\) is sum of all PV

\(r\) is the discount rate per period usually YTM/2

High coupon -> shorter duration Low Coupon -> Longer duration what is the duration of a zero coupon bond that matures in 5 years? just 5.

Modified Duration:

• adjust Macaulay duration to estimate the percentage price change of a bond for a 1% change in yield. Ignores convexity.
• Linear approx:
• Unitless (% price change per 1% change in yield)

\(D_{mod} \triangleq D_m/(1 + r)\)

\(D_{mod} = -1/P \cdot dP/dr\)

Convexity

• quantifies the curvature of the price yield relationship
• used to define the price change estimate given by duration alone

\(\text{Convexity} \triangleq 1/P \cdot d^2P/dr^2\)

\(\Delta P/P \approx -D_{mod} \Delta r + \frac{1}{2}\text{Convexity}(\Delta r)^2\)