A one-year, $100,000 loan carries a coupon rate and a market interest rate of 12 percent. The loan requires payment of accrued interest and one-half of the principal at the end of six months. The remaining principal and accrued interest are due at the end of the year.
a.What will be the cash flows at the end of 6 months and at the end of the year?
Cash flow in 6 months = $100,000 x .12 x .5 + $50,000 = $56,000 interest and principal.
Cash flow in 1 year = $50,000 x 1.06 = $53,000 interest and principal.
b.What is the present value of each cash flow discounted at the market rate? What is the total present value?
$56,000 ( 1.06 = $52,830.19 = PV of CF1
$53,000 ( (1.06)2 = $47,169.81 = PV of CF2
= $100,000.00 = PV Total CF
c.What proportion of the total present value of cash flows occurs at the end of 6 months? What proportion occurs at the end of the year?
Proportiont=.5 = $52,830.19 ( $100,000 x 100 = 52.830 percent.
Proportiont=1 = $47,169.81 ( $100,000 x 100 = 47.169 percent.
d.What is the duration of this loan?
TimeCash Flow PVof CFPV of CF x t
½ year $56,000 $52,830.19 $26,415.09
1 year $53,000 $47,169.81 $47,169.81
Duration = $73,584.91/$100,000.00 = 0.735849 years
Answer question 3
Two banks are being examined by the regulators to determine the interest rate sensitivity of their balance sheets. Bank A has assets composed solely of a 10-year, 12 percent, $1 million loan. The loan is financed with a 10-year, 10 percent, $1 million CD. Bank B has assets composed solely of a 7-year, 12 percent zero-coupon bond with a current (market) value of $894,006.20 and a maturity (principal) value of $1,976,362.88. The bond is financed with a 10-year, 8.275 percent coupon, $1,000,000 face value CD with a yield to maturity of 10 percent. The loan and the CDs pay interest annually, with principal due at maturity.
a.If market interest rates increase 1 percent (100 basis points), how do the market values of the assets and liabilities of each bank change? That is, what will be the net affect on the market value of the equity for each bank?
For Bank A, an increase of 100 basis points in interest rate will cause the market values of assets and liabilities to decrease as follows:
Loan:$120,000*PVIFAn=10,i=13% + $1,000,000*PVIFn=10,i=13% = $945,737.57.
CD:$100,000*PVIFAn=10,i=11% + $1,000,000*PVIFn=10,i=11% = $941,107.68.
Therefore, the decrease in value of the asset was $4,629.89 less than the liability.
For Bank B:
Bond:$1,976,362.88*PVIFn=7,i=13% = $840,074.08.
CD:$82,750*PVIFAn=10,i=11% + $1,000,000*PVIFn=10,i=11% = $839,518.43. The bond value decreased $53,932.12, and the CD value fell $54,487.79. Therefore, the decrease in value of the asset was $555.67 less than the liability.
b.What accounts for the differences in the changes of the market value of equity between the two banks?
The assets and liabilities of Bank A change in value by different amounts because the durations of the assets and liabilities are not the same, even though the face values and maturities are the same. For Bank B, the maturities of the assets and liabilities are different, but the current market values and durations are the same. Thus, the change in interest rates causes the same (approximate) change in value for both liabilities and assets.
c.Verify your results above by calculating the duration for the assets and liabilities of each bank, and estimate the changes in value for the expected change in interest rates. Summarize your results.
Ten-year CD Bank B(values in thousands of $s)
Par value = $1,000Coupon rate = 8.275%Annual payments
R = 10%Maturity = 10 years
Time Cash Flow PV of CF PV of CF x t
1 $82.75 $75.23 $75.23