Question

1. Variable Interest Rates Jodie is a medical student who has accumulated 24,000$ of credit card debt on numerous credit cards. She is looking to consolidate her debt. On 1/1/2010, she visits her bank and she is offered the following debt consolidation plan: the bank will lend her 24,000$ to pay off al1 her outstanding credit card balances on that day, in return for a fixed payment of 11,520S on 1/1/2011, plus a payment of 8,000*(1+0.01+Prime Rate on 1/1/2012)S on 1/1/2012, and a payment of 8,000 (1+0.01+Prime Rate on 1/1/2013)S on 1/1/2013 The Prime Rate is a market loan rate that is quoted on all the major financial newspapers It is the yearly rate for loans to prime borrowers, that is, borrowers who will almost certainly be able to pay back. Assume that the prime rate is 4.25% on 1/1/2012 and 7.50% on 1/1/2013. Compute the yield to maturity for Jodie's loan. Assume that the prime rate on 7.50% on 1/1/2012 and 4.25% on 1/1/2013. Compute the yield to maturity on Jodie's loan. Provide intuition on how the sequencing of interest payments affects the yield t<o maturity

Answer 1

Given that,

Accumulated Credit card debt = $ 24,000

For repaying the loan, the bank offered to lend $24,000 on 01/01/2010 with the conditions that Jodie makes specified payments on 1st Jan 2011, 2012 and 2013, that is, exactly after one year, two years and three years respectively

Jodie has to pay

=> On 1/01/2011, $ 11,520

=> On 1/01/2012, $ 8,000(1+0.01+0.0425) = $ 8,420 .....(Prime rate on 1/01/2012 is 4.25% = 0.0425)

=> On 1/01/2013, $ 8,000(1+0.01+0.0750) = $8,680 .....(Prime rate on 1/01/2012 is 7.50% = 0.0750)

YTM is the rate at which the present value of all the payments to be made by Jodie equal to $ 24,000 as paid by the bank to Jodie today. Let YTM = i

So, Amount of loan = PV of all future payments to be made by Jodie.

So, 24,000 = 11520/(1+i)¹ + 8420/(1+i)² + 8680/(1+i)³

Lets find the value of i by trial and error.

At i = 10%, Sum of Present value of all payments to be made by Jodie = $ 23,952

We are very close to the YTM.

At i = 9.85%, Sum of Present value of all payments to be made by Jodie = $ 24,013

By using interpolation, we find that i = 9.88% as the Sum of Present value of all payments to be made by Jodie = $ 24,000

Therefore YTM = 9.88%

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Assuming that the Prime rate is 7.50% on 01/01/2012 and 4.25% on 01/01/2013,

Jodie made the following payments:

=> On 1/01/2011, $ 11,520

=> On 1/01/2012, $ 8,000(1+0.01+0.0750) = $ 8,680 .....(Prime rate on 1/01/2012 is 7.50% = 0.0750)

=> On 1/01/2013, $ 8,000(1+0.01+0.0425) = $8,420 .....(Prime rate on 1/01/2012 is 4.25% = 0.0425)

Let YTM = i

So, 24,000 = 11520/(1+i)¹ + 8680/(1+i)² + 8420/(1+i)³

Lets find the value of i by trial and error.

At i = 10%, Sum of Present value of all payments to be made by Jodie = $ 23,972

We are very close to the YTM.

At i = 9.9%, Sum of Present value of all payments to be made by Jodie = $ 24,376

By using interpolation, we find that i = 9.93% as the Sum of Present value of all payments to be made by Jodie = $ 24,000

Therefore YTM = 9.93%

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We find that in the first case, YTM₁ = 9.88%. In the second case, YTM₂ = 9.93%

YTM₂ is slightly higher than YTM₁. Thus, the bank has earned more when the Prime rate was higher in the second year at 7.50% and then decreased to 4.25% (as in the second case), as compared to the first case.

This has happened because in the second case, the amount to be paid by Jodie in the second year was greater than the payment to be made in the third year.

From this, we understand that, if bigger amount is received in the early years, the YTM or the internal rate of return earned on the loan by the bank increases.

As the given example is of a loan term of just 3 years, the difference between interest rates is less. had the term of loan been 5 or 6 years, the difference in the interest rates would have been greater.