The present value of cash flows at an effective interest rate of i is given by the following function:

Determine which of the following expressions represents the modified duration of these cash flows.

• [[math]]\frac{1500(1+i)^{-4}-4000(1+i)^{-5}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]
• [[math]]\frac{1500(1+i)^{-3}-4000(1+i)^{-4}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]
• [[math]]\frac{-1500(1+i)^{-4}+4000(1+i)^{-5}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]
• [[math]]\frac{-1500(1+i)^{-3}+4000(1+i)^{-4}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]
• [[math]]\frac{-6000(1+i)^{-5}+20,000(1+i)^{-6}}{100+500(1+i)^{-3}-1000(1+i)^{-4}}[[/math]]

The current yield rates for zero-coupon bonds are as follows:

Calculate the implied two-year spot rate at the end of year two.

• 1.0%
• 4.0%
• 4.5%
• 5.0%
• 6.0%

A customer has the option to either buy a car for cash for 15,000 or lease it with a down payment of 1000 and monthly payments of X payable at the end of each month for three years. Under the purchase option, the car has a value to the customer of 8000 at the end of three years. At the end of the three-year lease, the car is returned and has no value to the customer.

The customer is indifferent to the options at an annual nominal interest rate of 12% convertible monthly.

Calculate X.

• 259
• 269
• 279
• 289
• 299

Current loan rates are based on the following term structure of interest rates:

A loan of 6000 is to be repaid with a single payment of principal plus interest at the end of five years. The borrower has two choices:

1. a 5-year loan.
2. a 3-year loan to be repaid with a single payment of principal plus interest by taking out a 2-year loan at the beginning of year 4.

Calculate the annual effective interest rate on the 2-year loan such that the borrower is indifferent between the two choices.

• 6.25%
• 7.25%
• 8.26%
• 9.54%
• 11.00%

A portfolio consists of two bonds. Bond A is a three-year 1000 face amount bond with an annual coupon rate of 6% paid annually. Bond B is a one-year zero-coupon bond. Both bonds yield an annual effective rate of 4%.

Calculate the percentage of the portfolio to invest in Bond A to obtain a Macaulay duration of two years.

• 44.5%
• 45.6%
• 50.0%
• 54.4%
• 55.5%

A bond is priced at 950, giving an annual effective yield to maturity of 9%. At 9%, the derivative of the price of the bond with respect to the yield rate is -4750 .

Calculate the Macaulay duration of the bond in years.

• 4.59
• 4.62
• 5.00
• 5.41
• 5.45

An investor purchases a portfolio consisting of three bonds. Bond A has annual coupons of 6% and matures for its face amount of 1000 in ten years. It is purchased for 1000. Bonds B and C are zero-coupon bonds, maturing for 1000 each in five and ten years, respectively. All three bonds have the same yield rate.

Calculate the Macaulay duration in years at the time of purchase of the portfolio with respect to the common yield rate.

• 7.23
• 7.43
• 7.60
• 8.33
• 8.38

The prices for four 1000 face amount zero-coupon bonds are as follows:

Determine which of the following statements describes the yield curve underlying these prices.

• The yield curve is increasing with constant slope.
• The yield curve is flat.
• The yield curve is inverted with constant slope.
• The yield curve is concave upward.
• The yield curve is concave downward

A five-year 1000 face amount bond with 10% annual coupons is purchased at time of issue for 1216.47 based on an annual effective interest rate of 5%.

Calculate the Macaulay duration

• 2.9
• 3.2
• 3.7
• 4.1
• 4.3

Two bonds have the same annual effective yield rate, r, where r > 0. The bonds have Macaulay duration of 5 years and 6 years with respect to r. One of the bonds has modified duration of 5.76 years while the other bond has modified duration of d years.

Calculate d.

• 4.760
• 4.800
• 5.208
• 5.240
• 6.912