BE631 – Risk Management and Financial Institutions
Seminar 1
Questions with analytical answers
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Question 1
Suppose that as a fixed income trader for a bank you currently are holding the following fixed income portfolio of assets and liabilities:
Assets: $1 million face value, 6-year coupon bond. 4.5% annual coupon payment, 3.5% yield to maturity.
$2 million face value, 2-year zero coupon bond. 2% yield to maturity.
Liability: $3 million face value, 1-year zero coupon bond. 1.75% yield to maturity.
a) Assuming that when you set up these three positions, the total purchase price of the two assets was exactly equal to the funding generated by the issuance of your liability, determine the current amount of profits for this portfolio. This is its net worth.
Solution:
Asset side
C = Coupon rate*face value = 1000000*0.045 = $45000
Price of coupon bond = PV(coupon bond) = C/(1+ytm)+ C/(1+ytm)^2+ C/(1+ytm)^3+ C/(1+ytm)^4 +C/(1+ytm)^5+ (C+face value)/(1+ytm)^6 =
= 45000/(1.035)+ 45000/(1.035)^2+ 45000/(1.035)^3+ 45000/(1.035)^4 +45000/(1.035)^5+ (1045000)/(1.035)^6 = $1,053,286
Market Price of zero-coupon bond = PV(zero coupon bond) = face value/(1+ytm)^2 = 2000000/(1.02)^2= $1,922,338
Market value of assets is therefore = market value (present value (PV) of coupon bond + market value (PV) of zero coupon bond = $1,053,286 +$1,922,338= $2,975,624
Liability side
Market value of liability = 3000000/(1+0.0175) = $2,948,403
Net worth = market value of equity = market value of assets – market value of liabilities= $2,975,624 - $2,948,403 = $27,220
(Assets = Equity + Liabilities => Equity = Assets – Liabilities)
b) Determine the total Macaulay durations of your assets and your liability. Comment on the discrepancy, specifically, what is the direction of interest rate changes that will lead to a reduction in this portfolio’s net worth?
Solution:
Asset side
Macaulay duration of coupon bond = [1*C/(1+ytm)+ 2*C/(1+ytm)^2+ 3*C/(1+ytm)^3+ 4*C/(1+ytm)^4 +5*C/(1+ytm)^5+6* (C+face value)/(1+ytm)^6 ] / Price of coupon bond =
=[1*45000/(1.035)+ 2*45000/(1.035)^2+ 3*45000/(1.035)^3+ 4*45000/(1.035)^4 +5*45000/(1.035)^5+6* (1045000)/(1.035)^6 ] /1,053,286 = 5.41 years
Macaulay duration of 2-year maturity zero coupon bond = 2 years (the duration of the zero coupon bond is always equal to its maturity)
Weighted average Macaulay duration of assets =( duration of coupon bond*price of coupon bond + duration of zero-coupon bond*price of zero-coupon bond)/ market value of assets (the sum of bond prices belonging to assets) = (5.41 * 1053286 + 2 * 1922338)/2975624 = 3.21 years
(2975624 = 1053286+1922338)
Liability side
Macaulay duration of liability = 1 year
So the portfolio will be hurt if interest rates rise, as assets have longer duration than the funding (liabilities). In the scenario of rising interest rates, assets (more interest rate sensitive) will fall more than liabilities (less rate sensitive). Hence the net worth (equal to assets-liabilities) will fall in value.
Question 2
We are currently in a global macro and political environment where there is widespread expectation that interest rates, starting from the USA, will rise significantly from their historic lows. Suppose that you are a member of the Asset-Liability Management (ALM) committee at a US commercial bank and are concerned about the effect of rapidly rising interest rates on your bank’s profitability. Discuss how each action listed below could be used to alter the interest rate sensitivity of your bank’s balance sheet and what they could mean for its interest margins.
a) Securitising a portion of the bank’s mortgage loan portfolio and investing the cash received in Treasury bills.
Solution: Securitising mortgages means bundling the loans into pools and selling them to investors. If the bank buys T-bills which by definition have shorter than one year maturities with the cash it received from the securitisation, then the net effect will be a lowering of asset duration in the bank’s balance sheet. So the bank will have less sensitivity to rates rising going forward.
The flip side is usually a decrease in interest income. Mortgage loans in general produce higher interest income than T-bills, so the bank would be giving that extra interest income up with this move.
b) Entering into pay-fixed, receive floating interest rate swaps with average maturity of 10 years.
Solution: this is equivalent to issuing a 10-year bond and placing the proceeds into 3 or 6-month LIBOR. So this action too will shorten asset duration, and it will at the same time lengthen liability duration. With an upward sloping yield curve, the receipts at least initially will be less than the payments on the swap, unless interest rates shift up considerably and fast.
c) Shorting Treasury bond futures.
Solution:
This will produce hedging income if medium to long-term interest rates rise. Recall that bond prices decrease if their yields rise and that the deliverable bonds in the various futures contracts that exist futures exchanges around the world usually have maturities that are in excess of 5-6 years. Margin payments will need to be made if rates go down, so the cash flow implications of that eventuality need to be taken into account. The yield curve may also twist in such a manner that while short term rates rise, thereby increasing funding costs for the bank, longer term rates that would apply to the deliverable bonds for the bond futures in question could stay the same or even fall. In that case the hedge would be grossly ineffective.
Question 3
Suppose that the one and two-year zero-coupon bond rates are 2% and 2.2%, respectively. Determine the rate on the 12x24 forward rate agreement (FRA) (12x24 means settlement is a year from now for delivery of the 1-year LIBOR at that date).
Solution:
From slide example: 1 + FRA12,24 = (1+z2)2 / (1+z1)
Z1: the one year zero coupon interest rate
Z2: the two year zero coupon interest rate
Under the law of no-arbitrage, the two year investment must be equal to the 1-year investment and the subsequent 1-year forward rate investment.
1* (1+Z1)(1+FRA12,24) = 1*(1+Z2)2 =>
1 + FRA12,24 = (1+z2)2 / (1+z1)
So 1+FRA = 1.0222/1.02 à FRA = 0.024 or 2.4%
Question 4
For the FRA contract in Question 3, determine the settlement payment that the buyer has to make on $100 million notional principal if at the settlement date (which is one year from the date of purchase of the FRA) one-year LIBOR turns out to be 1.9%
Solution:
The buyer pays FRA and receives LIBOR. Since LIBOR is less than the FRA rate in this case, the buyer needs to make a payment. The payment amount is 100,000,000*(0.024-0.019)/1.019 = $ 490,677. Note the present value adjustment in the payment calculation. This is necessary because the FRA is settled a year from now while the interest payments are due 2 years from now. See graph in lecture slides for timings.
