Accounting Assignment help on : Market value of the bonds

Section 1:

1)       The exact change in the market value of the bonds is calculated by calculating the price of each bond for three different interest rates, i.e. the existing interest rates and the increased and decreased rates. The given market interest rate is 1.7%, therefore, the price changes for both increase and decrease of 1 percentage point (1%) is calculated for all the three bonds using the PRICE() function in MS Excel. However, in case of 5 percentage points (5%) change, only the price for increase could be calculated as negative yield is not allowed in the PRICE() function. A negative yield indicates that the present value is less than the future value of the bond which is rarely possible, unless the economy is in a recession. The table below shows the calculated values:

Macaulay Duration indicates the average time (years) in which the present value of the investment is received in a series of coupon payments. Duration is also a measure of sensitivity of the bond’s price to interest rates. The Duration of each of the three bonds was calculated using the DURATION() of MS Excel. The price change in percentage for every ∆i change in interest rate (i) is given as:

% ∆P = (-)D*∆i/(1+i)

Where D- Duration, ∆P: Change in Price

The table below shows the calculated amount of change in the values of the bonds due to the change in interest rates:

2)      In the question above we saw the effect of change in interest rates on the price of the bonds. From the tables we can clearly see that the exact change in the bond prices are not equal to the changes arrived at using Macaulay Duration. This can be explained as shown in the figure below

Figure 1: Price/Yield Relationship and Duration, Bond Education: Understanding a Callable, Sherwin (2010)

We can clearly see that Duration is a straight line tangent to the price curve of a bond. This means that Duration of a bond cannot always predict the exact change in the value of the bond. After careful observation of the above graph we can infer that at the point of intersection and at the yield closer to the point of intersection, duration tends to predict the exact change. Therefore, only for very small changes in the yield the Duration method can predict the exact change. Thus, in our case both 1 and 5 percentage points are large amount of changes in the interest rates due to which the duration method could not predict the exact changes in the price. Further, the time to maturity of the bond also affects the duration and indirectly affects the change in price of the bond. We can see that Duration is higher for a bond with longer time to maturity than that of the lower time to maturity keeping all other parameters same. This is due to the fact that for longer term to maturity bonds tend to repay over a longer period of time than that of the shorter counterparts. Thus, Duration alone is not sufficient to predict the price sensitivity of bonds to interest rates.

Section 2

3)      Duration gap is calculated as given below (Mishkin):

The percentage change in the value of equity is calculated as (Mishkin):

Prior to using these formulas, the market value of assets and liabilities was calculated using the PRICE() function  of MS Excel. This was followed by calculating Duration of each of the bonds in assets and liabilities and finally the duration of assets and liabilities were calculated as weighted average of duration of individual bonds in assets and liabilities. The table below shows the change in equity calculated from duration gap:

4)      The duration gap can be used to assess the sensitivity of the value of equity to change in interest rates. In order to calculate the size of the interest rate shock that could result in insolvency of the bank, a linear programming method was used. MS Excel provides a data analysis tool named Solver using which the equity value was equated to zero and the interest rate shock was allowed to take any integer between 0 and 1. The output is shown below:

The output from the model is 36.22 percentage points, i.e. the interest rate needs to increase by more than36.22 percentage points in order to make the bank insolvent. Such a situation is not normal as interest rates under normal conditions vary mostly in basis points (1 hundredth of a percentage point). Thus, such huge changes in the interest rates are not expected. However, in such conditions the bank will not be able service their liability and will eventually become insolvent as the value of its liabilities would be greater than that of the assets.

Section 3

5)      An interest swap consists of two parties who have agreed upon to exchange interest rates on the basis of a notional principal. However, the notional principal is never actually paid by one party to the other. As we know the swap rate, we can calculate the amount of principal needed to be set as notional principal so that the bank is able to hedge against a 1% increase in the interest rates. The principal was calculated by dividing the change in the value of the equity for 1% increase in interest rate by the swap rate. The fictive principal was found to be 3102.51

6)      The bank enters into a swap to receive the fixed change (swap rate) in the value of equity in exchange for the percentage change in the interest rate. This enables the bank to stem the volatility in the equity. Since the Duration Gap is positive, the value of assets will reduce more as compared to the value of liabilities, thus, decreasing the equity.  The swap’s payments for the next three years will be calculated as per the notional amount (15069.32). Settlements in a swap contract are netted out and the party which has the net-liability pays the required amount to the other party. In case of 1% change in interest rates, there will be a net payment of amount equal to that of the change in equity due to the same amount of change in interest rates, as the notional amount was fixed so as to offset the effect of 1% increase rates. However, in case of a 5% change, the bank will pay the difference between the change in equity after 5% increase and the fixed change of value in equity (at swap rate). There is an outflow from the bank as the interest rates have risen sharply and the swap rate is lower than the new market rate.

Section 4

7)      The spot rates calculation using bonds is known as bootstrapping and it is done in order to draw the term structure of interest rates. The term structure represents the spot rates on zero coupon bonds for different maturities and its shape represents the state of the business cycle an economy currently is. In our example, 4 government bonds are given with term to maturity of 1 through 4 years respectively. Thus, the term structure for 4 years can be calculated using the Bootstrapping method. The table below shows the calculated spot rates :

8)      In order to determine if the swap rate offered to the bank was arrived at by fair valuation, we need to verify by taking a nominal amount and calculating the implied swap rate (3.4%). It was calculated as shown below using the spot rates calculated in the previous question. Thus, the swap rate should have been fixed at 5.379% rather than 3.400%, this implies non-fair value pricing of the swap. Such conditions would always result in one of the parties paying most of the settlements during the life of the swap. In this case, if there is a 1% increase in the interest rate (yield = 2.70%) the bank’s equity reduces by 105.49, this amount will be exactly offset by the swap as it has been specifically designed to do so. However, if there is a 5% increase in the interest rates (yield = 6.70%), the bank’s equity reduces by 507.33 and this amount cannot be recovered through the swap, rather the bank has to make a further payment of 497.29 as a result of the swap agreement. If the swap rate had been 5.379% instead, then the bank would have been paying only 52.01 (nominal principal = 3937.49) as the swap rate is lesser than the market interest rate (yield). As we have seen that in both the cases bank would have been paying the counterparty, but the second case exhibits fair pricing of swap.

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