Tutorial assignment: week 5 2022
Answer guide
1. Consider a bond market where the demand for a particular bond is given by the equation
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where is the number of bonds demanded and P is the price of the bond.
Suppose the supply of this bond is fixed at 1400.
(a) Calculate the price of the bond
In equilibrium,
(b) Suppose the bond is a consol with an annual coupon of 45. What is the yield on this security?
The formula for the price of a consol is
Thus the yield is 3.75%
2. Suppose there is a one-year discount bond trading in the Australian market with a face value of $1000 and supply and demand curves given by the following equations:
where P is the price of the bond, and represent the number of bonds supplied or demanded.
Calculate the following (half mark each):
(a) The equilibrium price of the bond
In equilibrium,
(b) The yield to maturity
Since this is a one-year discount bond,
(c) The number of bonds supplied and demanded
Note that supply and demand are equalised at the equilibrium price.
(d) The yield to maturity if this had been a two-year bond rather than one-year.
If this were a two-year security the appropriate formula would have been:
3. Using the same supply and demand equations as the previous question, and assuming this is a one-year bond, what would be the effect on (a) the price and (b) the yield to maturity if the RBA decided to sell 120 of these bonds from its own holdings to the market? (1 mark each)
The supply of bonds is now given by the equation:
The demand equation is unchanged.
Equilibrium is given by,
Rearranging:
Calculating the yield to maturity by the same formula as previously:
An increase in the supply of bonds lowers the price and raises the interest rate.
4. Consider a simple economy with only one form of money, which is currency. In this economy, the demand for money is given by the equation:
where r is the interest rate. Calculate (a) the equilibrium interest rate if the supply of currency is fixed at $2000 and (b) the change in the interest rate that would occur if the central bank increased the supply of currency by 10% (calculate to two decimal places).
(b) In this case,
Using the same formula,
An increase in the money supply lowers the interest rate when money demand is stable.
5. Consider the same economy as in the previous question with the supply of money fixed at $2000. Now suppose there is a shift in the money demand equation such that households in aggregate desire to hold an additional $150 in cash balances for any given level of interest rates. (a) Calculate the effect this has on the equilibrium interest rate (to two decimal places). (b) What would the central bank have to do to offset this effect?
Equilibrium requires that:
(b) An increase in money demand causes the interest rate to rise when the money supply is fixed. To offset this, the central bank would need to expand the money supply by the same amount (that is, by 150). It is easy to see that in the above calculations, this restores the initial equilibrium.
6. Why does the demand curve for bonds slope downwards? What factors could cause this curve to shift to the left and what effect would this have on the interest rate? In your answer, explain how this could be equivalently described in terms of the money market rather than the bond market.
When the price of bonds rises, fewer are demanded because they are more expensive or, equivalently, they have a lower yield. Movements along the demand curve describe how the amount demanded changes in response to a change in the price, all other things being unchanged. If this curve shifts to the left and the supply curve stays the same, the price of bonds will fall and hence the interest rate (or yield) rises.
Things that could cause the bond demand curve to shift to the left are anything that lowers the relative attractiveness of holding a bond for any given price or interest rate, or lowers the demand for assets in general. Examples:
· A fall in aggregate investor wealth (for example, an economic recession that lowers the value of all assets)
· Higher returns on an alternative asset (for example, a rise in expected returns in the equity market)
· Loss of liquidity in the bond market (example, a general rise in brokerage costs)
· Increase in riskiness of holding bonds (example, inflation risk is perceived to be higher for some reason associated with a change in economic conditions)
Equivalent money-market explanation
In the simple two-asset model of interest rates, we can also think of the interest rate as being determined by the interaction of supply and demand for money. The money demand curve slopes downwards because the interest rate is the opportunity cost of holding money. As the interest rate rises the opportunity cost of holding money increases, so less will be demanded (or equivalently, more bonds are demanded.)
A shift in the bond demand curve to the left means that the money demand curve shifts to the right. Bonds have become less attractive, so households increase their desired holdings of money at any given interest rate. This might occur in a period of extreme uncertainty when money is perceived as the safe asset (a ‘flight to cash’). As we have seen, an increase in money demand causes the interest rate to rise when money supply is fixed – effectively, households are prepared to pay more to hold the same available supply.
7. Consider two hypothetical economies (Country A and Country B) where the central bank undertakes operations to increase the money supply by 5 per cent. In Country A the interest rate falls, but in Country B the interest rate rises. What sorts of differences between conditions in the two countries could explain these differing responses?
Here we need to remember that our supply-demand diagrams depict what happens at a point in time when nothing else is changing. If nothing else changes to affect the position of the money demand curve, an increase in money supply causes the interest rate to fall.
However, it is likely that other things would change over time if the money supply is increased, specifically:
· The price level of goods and services would rise
· Incomes and employment might rise
· Expectations of future inflation might rise, especially if the country has a history of repeatedly expanding the money supply and generating ongoing inflation
The net overall impact will depend on the relative strength of these different factors. The first two factors will cause the money demand curve to shift to the right – more money is demanded at a given interest rate when prices and incomes are higher, because higher monetary transactions are occurring. This will tend to push interest rates back up, but will probably happen slowly over time.
The expectations effect can potentially work more quickly – there might be an immediate jump in inflation expectations which could be large enough to more than offset the initial fall in the interest rate.
The difference between Country A and Country B could therefore be explained if the inflation expectations effect is larger in Country B. This could occur if Country B has a relatively poor history of inflation control, whereas in Country A, households may have greater confidence that inflation will remain under control.
8. Suppose you are seeking to analyse the influence of interest rates on Australian homebuyers during the one-year period from December 2019 to December 2020. Use data from the RBA website to obtain measures of the nominal interest rate and the ex ante and ex post real interest rates that you think would have been most relevant to these decision makers. Explain the reasons for your choices and identify any possible shortcomings in these measures.
[Note: Given the range of indicators available, marks will be awarded for any well-reasoned choice of indicators and for correct application of the appropriate real interest rate definitions.]
Nominal interest rates
Relevant housing loan rates can by found in Table F6 on the RBA site in the Statistics section.
There are a number of different rates to choose from.
A good choice of nominal interest rate in this context would be: new (rather than outstanding) loans, all lenders (not just large), owner occupiers, variable rate. Combining these gets you to Column M. In December 2019 this interest rate was 3.28%
For inflation measures, we need the inflation rate (actual or expected) over the given one-year period (Dec 2019 – Dec 2020).
Actual inflation for the year Dec 2019 – Dec 2020 (CPI definition) can be found in Table G1 Column C. The figure is 0.9 per cent. Some students may want to choose one of the ‘underlying’ inflation measures for this purpose (Columns E, J or K), which is also a valid choice.
Expected inflation in December 2019 for the year ahead can be found in Table G3, Column B. The figure for December 2019 was 4.0 per cent, which covers expectations over the year Dec 2019 – Dec 2020).
Using these figures we get the following:
Nominal interest rate = 3.3% (rounded)
Ex ante real interest rate = (3.3 – 4.0) = minus 0.7%
Ex post real interest rate = (3.3 – 0.9) = 2.4%
9. In recent years, when the RBA has made a change in the cash rate, the change has been 25 basis points (0.25 percentage point) or less. Using information from the RBA web site, when was the last time the RBA changed the cash rate by 100 basis points (1.0 percentage point) in a single move? What was happening at the time and what reasons were given for the decision?
The easiest way to find this is in the Statistics section using the link to Cash Rate Target in the left-hand menu. Here is the link:
https://www.rba.gov.au/statistics/cash-rate/
Scrolling down the page, you can see that the last time the cash rate was changed by 100 basis points (1.00 percentage point) was on 4 February 2009. Click the Statement link on that entry to find the announcement of the decision. The statement indicates that the global economy was in a severe downturn as a result of the financial crisis. The interest rate was reduced by that amount to support demand in the Australian economy so as to cushion the economy from contractionary forces coming from abroad.
10. In the period following the global financial crisis government deficits increased in most of the major economies, resulting in a substantially increased supply of bonds, yet government bond yields fell. How could this be explained?
An increase in the supply of bonds will cause bond prices to fall (yields or interest rates to rise) if the position of the bond demand curve is unchanged. However, a fall in government bond yields could be explained if, at the same time, there was an increase in demand for government bonds more than sufficient to offset the increase in supply. Factors that could have contributed to this would have been:
· lower expected returns on alternative assets due to the economic recession (equity prices and real estate values were falling rapidly at that time)
· general economic uncertainty causing an increased demand for safe assets (government bonds were perceived as a safe asset compared with equities and real estate)
· lower expectations of future inflation, due to the economic recession
· increased supply of money by central banks, causing low money market interest rates and hence increased demand for longer-term government bonds
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