Financial Markets

by Professor Throckmorton
for Intermediate Macro
W&M ECON 304
Slides

Summary

• Define present value and arbitrage
• Arbitrage $\rightarrow$ equal expected returns, i.e., a financial market equilibrium
• Define Yield-to-maturity and interpret the yield curve
• Derive relationship between interest rates and stock prices

Present Value

• Present value is the current worth of a future payment or stream of payments, discounted back using an appropriate interest rate to reflect the time value of money.
• Applying present value allows us to price any asset, e.g., bond, stock, mortgage
• Since present value depends on market expectations, current asset prices have information about expectations

Today ($t$)	Next year ($t+1$)	in 2 years ($t+2$)
$\$1$	$\$(1+i_t)$	$\$(1+i_t)(1+i_{t+1}^e)$
$\$\frac{1}{1+i_t}$	$\$1$
$\$\frac{1}{(1+i_t)(1+i_{t+1}^e)}$		$\$1$

• Asset pays principal plus (expected) interest, compounding over time
• Present value of $\$1$ received next year is discounted by $1+i_t$
• Present value of $\$1$ received in 2 years is discounted by $(1+i_t)(1+i_{t+1}^e)$
• What's important is to discount future cash flow according to when it's received.

• Recall a simple loan/one-period zero-coupon discount bond (e.g., T-Bill) \begin{gather*} P_B = \frac{\textrm{Face Value}}{1+i} \end{gather*}
• Current market price equals face value (future cash-flow) discounted to present by interest rate
• $P_B$ is initial investment/loan given by market price, Face Value is known, this implies an interest rate, $i$ (a.k.a. yield to maturity)

Arbitrage

• General use: buy thing at low price in one market and sell it at a higher price in a different market
• For example, you could buy laptops for cash on college campuses and then list them on ebay at a higher price.
• In finance, arbitrage refers to buying/selling assets depending on relative expected returns
• For example, sell an asset with a high price/low expected return, then buy a different asset with low price/high expected return.
• Goal: maximize expected return on a portfolio of assets

Portfolio Choice

• Properties of bonds
  • Credit/default risk: who issued them and what's the risk premium/spread?
  • Maturity: how long until bond pays face value?
  • Discount/Coupons: does bond also pay coupons?
• Consider a small portfolio
  • Credit/default risk: bonds issued by U.S. government, so no default/credit risk
  • Maturity: either 1 or 2 years
    • 1-year bond has price $P_{1,t}$ and pays $F.V.$ in $1$ year
    • 2-year bond has price $P_{2,t}$ and pays $F.V.$ in $2$ years
  • Discount/Coupons: discount bonds only, no coupons, e.g., Treasury bills
• 2-year bond is transferable and could be sold after $1$ year
  • Q: What is price of 2-year bond with 1-year left?
  • A: The price of a 1-year bond at that time.
• Q: How do we manage this portfolio?

Choice	This year ($t$)	Next year ($t+1$)	in 2 years ($t+2$)
A: Buy 1-year bond	$\dfrac{P_{1,t}}{P_{1,t}} = \$1$	$\dfrac{F.V.}{P_{1,t}}$	—
B: Buy 2-year bond	$\dfrac{P_{2,t}}{P_{2,t}} = \$1$	$\dfrac{P^e_{1,t+1}}{P_{2,t}}$	$\dfrac{\text{F.V.}}{P_{2,t}}$

• To compare cashflow next year, sell the 2-year bond next year at expected 1-year bond price, $P^e_{1,t+1}$.
• Divide by bond price to "invest" $\$1$ in each choice.
• The return on $\$1$ worth of a 1-year bond is $F.V./P_{1,t}$ vs. the expected return on $\$1$ worth of a 2-year bond, which is $P^e_{1,t+1}/P_{2,t}$

• Goal: maximize expected return on portfolio
  • $1$-year bond return: $F.V./P_{1,t}$
  • $2$-year bond expected return: $P_{1,t+1}^e/P_{2,t}$
• Three possible cases:
  1. Disequilibrium (arbitrage opportunity):
     $F.V./P_{1,t} < P_{1,t+1}^e/P_{2,t}$
     $\rightarrow$ sell 1-year ($\downarrow P_{1,t}$) and buy 2-year ($\uparrow P_{2,t}$)
     $\rightarrow$ expected returns equalize
  2. Disequilibrium (arbitrage opportunity):
     $F.V./P_{1,t} > P_{1,t+1}^e/P_{2,t}$
     $\rightarrow$ buy 1-year ($\uparrow P_{1,t}$) and sell 2-year ($\downarrow P_{2,t}$)
     $\rightarrow$ expected returns equalize
  3. Equilibrium: $F.V./P_{1,t} = P_{1,t+1}^e/P_{2,t}$ (equal returns)
     $\rightarrow$ no incentive to rebalance portfolios
• Cases 1) and 2) are arbitrage opportunities that lead to the equilibrium where there is no more arbitrage.

Equilibrium 2-year Bond Price

• In equilibrium, 1-year return $=$ expected 2-yr return \begin{gather*} \frac{F.V.}{P_{1,t}} = \frac{P_{1,t+1}^e}{P_{2,t}} \rightarrow P_{2,t} = \frac{P_{1,t}P_{1,t+1}^e}{F.V.} \end{gather*}

• Recall present value formula for 1-year bond price \begin{gather*} P_{1,t} = \frac{F.V.}{1+i_{1,t}} \end{gather*}

• Combine those \begin{gather*} P_{2,t} = \frac{P_{1,t+1}^e}{1+i_{1,t}} \end{gather*}

• Update 1-year bond price forward one year, take expectation \begin{gather*} P_{1,t+1}^e = \frac{F.V.}{1+i_{1,t+1}^e} \end{gather*}

• Substitute that into 2-year bond price \begin{gather*} P_{2,t} = \frac{P_{1,t+1}^e}{1+i_{1,t}} = \frac{F.V.}{(1+i_{1,t})(1+i_{1,t+1}^e)} \end{gather*}

• This is the present value (i.e., asset pricing) formula for a 2-year discount bond.

Yield to Maturity/Curve

• Yield to Maturity (YtM): the annual interest rate a bond holder receives if bond is held to maturity

• I.e., YtM is the constant interest rate that equates current bond price with present value of all future cash-flow, e.g., for an $N$-year discount bond,

  \begin{gather*} P_{N,t} = \frac{F.V.}{(1+i_{N,t})^N} \end{gather*}

  where $i_{N,t}$ is $N$-year YtM

• We know $P$ and cash flow, so we can calculate all YtM

• Yield Curve: a graph/table of YtM as function of maturity for bonds that have same credit/default risk, e.g., U.S. Treasuries

• Combine bond market equilibrium with YtM definition

  \begin{gather*} P_{2,t} = \frac{F.V.}{(1+i_{1,t})(1+i_{1,t+1}^e)} = \frac{F.V.}{(1+i_{2,t})^2} \end{gather*}

• Since numerators are equal, then denominators are equal

  \begin{gather*} (1+i_{1,t})(1+i_{1,t+1}^e) =(1+i_{2,t})^2 \end{gather*}

• Equation is nonlinear, so let's linearize it! Recall

  \begin{gather*} \log((1+x)(1+y)) = \log(1+x) + \log(1+y) \log(1+x) \approx x \textrm{ if $x$ is small} \log((1+y)^b) = b \log (1+y) \end{gather*}

  where $\log$ is the natural logarithm

• Take logs and use properties

  \begin{gather*} (1+i_{1,t})(1+i_{1,t+1}^e) =(1+i_{2,t})^2 \rightarrow \log(1+i_{1,t}) + \log(1+i_{1,t+1}^e) = \log((1+i_{2,t})^2) \rightarrow \log(1+i_{1,t}) + \log(1+i_{1,t+1}^e) = 2 \log(1+i_{2,t}) \rightarrow i_{1,t} + i_{1,t+1}^e = 2 i_{2,t} \end{gather*}

• 2-year YtM is average of 1-year rates

  \begin{gather*} i_{2,t} = (i_{1,t} + i_{1,t+1}^e)/2 \end{gather*}

• Expected 1-year rate (predicted)

  \begin{gather*} i_{1,t+1}^e = 2i_{2,t} - i_{1,t} \end{gather*}

Expectations Hypothesis

\begin{gather*} i_{2,t} = (i_{1,t} + i_{1,t+1}^e)/2\\ i_{1,t+1}^e = 2i_{2,t} - i_{1,t} \end{gather*}

• Expectations hypothesis: the long-term rate is determined purely by current and future expected short-term rates
• Since we can calculate YtM from current bond prices, the expectations hypothesis leads to a prediction about expected future short-term rates
• E.g, if $i_{2,t}$ falls, then $i_{1,t+1}^e$ also falls
• Q: When would $i_{1,t+1}$ actually decrease?
  A: When the central bank sets a lower rate (in a recession).

Expectations hypothesis gives three cases for interpreting yield curve

• Upward sloping: $i_{2,t} > i_{1,t}$
  $\rightarrow$ $i_{1,t+1}^e > i_{1,t}$, i.e., central bank expected to raise rate
  $\rightarrow$ bond market expects boom/expansion

• Downward sloping ({\color{red} i.e., inverted}): $i_{2,t} < i_{1,t}$
  $\rightarrow$ $i_{1,t+1}^e < i_{1,t}$, i.e., central bank expected to lower rate
  $\rightarrow$ bond market expects bust/recession

• Flat (typically before recessions): $i_{2,t} = i_{1,t}$
  $\rightarrow$ $i_{1,t+1}^e = i_{1,t}$
  $\rightarrow$ bond market expecting move from boom to bust

It's usually upward sloping, but downward sloping ($i_{10,t} - i_{2,t} < 0$)before recessions?
10-Year Treasury Constant Maturity Minus 2-Year Treasury Constant Maturity (T10Y2Y)

Stock Market

• Bonds and stocks both store wealth (i.e., they are substitutable)
• New goal: maximize expected return on portfolio of 1-yr bond and a stock
• Stock characteristics
  - $Q_t$ is current price
  - $D_{t+1}^e$ is expected dividend (e.g., random, could be zero)
  - $Q_{t+1}^e$ is expected future price
• Q: What's the cash flow on a stock if you sell it after a year?
  A: The expected dividend plus the expected sale price.

Portfolio Choice Problem

Choice	$t$	$t+1$
A: Buy 1-yr bond	$\frac{P_{1,t}}{P_{1,t}} = \$1$	$\frac{F.V.}{P_{1,t}} = 1 + i_{1,t} + x$
B: Buy stock	$\frac{Q_t}{Q_t} = \$1$	$\frac{D^{e}_{t+1} + Q^{e}_{t+1}}{Q_t}$

Normalize

• Time horizon $= 1$ year
• Initial investment $= \$1$
• $x$ is the risk premium, i.e., the additional return the portfolio manager wants to compensate them for the price risk of holding a stock

Goal: Maximize expected portfolio return

1. Disequilibrium (arbitrage opportunity):
   1-yr bond return $<$ expected stock return
   $\rightarrow$ sell 1-yr bond ($\downarrow P_{1,t}$) and buy stock ($\uparrow Q_t$)
   $\rightarrow$ expected returns equalize
2. Disequilibrium (arbitrage opportunity):
   1-yr bond return $>$ expected stock return
   $\rightarrow$ buy 1-yr bond ($\uparrow P_{1,t}$) and sell stock ($\downarrow Q_t$)
   $\rightarrow$ expected returns equalize
3. Equilibrium: 1-yr bond return $=$ expected stock return
   $\rightarrow$ no incentive to reallocate portfolio
   (equal expected returns / no more arbitrage opportunties)

Stock Price

• Assume equilibrium and solve for $Q_t$ (with risk premium)

  \begin{gather*} Q_t = \frac{D_{t+1}^e + Q_{t+1}^e}{1+i_{1,t}+ x} \end{gather*}

• Update one period and take expectation

  \begin{gather*} Q_{t+1}^e = \frac{D_{t+2}^e + Q_{t+2}^e}{1+i_{1,t+1}^e + x} \end{gather*}

• Substitute 2 into 1

  \begin{gather*} Q_t = \frac{D_{t+1}^e}{1+i_{1,t}+ x} + \frac{D_{t+2}^e + Q_{t+2}^e}{(1+i_{1,t}+ x)(1+i_{1,t+1}^e + x)} \end{gather*}

• This is a present value formula for a stock.

Fundamental Value

• Keep substituting out expected stock price

  \begin{gather*} Q_t = \frac{D_{t+1}^e}{1+i_{1,t}+ x} + \cdots + \frac{D_{t+N}^e + Q_{t+N}^e}{(1+i_{1,t}+ x)\times \cdots \times (1+i_{1,t+N-1}^e + x)} \end{gather*}

• As $N \rightarrow \infty$, PV of expected stock price goes to zero

• Prediction: the fundamental value of a stock is just the present value of all future dividends (for a given risk premium)

Stock Price Summary

\begin{gather*} Q_t = \frac{D_{t+1}^e}{1+i_{1,t}+ x} + \cdots + \frac{D_{t+N}^e + Q_{t+N}^e}{(1+i_{1,t}+ x)\times \cdots \times (1+i_{1,t+N-1}^e + x)} \end{gather*}

• Arbitrage $\rightarrow$ equal expected returns between a bond and stock (for a given risk premium)
• Stock price is positively related to expected dividends, i.e., expected future profitability
• Stock price is negatively related to (current or expected) short-term interest rates, e.g., news of higher future short-term interest rates lowers stock price
• If managers become more risk averse, then stock prices fall.
• The fundamental value of a stock is just the present value of all future dividends.