FNCE 4820 Fall 2013 NAME__________________David M. Gross, Ph.D.

Midterm 1 with Answers

Answer the questions in the space below. Written answer requires no more than a few sentences. Show your work to receive partial credit. Points are as indicated.

1. (9 Points) Briefly define the following in the context of holding a bond.

(a) Interest-Rate RiskRisk of price change due to changes in the bond’s yield.

(b) Inflation RiskRisk of earning a lower-than-expected real return if inflation exceeds expectations.

(c) Liquidity RiskThe risk of a large price drop if the bond must be sold quickly or the inability to sell quickly without incurring a large price drop.Note: Liquidity Risk can also refer to the inability of a portfolio manager to determine the “correct” market value for a bond (for mark-to-market purposes) due to infrequent trading of the bond.

2. (7 Points) Answer the following questions about embedded conversion options in bonds.

(a) To which party (the borrower or the lender) is a conversion option granted?The lender is granted the convert option.

(b) Why might the party granted the option exercise the convert option?A convert option allows the holder of the bond to give up the bond and get common stock (convert the bond into stock) at a pre-set price per share. The holder would do this if the market price of the stock (far) exceeds the pre-set price.

(c) All else equal, will the inclusion of a convert option increase or decrease the yield on the bond?Decrease the yield since lender must compensate the borrower for granting the option.

3. (6 Points) Assume a floating rate bond makes semi-annual coupon payments and has exactly 20 years to maturity. The coupon rate is the reference rate plus 250 bps. The reference rates is currently 2.50%. Compute the discount margin for the bond if the price is 98.25. N = 40 PMT = (2.5 + 2.5)/2 = 2.5 PV = -98.25 FV = 100 I/Y = 2.57 YTM = 5.14% Discount Margin = 5.14% - 2.50% = 2.64% = 264 bps > 250 bps

4. (6 Points) A portfolio manager with a four-year investment horizon is considering purchasing a newly issued ten-year, 6% semi-annual coupon bond selling at par. The portfolio manager has computed the applicable

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forward rates and finds that the all the coupons can be reinvested until the end of the investment horizon at 5.00%. The portfolio manager has also computed the expected six-year YTM in four years for the bond and finds that it will be 6.50%. Compute the total return for holding this bond over the four-year investment horizon.Coup + I on I: N = 4 x 2 = 8 Sale Price: N = 6 x 2 = 12

I/Y = 5.00/2 = 2.5 I/Y = 6.50/2 = 3.25 PMT = 6/2 = 3 PMT = 6/2 = 3 FV = 26.21 FV = 100

PV = 97.55Total Future Dollars = 26.21 + 97.55 = 123.76Total Return = (123.76/100)1/8 – 1 = 0.0270 2 x 0.0270 = 5.40%Note must be reported in BEY terms: Total Return ≠ (123.76/100)1/4 – 1 = 5.47%

5. (5 Points) In class we used the St. Louis Federal Reserve Economic Data (FRED) Excel add-in to download data and then compute the monthly TED spread over the last 13 years. Recall the TED spread is the difference between the rate paid by banks on off-shore US dollar deposits (the Eurodollar rate or LIBOR) and the rate paid by the US Government (the T-bill rate). Given that both types of loans are very liquid, what risk does the TED spread measure? Why might the TED spread increase?The TED spread measure the default risk of banks relative to the default risk of the US Government. The TED spread would increase if default risk increases. (See the spreadsheet on the course website. Note the spikes near the collapse of Bear Sterns and Lehman. )

6. (6 Points) A corporate bond with a 5.00% coupon matures on March 15, 2030. Its current price is 103-18. It is putable at par on March 15, 2015 at par and has the call schedule (call dates and redemption prices - also known as call prices) shown in the table below. You have computed the yield for each of these events.

Event Settle Maturity Redemption YieldPut 10/3/2013 3/15/2015 100.00 2.4824%Call 10/3/2013 3/15/2020 100.50 4.4264%Call 10/3/2013 3/15/2026 100.40 4.6438%Call 10/3/2013 3/15/2027 100.30 4.6566%Call 10/3/2013 3/15/2028 100.20 4.6679%Call 10/3/2013 3/15/2029 100.10 4.6778%

Maturity 10/3/2013 3/15/2030 100.00 4.6867%

(a) What is the Yield to Maturity for the bond?YTM = 4.6867%

(b) What is the Yield to First Call for the bond?YTC = 4.4264%

(c) What is the Yield to Worst for the bond?YTW = 4.4264%

7. (27 Points) A $1,000 face value bond makes semi-annual coupon payments. It has exactly 20 years to maturity. The yield to maturity is 7.50% and the coupon rate is 7.50%.

(a) Calculate the price of the bond.

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N = 20 x 2 = 40; I/Y = 7.50/2 = 3.75; PMT = 0.075/2 x 1000 = 37.5; FV = 1000; PV = 1,000

(b) Compute BOTH the new lower price of the bond if the YTM increases by 100 basis point and the new higher price of the bond if the YTM decreases by 100 basis point. N = 20 x 2 = 40; I/Y = (7.50 + 1.00)/2 = 4.25; PMT = 0.075/2 x 1000 = 37.5; FV = 1000; PV = 904.61 P+ = 904.61

N = 20 x 2 = 40; I/Y = (7.50 - 0.01)/2 = 3.25; PMT = 0.075/2 x 1000 = 37.5; FV = 1000; PV = 1,111.04 P- = 1,111.04

(c) Use your calculations from parts (a) and (b) to compute the approximate Modified Duration (D*) for the bond. D* ≈ (P- - P+)/(2 x P0 x Δy) = (1,111.04 – 904.61)/(2 x 1000 x 0.01) = 10.32

(d) Calculate the actual dollar change in price of the bond for a 100 bps increase in yield.N = 20 x 2 = 40; I/Y = (7.50 + 1.00)/2 = 4.25; PMT = 0.075/2 x 1000 = 37.5; FV = 1000; PV = 904.61 ΔP = P+ – P0 = 904.61 – 1,000 = -95.39

(e) Use the bond’s modified duration calculated in part (c) to estimate the dollar change in price of the bond for a 100 bps increase in yield.ΔP ≈ -D* x P x Δy = -10.32 x 1,000 x 0.01 = -103.22

(f) Use your calculations from parts (a) and (b) to compute the approximate Convexity (CX) for the bond. CX ≈ (P- + P+ – 2P0)/(P0 x Δy2) = (1,111.04 + 904.61 – 2 x 1,000)/(1,000 x 0.012) = 156.50

(g) Use the bond’s modified duration calculated in part (c) AND the convexity calculated in part (f) to estimate the dollar change in price of the bond for a 100 bps increase in yield.ΔP ≈ (-D* x P0 x Δy) + (½CX x P0 x Δy2) = (-10.32 x 1,000 x 0.01) + [½(156.50) x 1,000 x (0.01)2] = -103.22 + 7.82 = -95.39

(h) Compare your estimations from parts (e) and (g) to the actual change computed in part (d). Briefly explain why your estimation from part (g) is better than your estimation from part (e). The estimation using both Duration and Convexity from part (g) is better since it corrects for the “curvature” or “change in slope” of the price/yield relationship. See Exhibit 4-13 on page 75 of the text.

8. (6 Points) A $100 million bond portfolio has the following three bonds.

Bond Market Value D*A $25,000,000 5B $25,000,000 7

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C $50,000,000 2(a) Estimate the PVPB (also called the DV01) for the portfolio.

Port D*= ∑ w x D* = 0.25(5) + 0.25(7) + 0.5(2) = 4PVBP = ΔP ≈ (-D* x P0 x Δy) = D* x P x 0.0001 = 4 x $100,000,000 x 0.0001 = $40,000

(b) Assume the portfolio manager is thinking of replacing Bond B with either Bond D with D* equal to 4 or Bond E with D* equal to 9. If the manager believes interest rates will rise, which bond will the manger choose to replace Bond B? EXPLAIN.Choose Bond D with lower D* to lower interest rate risk and therefore the price loss if rates

increase.

9. (6 Points) Bond X and Bond Y are roughly equivalent except that Bond X has greater convexity than Bond Y.

(a) Which bond would you expect to have the greater yield? Bond Y. Bond holders pay a premium and therefore earn less (or earn lower yield) for higher

convexity.

(b) Explain why greater convexity is better or worse.

It is better because the greater the convexity, the greater the price gain for a decrease in yield relative to the price loss for a symmetric increase in yield.

10. (6 Points) You have just looked up the coupon rate, YTM and price for the on-the-run ½ year, 1 year and 2 Treasury bills and bonds. In addition, you used linear interpolation to compute the YTM for the1½ year bond and then used that yield and its coupon to calculate its price. Use this information to compute the implied zero-coupon spot rate maturing in 1½ years. All rates are in BEY terms.

Years Periods Coup % YTM Price Spot Rate 0.5 1 3.00% 3.00%1 2 4.00% 4.00%

1.5 3 6.00% 5.00% $101.432 4 7.00% 6.00% $101.86

P = C/(1 + Z1) + C/(1 + Z2)2 + (C + M)/(1 + Z3)3

101.43 = 3/(1 + 0.03/2) + 3/(1 + 0.04/2)2 + 103/(1 + Z3)3

101.43 – 2.96 + 2.88 = 103/(1 + Z3)3

95.59 = 103/(1 + Z3)3

Z3 = (103/95.59)1/3 – 1 = 0.02522 x 0.0252 = 5.04%

11. (6 Points) You have just looked up the STRIPS rates in the table below. Recall that STRIPS rates are “tradable” while implied spot rates are not. In one year you will collect some money which you will not need to use for six months, so you will lend the money in one year for six months. Compute the rate can you lock in today for the forward loan. All rates are in BEY terms.

Years Periods STRIPS Rates

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0.5 1 2.50%1 2 3.10%

1.5 3 4.20%2 4 5.00%

(1 + Z2)2(1 + f) = (1 + Z3)3

f = (1 + Z3)3/(1 + Z2)2 – 1 = (1 + 0.0320/2)3/(1 + 0.0310/2)2 – 1 = 1.0213/1.01552 – 1 = 0.03212 X 0.0321 = 6.42%

12. (5 Points) You believe that in the near future bond market participants will increase their portfolio weights of short-term two-year bonds and decrease their portfolio weights of long-term ten-year bonds. What affect will this “moving down the curve” have on the shape of the yield curve?

If managers buy short-term bonds and sell long-term bonds:The price of short-term bonds will increase and the price of long-term bonds will decrease. The yield on short-term bonds will decrease and the yield on long-term bonds will increase. Therefore the yield curve will steepen.

13. (5 Points) You observe long-term interest rates falling and short-term interest rates rising. You attribute this change to market participants believing long-term rates will be lower in the future. Which yield curve theory covered in class, the text and homework best supports your reason for the change? EXPLAIN.

The Pure Expectations Theory supports this reason for the change in the yield curve.The Pure Expectations Theory claims current prices and yields (and therefore implied spot rates and implied forward rates) exclusively represent expected future rates. There are no “biases” (other factors) in the yields of current bonds.

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