EC 380: Lecture 17

.title[
# EC 380: Lecture 17
]
.subtitle[
## Global Finance: Exchange Rates SR-Med
]
.author[
### Philip Economides
]
.date[
### Winter 2024
]

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# Prologue

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# Recap

* Exchange rates determine price of foreign goods

* Numerous reasons to hold foreign reserves

* Shifts in exchange rates often mirror changes in demand and supply of currencies

* PPP holds only in LR

### Today

* SR, Medium exchange rates and ExR Systems

---

# Topics

<br>

* Reasons for holding foreign reserves, main institutions

* Effect of `$\Delta S, \Delta D$` of foreign currency on home currency

* __Identify short, medium and long term forces that affect currency value__

* Three rules of gold standard

* Compare and contrast various exchange rate systems

* Price changes and real exchange rate interactions

* List conditions necessary to form single currency area

---

# FX Market: Flexible

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# FX Market: Flexible

---

# FX Market: Flexible

Decreased demand for GBP in USA (D2 `$\leftarrow$` D1)

---

# FX Market: Medium Run

<br>

Usually a 4-5 year range, where countries experience ebbs and flows of .hi-pink[business cycle] booms and busts.

During boom periods, incomes tick up, .hi-pink[aggregate demand rises] across consumers

Some of that spills over into .hi-pink[increased demand for imports].

This implies a greater need for foreign currency.

---

# FX Market: Medium Run

Increased demand for GBP in USA (D1 `$\rightarrow$` D2)

---

# FX Market: Medium Run

Suppose UK sees economic boom, increased import of US goods, higher GBP reserves (S1 `$\rightarrow$` S2)

---

# FX Market: Short Run

Short run is a time period of one year or less.

We observe continuous day-to-day fluctuations in exchange rates.

---

# FX Market: Short Run

<br>

* Interest rate adjustments, which can occur 4-5 times in a year.

* Speculation, driven by exogenous shocks to the state of the economy (e.g. political scandal in Brazil, discovery of natural gas deposits off Cypriot coast, COVID outbreak at Chinese ports)

<br>

This latter cohort contribute to a mechanism known as .hi-pink[price discovery] in exchange rates.

---

# FX Market: Short

<br>

> Price discovery is the trial-and-error process of discovering the equilibrium price in a market. If speculators believe a currency is overvalued, they sell that currency, driving its value down.

If correct, currency moves towards equilibrium and speculators are rewarded through arbitrage profits.

If wrong, large losses are realized. This can often lead to bankruptcy if these risks were taken based on borrowed funds.

.hi-pink[These stakes deliver a strong incentive to make the correct decision, bolstering the speed of market adjustments in the process.]

---

# FX Market: Interest Parity

<br>

.hi-pink[Interest Parity Condition]: Difference between the home and foreign interest rates should be equal to the expected change (appreciation or depreciation) of the exchange rate.

<br>

`$$i-i^* = \frac{\left(F-R\right)}{R}$$`

<br>

where `$i$` is the home interest rate, `$i^*$` is the foreign interest rate, `$F$` is expected future exchange rate and `$R$` is the current exchange rate.

---

# FX Market: Interest Parity

<br>

Suppose an investor has the choice to either invest locally or abroad in one-year bonds.

> X year bond: After maturing for X year, the bond is paid back to the investor in full principal amount and includes a yield of interest accrued for X year(x)

Respective rates of return are `$i=0.03$` and `$i^*=0.02$`. Each payoff is in a different currency.

Suppose the 1-year US bond has a price of 1000 USD.

---

# FX Market: Interest Parity

<br>

The bond's payoff is equal to

`$$\text{P}\times(1+i)^n$$`

where `$P$` is the principal amount invested, `$i$` is the interest rate and `$n$` represents the number of years the bond accrues interest for before maturing.

`$$\text{US bond} = 1000 \times (1.03)^1 = 1030 \  \text{USD}$$`
--

---

# FX Market: Interest Parity

<br>

Bond principal plus __2% interest__ is paid in euros.

The dollar value of this investment choice depends on the .hi-pink[future exchange rate one year from now].

Suppose exchange rate today is __1.2 USD per EUR__.

Investor can buy __833.33 EUR__ in exchange for __1000 USD__.

<br>

`$$\text{EUR bond} = 833.33 \times (1.02)^1 = 850 \  \text{EUR}$$`

---

# FX Market: Interest Parity

<br>

To accurately compare the two choices, investor must .hi-pink[forecast the exchange rate] in one years time (F).

If `$E(F)=1.3 \ \text{USD/EUR}$`, implies expected foreign value of `$850 \times 1.3 = 1105 \ \text{USD}$`

US Bond worth `$1030$`, so foreign return of `$1105$` is more profitable.

Expect return of foreign investment: `$i' = \left[\frac{E(F)}{R}\times (1+i^*)\right]-1$`

---

# FX Market: Interest Parity

<br>

To guarantee `$E(F) = F$` is to remove uncertainty from the setting.

Hedge against exchange rate risk by signing contracts on the .hi-pink[forward exchange rate market.]

Today's spot rate `$R$` and market rate on one year exchanges `$F$` indicate where market sees ExR moving in one year.

If `$F>R$`, the currency is expected to depreciate.

If `$F<R$`, the currency is expected to appreciate.

---

# FX Market: Interest Parity

<br>

In our example, investors were willing to sell off US bonds in order to purchase foreign bonds.

Funds will flow from the US to Germany, causing German money supply to rise.

A greater money supply implies lower interest rates, so `$i*$` falls.

Demand for German currency used to buy the bonds rises, so `$R$` rises.

Market moves towards interest rate parity `$i - i^* = \frac{\left(F-R\right)}{R}$`

---

# FX Market: Interest Parity

<br>

The future dollar value of German bonds in USD changes too.

The scale and direction of the change dictate the choice investors make.

Let us return to our previous example to see this in action.

---

# FX Market: Interest Parity

<table class="table" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> US bond, USD </th>
   <th style="text-align:right;"> German bond, EUR </th>
   <th style="text-align:right;"> F (USD per EUR) </th>
   <th style="text-align:right;"> German bond, USD </th>
   <th style="text-align:left;"> Purchase </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 0.7 </td>
   <td style="text-align:right;"> 595 </td>
   <td style="text-align:left;"> US </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 0.8 </td>
   <td style="text-align:right;"> 680 </td>
   <td style="text-align:left;"> US </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 0.9 </td>
   <td style="text-align:right;"> 765 </td>
   <td style="text-align:left;"> US </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.0 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:left;"> US </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.1 </td>
   <td style="text-align:right;"> 935 </td>
   <td style="text-align:left;"> US </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.2 </td>
   <td style="text-align:right;"> 1020 </td>
   <td style="text-align:left;"> US </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.3 </td>
   <td style="text-align:right;"> 1105 </td>
   <td style="text-align:left;"> German </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.4 </td>
   <td style="text-align:right;"> 1190 </td>
   <td style="text-align:left;"> German </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.5 </td>
   <td style="text-align:right;"> 1275 </td>
   <td style="text-align:left;"> German </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.6 </td>
   <td style="text-align:right;"> 1360 </td>
   <td style="text-align:left;"> German </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.7 </td>
   <td style="text-align:right;"> 1445 </td>
   <td style="text-align:left;"> German </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.8 </td>
   <td style="text-align:right;"> 1530 </td>
   <td style="text-align:left;"> German </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1030 </td>
   <td style="text-align:right;"> 850 </td>
   <td style="text-align:right;"> 1.9 </td>
   <td style="text-align:right;"> 1615 </td>
   <td style="text-align:left;"> German </td>
  </tr>
</tbody>
</table>

---

# FX Market: Interest Parity

<br>

Suppose `$i<i^*$` and `$F<R$`. US yields are lower and world expects USD to appreciate.

Fed decides to set `$i$` equal to `$i^*$`. Increased returns on US bonds .hi-pink[(Foreign Reserves rise)].

Implies US foreign currency supply increases _(shifts right)_ from `$S1$` to `$S2$`.

Demand for foreign bonds falls, shifts demand down _(shifts left)_ from `$D1$` to `$D2$`.

Think of GBP market, expect USD to .hi-pink[appreciate] under these conditions.

---

# FX Market: Interest Parity

<br>

These mechanisms playout somewhat like a .hi-pink[self-fulfilling prophecy].

If investors think USD will depreciate in future, `$F$` rises.

This causes return on foreign bonds to improve, causing demand on foreign currency to rise and the dollar's spot rate `$R$` depreciates.

<br>

.hi-pink[A change in expectations about future exchange rates can lead to a similar change in the spot rate].

---

# FX Market: Interest Parity

<br>

We can think about the market playing out in two manners:

1) Covered interest parity (CIP)

> Assume the investor has .hi-pink[perfect foresight] such that `$E(F)=F$`.

2) Uncovered interest parity (UIP)

> Assume investor accepts risk involved and purchases with E(F) in mind.

---

# FX Market: CIP

<br>
--

Investors will trade until returns from either bond equalize, and .hi-pink[all arbitrage opportunities are exhausted].

`$$\text{CIP}: (1+i) = (1+i^*)\frac{F_{\text{USD}/\text{EUR}}}{R_{\text{USD}/\text{EUR}}}$$`

In that case, investors are .hi-pink[indifferent] between either bond since their real returns are the same.

Allows us to pin down the `$F$` necessary to achieve CIP
`$$F_{\text{USD}/\text{EUR}} = R_{\text{USD}/\text{EUR}} \frac{1+i}{1+i^*}$$`

---

# FX Market: UIP

This alternative method of investment allows us to determine how spot rates are established.

The no-arbitrage condition for UIP is written as follows:

`$$\text{UIP}: (1+i) = (1+i^*)\frac{E(F)_{\text{USD}/\text{EUR}}}{R_{\text{USD}/\text{EUR}}}$$`

Allows us to pin down the `$R$` necessary to satisfy UIP
`$$R_{\text{USD}/\text{EUR}} = E(F)_{\text{USD}/\text{EUR}} \frac{1+i^*}{1+i}$$`
--

We can calculate today's spot rate if we know market expected exchange rate and these two respective interest rates.

---

# Evidence of UIP

<br>

Taking both our previous equations, and dividing one into the other on __both sides__.

`$$\text{CIP}: (1+i) = (1+i^*)\frac{F_{\text{USD}/\text{EUR}}}{R_{\text{USD}/\text{EUR}}}\\
\text{UIP}: (1+i) = (1+i^*)\frac{E(F)_{\text{USD}/\text{EUR}}}{R_{\text{USD}/\text{EUR}}},\\
\implies 1 = \frac{F_{\text{USD}/\text{EUR}}}{E(F)_{\text{USD}/\text{EUR}}}$$`

Under the assumption of both types of investors (risky and riskless) exhausting all arbitrage opportunities, expected exchange rates should be equal to forward exchange rates.

---

# Evidence of UIP

<br>

For both relationships to hold, these items _must be equal to one another empirically_.

_In equilibrium_, if investors do not care about risk, then they have no reason to prefer to avoid risk by using the forward market rate.

Using previous equation, test described as checking whether the .hi-pink[expected rate of depreciation] is equal to the .hi-pink[forward premium]

`$$\underbrace{\frac{F_{\text{USD}/\text{EUR}}}{R_{\text{USD}/\text{EUR}}}-1}_{\text{Forward Premium}} = \underbrace{\frac{E(F)_{\text{USD}/\text{EUR}}}{R_{\text{USD}/\text{EUR}}}-1}_{\text{Expected Rate of Depreciation}}$$`

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# Evidence of UIP

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# Summary

### Recap

* FX market mechanisms in the medium run driven by business cycles

* Short run variation in exchange rate attributed to monetary policy and speculation

* Parity relationships allow us to identify breakeven points at which investment decisions are made

### Next Time

* ExR Systems and single currency areas

---