The Solutions to Ch s 7-9 problems

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1 The Solutions to Ch s 7-9 problems Chapter 7 Ch 7-Q2: For each of the following, calculate: (i) the exact real interest rate; (ii) the approximate real interest rate (a) i t =6%, π t =%;(b)i t = 0%, π t =5%;(c)i t = 50%, π t = 45% Solution: formula: r t = +it ; r +πt e t i t πt e (a) r =4.95%; 5% (b) r =4.76%; 5% (c) r =3.45%; 5% Ch 7-Q3: Can the real interest rate ever be negative? Under what circumstances? Explain in words what this implies about borrowing and lending. Solution: Yes, r t canbenegativeifi t < πt e. In this case, borrowers gains while lenders lose Ch 7-Q4: Suppose that the monthly inflation rate in a country is expected to remain constant at 30%. (a)whatistherateofinflation per year? (b) if the yearly nominal interest rate is 2340%, calculate the yearly real interest rate using: (i) the exact formula, (ii) the approximate formula. Solution:πt e =30%where t denotes month, Pt+2 e = P t ( + πt e ) 2 (let y denote yearly ) πy e = P t+2 e P t =(+πt e P )2 t (a) πy e =(+0.3) 2 =22.3 (b) (i) i y =23.4; r y = +iy =4.72% (ii) r +πy e y i y πy e = 0% The large descrepancy is due to the fact that the dropped term r y πy e is too large to be dropped. Ch 7-Q5: Suppose that you win the lottery and will receive $00,000 now, another $00,000 one year from now, and a final $00,000 two years from now. Calculate the present discounted value of these payments when the interest rate is expected to remain constant at (a) 0%, (b) 5%, (c) 0%. Solution: PV = 00, 000 h + + i +i (+i) 2 (a) PV = 300, 000; (b)pv =285, 940; (c)pv =73, 550 Ch 7-Q7: Using the IS-LM model, determine the impact on C, I, Y, i (nominal rate), and r (real rate) for each of the following: (a) a decrease in expected inflation (b) a decrease in expected inflation combined with a contractionary monetary policy

2 Solution: Assume that M P is fixed. Note that Y = C (Y T )+I (Y,r)+G M P = Y L (r + πe ),wherei = r + π e Use a graph with r rather than i plotted on the vertical axis. (a) π e LM shifts up r & Y C & I From the figure you draw, 4r < 4π e = 4π e 4i = 4r + 4π e < 0 i Why? Y but M is fixed M d is fixed L (i P ) i (b) π e & M LM shifts up to LM and then up to LM r & Y C & I. Since M = Y P L (i ), L and i if in Y is greater than in M; while L and i if in Y is smaller than in M. Ch 7-Q9: Suppose you are about to enter into a business venture from which you expect to earn $0,000 in the current year, and the same amount next year and the year after. The one-year nominal interest rate is 5% and you expect it to remain unchanged in the future. (a) what is the expected PV of your future earnings? (b) calculate the new expected PV of future earnings under each of the following changes: (i) a transitory (current year only) increase in earnings from $0,000 to $20,000 (ii) an increase in earnings from $0,000 to $20,000, expected to be permanent (all three periods) (iii) a transitory (current year only) increase in the nominal interest rate from 5% to 0% (iv) an increase in the current nominal interest rate from 5% to 0%, expected to be permanent. Solution: z t = zt+ e = zt+2 e =0, 000; i t = i e t+ = i e t+2 =0.05 (a) PV =0, 000 ³ = 2 (b) (i) PV 2 =0, PV = ; (ii) PV 3 =2PV = (iii) PV 4 =0, 000 ³ = 2 ;(iv)pv5 =0, 000 ³ = 2 Chapter 8 Ch 8-Q4: A pretzel manufacturer is considering buying another pretzel-making machine that costs $50,000. The machine will depreciate by 0% per year. It will generate real profits equal to $0,000 from this year on. Determine whether the manufacturer should buy the machine if the real interest rate is assumed to remain constant at (a) 5%, (b) 0%, (c) 5%. Solution:P K =50, 000; δ = 0%; r t = rt+i e = r; π t = πt+i e =$0, 000 2

3 PV =0, 000 h + δ + ( δ)2 + i =0, 000 +r (+r) 2 δ +r (a) PV =0, =70, 000 >P 0.5 K invest (b) PV 2 =0, 000. =55, 000 >P 0.2 K invest (a) PV =0, =46, 000 <P 0.25 K do not invest =0, 000 +r r+δ Ch 8-Q5: A consumer with nonhuman wealth of $00,000 will earn $50,000 this year, and expects her salary to rise by 5% in real terms each year for the following two years. She will then retire. The real interest rate is equal to zero and is expected to remain equal to zero in the future. Labor income is taxed at the rate of 40%. (a) what is this consumer human wealth? (b) If this consumer expects to live for another ten years and wants her consumption to remain the same every year, how much should she consume this year? (c) if this consumer were given a bonus of $20,000 in the current year only, with all future salary payments remaining as stated earlier, by how much would her current consumption rise? Solution: Non-human wealth: W nh =00, 000. Assume: consumption smooth (a) human wealth: " W h = V (YL e T e )=( 0.4) 50, # ( + 0) 2 = (b) total wealth: W T = W nh + W h =; constant consumption: C = W T 0 = (c) W 0 T = W T +20, 000 =; constant consumption: C 0 = W 0 T 0 =; 4C = C0 C = Ch 8-Q6: An investor can sell a bottle of wine today for $7,000, or put it in storage and sell it in 30 years for a real (constant dollar) price of $20,000. If the real interest rate is expected to remain constant at 4%, what should the investor do? Solution: PV = 20,000 (+0.04) 30 = <P current =7, 000; sell it now Ch 8-Q7: A worker signs a contract that freezes her salary at $40,000 for the next three years. The real interest rate is expected to remain constant at 3%, and the inflation rate to remain constant at 5%. What is the present discounted value of her three-year salary? Solution: i = r + π e =0.08; $PV =40, 000 ³ =; 3 PV =40, 000 ³ =; 3 Chapter 9 Ch 9-Q: Determine the yield to maturity of each of the following bonds. (a) a discounted bonds with a face value of $,000, a maturity of two years, and a price of $800 (b) a discounted bonds with a face value of $,000, a maturity of two years, and a price of $900 3

4 (c) a discounted bonds with a face value of $,000, a maturity of three years, and a price of $900 (d) Using your answers to parts a and b, what happens to a bond s yields when everything else remains the same but the price of the bond increases by $00? Solution: P = F (+i nt) n (a) 800 == 000 ; i (+i 2t ) 2 2t = q 000 =; 800 (b) 900 == 000 ; i 0 (+i 0 2t) 2 2t = q 000 =; 900 (c) 900 == 000 ; i (+i 3t ) 3 3t = ³ =; 900 (d) compare (a) and (b): P i 2t Ch 9-Q2: If the interest rate for the current year is 0% and the interest rate expected for next year is 5%, determine the yield on a two-year bond using: (a) the approximate formula (b) the exact formula Solution: i t =0.; i e,t+ =0.05 r i t2 = ( + i t ) ³ +i,t+ e =..05 =; i t2 i t+i e,t+ 2 = =0.3 Ch 9-Q3: Suppose that the interest rate this year is 4% and financial markets expect the interest rate to increase by % each year thereafter. Determine the yield to maturity on a (a) one-year bond (b) two-year bond (c) three-year bond Solution: i t =0.04; i e,t+ =0.05; i e,t+2 =0.06 (a) i t =0.04; (b) i 2t =0.045; (c) i 2 2t =0.05; 3 so, the yield curve is upward sloping Ch 9-Q4: Suppose that a country s central bank unexpectedly pursues an expansionary monetary policy that financial markets believe will be reversed in the future. What effect will the expansionary monetary policy have on the LM curve? On the yield curve? Why? Solution: The monetary expansion leads the LM curve to shift down, which increases income and decreases the interest rate in equilibrium. Since this monetary expansion is believed to be short-lived in such a way that the money supply will soon returned to the intinial level, the LM curve will shift back to where it was and the interest rate will rise back to its initial level. However, in this process of changes (i & but < initial i), the interest rate (i.e., the short rate i S ) is expected to be below the initial level: i e S <i S. Thus, the current long rate i L is lower than the current short 4

5 rate since i L = i S+i e S < i S+i S = i 2 2 S. This suggests that the yield curve is downward sloping during the policy change, and becomes flatter by a bit since the monetary change is expected not to last for long. Empirical evedence shows that 4i L = ζ4i S where 0 < ζ (is small) <. You may use graphs. Ch 9-Q5: A share will pay a dividend of $00 in one year and will be sold for an expected price of $,000 at that time. Determine the current price of the stock if the current one-year interest rate is (a) 5%, (b) 0% Solution: $Q t = $00 + $00 + $00 + +i (+i) 2 (+i) 3 = $00 + ³ $00 + $00 + = $00 + $Qe +i +i +i (+i) 2 t+ +i +i (a) $Q t = $00 + $000 =; (b) $Q t = $00 + $000 =.. Ch 9-Q6: A share is expected to pay a real dividend of $00 next year, and thereafter the real value of dividend payments is expected to increase by 3% per year, forever. Determine the current price of the stock if the real interest rate is expected to remain constant at (a) 5%, (b) 0% Solution: real Q = 00 h +r +.03 (+r) (+r) 3 + i = 00 +r (a) Q = =; (b) Q = =..03 +r = 00 r 0.03 Ch 9-Q7: Using the IS-LM model, determine the impact of each of the following on stock prices P S. If the effect is ambiguous, explain what additional information would be needed to reach a conclusion. (a) An unexpected contractionary monetary policy with no change in fiscal policy (b) An unexpected contractionary fiscal policy with no change in monetary policy (a) An unexpected contractionary fiscal policy with a monetary policy that keeps output unchanged (a) An unexpected contractionary fiscal policy with a monetary policy that keeps interest rates unchanged Solution: use graphs and the pricing formula of stocks (a)( An unexpected monetary contraction M the LM Y e sales,profit Π e dividends D e P S i e P P S S (b)( An unexpected fiscal contraction T or G the IS Y e profit Π e dividends D e P S i e the net effect on P P S S is unclear If the LM is flat enough, the effect of Y e domonates that of i e and the net effect is P S ; If the LM is steep enough, the effect of Y e is domonated by that of i e and the net effect is P S 5

6 (c) An unexpected fiscal contraction T or G the IS. A monetary expansion must be used to keep Y from declining due to the fiscal contraction. Then, this plocy mix i e P S (d) An unexpected fiscal contraction T or G the IS. A monetary contractionmustbeusedtokeepi from declining due to the fiscal contraction. Then, this plocy mix Y e profit Π e dividends D e P S 6