§17 Financial Markets and Expectations

Expected Present Discounted Value

• Key issue: How do we decide that the current price of an asset (or a real investment) is consistent with the future cash flows generated by that asset?
• An answer to that question involves:
  • Expectations (the Expected part of EPDV);
  • Some method to compare payments received in the future with payments made today (the PDV part of EPDV).
  • Note: the word Discounted, suggests that payments received in the future are worth less that payments received in the present…

Expected Present Discounted Value

• A general formula (PDV)

  $$\begin{aligned} \$V_t = &\ \$z_t + \frac{\$z_{t+1}}{(1+i_t)} + \frac{\$z_{t+2}}{(1+i_t)(1+i_{t+1})} + \cdots\\ & + \frac{\$z_{t+n}}{(1+i_t)(1+i_{t+1})\cdots(1+i_{t+n-1})} \end{aligned}$$

• What if we do not know the future? (EPDV)

  $$\begin{aligned} \$V_t = &\ \$z_t + \frac{\$z^e_{t+1}}{(1+i_t)} + \frac{\$z^e_{t+2}}{(1+i_t)(1+i^e_{t+1})} + \cdots\\ & + \frac{\$z^e_{t+n}}{(1+i_t)(1+i^e_{t+1})\cdots(1+i^e_{t+n-1})} \end{aligned}$$

  where the expectation is taken as of time $t$: $E_t[x_{t+s}]$.

Expected Present Discounted Value: Examples

• Constant interest rate

  $$\$V_t = \$z_t + \frac{\$z^e_{t+1}}{(1+i)} + \frac{\$z^e_{t+2}}{(1+i)^2} + \cdots + \frac{\$z^e_{t+n}}{(1+i)^n}$$

• Constant payments

  $$\begin{aligned} \$V_t = &\ \$z \left[ 1 + \frac{1}{(1+i_t)} + \frac{1}{(1+i_t)(1+i^e_{t+1})} + \cdots \right. \\ & \left. + \frac{1}{(1+i_t)(1+i^e_{t+1})\cdots(1+i^e_{t+n-1})} \right] \end{aligned}$$

• Constant interest rate and payments

  $$\$V_t = \$z \frac{1 - \left[1 / (1+i)^n \right]}{1 - \left[1 / (1+i) \right]}$$

• Constant interest rate and payments forever

  $$\$V_t = \$z \frac{1 + i}{i}$$

• If payments start next year (rather than in the current year)

  $$\$V_t = \frac{\$z}{i}$$

• What happens as $i$ goes to zero?

Bond Prices and Bond Yields

• Bonds differ in their maturity.
  • A bond that promises to make a $1k (final) payment in 6 months, has a maturity of 6 months.
  • A bond that promises to pay $100/y for 20y and $1k final payment in 20 years, has a maturity of 20 years.
• Bonds of different maturities each have a price and an associated interest rate, called the yield to maturity or, simply the yield.
• The relationship between maturity and yield is called the yield curve, or the term structure of interest rates.

Bond Prices as Present Values

• Example 1: A bond that pays $100 in 1y

  $$\$P_{1t} = \frac{\$100}{1 + i_{1t}}$$

  where the subscript 1 means 1y.

  • Note that the price of the 1y bond varies inversely with the current 1y nominal interest rate. Why?

• Example 2: A bond that pays $100 in 2y

  $$\$P_{2t} = \frac{\$100}{(1 + i_{1t})(1 + i^e_{1t+1})}$$

  where the subscript 2 means 2y… and note that the price of the 2y bond varies inversely with the current 1y nominal interest rate and the 1y rate expected for next year.

Arbitrage and Bond Prices

• Suppose you are considering investing $1 for one year and you have two options:

  • Invest in the 1y bond;
  • Invest in the 2y bond and sell it after the first year.

• That is

  $$1 + i_{1t} = \frac{\$P^e_{1t+1}}{\$P_{2t}}$$

  or

  $$\$P_{2t} = \frac{\$P^e_{1t+1}}{1 + i_{1t}}$$

• But

  $$\$P^e_{1t+1} = \frac{\$100}{1 + i^e_{1t+1}}$$

  so that

  $$\begin{aligned} \$P_{2t} &= \frac{1}{1 + i_{1t}} \$P^e_{1t+1} \\ &= \frac{\$100}{(1 + i_{1t})(1 + i^e_{1t+1})} \end{aligned}$$

  which is the same expression we obtained from the EPDV approach.

From Bond Prices to Bond Yields

• The yield to maturity on an $n$-year bond (or the $n$-year interest rate) is defined as the constant annual interest rate that makes the bond price today equal to the PDV of future payments of the bond.

• For example, for our 2y bond, we have or

  $$\$P_{2t} = \frac{\$100}{(1 + i_{2t})^2}$$

• This implies that

  $$\frac{\$100}{(1 + i_{1t})(1 + i^e_{1t+1})} = \frac{\$100}{(1 + i_{2t})^2}$$

• Rearranging:

  $$(1 + i_{2t})^2 = (1 + i_{1t})(1 + i^e_{1t+1})$$

  to imply:

  $$i_{2t} \approx \frac{1}{2} (i_{1t} + i^e_{1t+1})$$

Adding Risk

• Bonds have two types of risks:

  • Default risk: the risk that the issuer of the bond will not pay back the full amount promised by the bonds.
  • Price risk: the uncertainty about the price at which you can sell the bond in the future if you want to sell it before maturity.

• Suppose we are dealing with highly rated bonds so the default risk is minimal, and we still have the option to invest in a 1y or a 2y bond.

  $$1 + i_{1t} + x^{b} = \frac{\$P_{1t + 1}^{e}}{\$P_{2t}}$$

  or

  $$\$P_{2t} = \frac{\$100}{\left(1 + i_{1t} + x^{b}\right)\left(1 + i_{1t + 1}^{e}\right)}$$

• Note that this also implies that the yield to maturity includes a risk premium (term premia):

  $$i_{2t} \approx \frac{1}{2}\left(i_{1t} + i_{1t + 1}^{e} + x^{b}\right)$$

— Apr 25, 2025

§17 Financial Markets and Expectations by Lu Meng is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Permissions beyond the scope of this license may be available at About.