NUMERICAL EXAMPLE: CALCULATING THE FAIR VALUE UF A LOAN
To illustrate the points mentioned in the previous section let us try to calculate the fair value of a set of four loans issued by a development institution to four borrowers, in this case China, Argentina, Uruguay, and Ukraine.
In Figure 3.4a we show the term structure of CDS rates used for each of the borrowers. We use these, as shown in Section 3.2.2, to find the hazard rates and survival probabilities of each borrower. We assumed that we have issued to each of the four borrowers on September 30, 2012, a 10year loan with a principal of USD 100M linearly amortizing every six months and with interest payments of USD sixmonth LIBOR plus 30 bps.
FIGURE 3.4 a) The term structures of CDS rates for the borrowers used in the example; b) the fair value of the loan as a percentage of the principal (primary axis) and the value, in basis points, of the prepayment option (secondary axis).
TABLE 3.1 The detailed output of the fair value of the loan to China.
Date 
Principal amount (USD M) 
USD discount factor 
Survival probability 
Interest amount 
Principal repayment 
Principal repayment in case of default 
3/31/2013 
100 
0.9981 
0.9864 
333,624 
4,922,833 
233,878 
9/30/2013 
95 
0.9954 
0.9804 
391,050 
4,879,599 
198,512 
3/31/2014 
90 
0.9908 
0.9739 
535,299 
4,824,587 
203,161 
9/30/2014 
85 
0.9847 
0.9671 
617,673 
4,761,839 
196,156 
3/31/2015 
80 
0.9756 
0.9592 
812,351 
4,679,175 
213,757 
9/30/2015 
75 
0.9647 
0.9507 
884,707 
4,585,389 
215,034 
3/31/2016 
70 
0.9510 
0.9407 
989,924 
4,473,151 
229,360 
9/30/2016 
65 
0.9357 
0.9299 
1,012,386 
4,350,330 
224,532 
3/31/2017 
60 
0.9187 
0.9200 
1,011,958 
4,226,389 
184,876 
9/30/2017 
55 
0.9003 
0.9112 
991,416 
4,102,067 
147,201 
3/31/2018 
50 
0.8824 
0.9025 
866,369 
3,981,816 
129,997 
9/30/2018 
45 
0.8638 
0.8939 
803,866 
3,860,485 
110,873 
3/31/2019 
40 
0.8435 
0.8846 
760,387 
3,731,093 
101,774 
9/30/2019 
35 
0.8227 
0.8751 
676,458 
3,599,830 
86,918 
3/31/2020 
30 
0.8047 
0.8656 
497,733 
3,482,845 
71,536 
9/30/2020 
25 
0.7866 
0.8563 
412,509 
3,368,032 
54,751 
3/31/2021 
20 
0.7578 
0.8470 
338,312 
3,251,605 
40,110 
9/30/2021 
15 
0.7488 
0.8378 
252,494 
3,137,073 
25,526 
3/31/2022 
10 
0.7290 
0.8274 
172,847 
3,016,060 
14,210 
9/30/2022 
5 
0.7093 
0.8162 
84,918 
2,894,763 
14,211 
In Table 3.1 we show the results of the calculations in detail for one of the borrowers, China. In the first column we show the date on which the interest is paid and the principal amortizes; in the second column we show the principal on which, on that date, the interest is paid; in the third and fourth columns we show the USD discount factor and China's survival probability on that day; in the fifth and sixth we show the present value of the interest payment and the principal repayment contingent on China's survival; in the seventh column we show the present value of the payout in case of default.
Breaking down Equation 3.23 we can see that the interest payment at time Tj is given by
Using actual numbers for the interest payment due on March 31, 2014, we have
(3.30)
as shown in Table 3.1. For the interest payment we have which, using actual numbers also for March 31, 2014, we have
(3.31)
as shown in the table. The calculation of the payment in case of default is slightly more complicated. As seen in Equation 3.23 from the fact that the summation has a different index, for each payment date (if we want to visualize it as done in the table) we are supposed to integrate over the period to that date from the previous one and calculate the payout in case the borrower defaults on the amount outstanding in that time interval. In order to do that, we start at the end of the period and take two points, the last day Tn, and another point Tn1 a few days earlier (for example, two days, the shorter the time difference the more accurate the calculation); we calculate the probability of the borrower defaulting between Tn1 and Tn, conditional on survival up to Tn1 by taking (); we multiply this by the outstanding principal amount in that interval; we multiply this by the assumed recovery rate; finally we discount this value.^{[1]} We repeat the process now between Tn2 and Tn1 and so forth until we reach the beginning of the interval and then we sum all the values obtained. This is what is shown in the seventh column of Table 3.1.
Table 3.2 shows a summary of the same calculations done for the four loans. The value of the loan (without prepayment option) is also shown, as percentage of the loan principal, on the primary axis of Figure 3.4b. We immediately see what we meant by saying that the lender is holding assets of very different values. Not only is none of the loans worth par (the principal amount), but, for example, the loan to Ukraine is worth more than 10% less
TABLE 3.2 The summary of the fair values of the four loans.
China 
Argentina 
Uruguay 
Ukraine 

Interest payment 
12,446,282 
12,229,563 
11,656,080 
8,788,762 
Principal repayment 
82,852,333 
81,847,125 
79,193,556 
66,210,098 
Total value without option 
95,271,615 
94,076,688 
90,849,636 
74,998,860 
Prepayment option value 
(333,332) 
(63,174) 
(32,773) 
(11,630) 
than the one to China (which implies, as we would expect, that the credit risk associated with China is a lot smaller than the one associated with Ukraine).
In Table 3.2 we present the value of the prepayment option, which is also shown on the secondary axis of Figure 3.4b. We have implemented the model mentioned in the previous section (and shown in Appendix C) by solving the PDE using an alternating direction explicit method. Essentially, the value of the option, like in any situation involving a prepayment or callable feature, is the difference between the value of the loan should the borrower prepay it and the value should the borrower hold it to maturity. Since it is the borrower who holds the option, the value is negative as we price the loan from the development bank's point of view.
Observing Figure 3.4b, we can immediately answer pictorially the question about the sensitivity of the option value to the credit standing of the borrower. If we assume that the par level (i.e., unity in our graph on the primary axis) is the level at which a borrower could get a loan in the market outside development institutions, how does this relate to the option value shown in basis points on the secondary axis? A borrower who could realistically obtain another loan somewhere else would attribute value to the option to do so; a borrower who cannot will consider the option as having almost no value at all. In the graph shown in Figure 3.4b, China's loan value is close to par, meaning that it is not inconceivable that its own credit standing might one day improve to the level of pushing the loan value above par (the trigger indicating that a loan in the forprofit market is an attractive alternative). Ukraine has almost no such hope. This is in line with the values of the four different options, which show that the optionality embedded in the loan to China is far more valuable than the one embedded in the loan to Ukraine. It is probably not a surprise to the reader the fact that, in general and in particular as we can see from our realistic example, the optionality embedded in loans extended by development organizations to developing countries is very small.
 [1] The choice of which discount factor to use is an open one. Since we are assuming the time between the two points tending to zero we could, for example, take the discount factor at a time .