Rust Bucket Motor Credit Corporation (RBMCC), a subsidiary of Rust Bucket Motor,
ID: 2814305 • Letter: R
Question
Rust Bucket Motor Credit Corporation (RBMCC), a subsidiary of Rust Bucket Motor, offered some securities for sale to the public on March 28, 2008. Under the terms of the deal, RBMCC promised to repay the owner of one of these securities $100,000 on March 28, 2039, but investors would receive nothing until then. Investors paid RBMCC $23,399 for each of these securities; so they gave up $23,399 on March 28, 2008, for the promise of a $100,000 payment 31 years later.
Based on the $23,399 price, what rate was RBMCC paying to borrow money? Suppose that, on March 28, 2018, this security’s price is $41,680. If an investor had purchased it for $23,399 at the offering and sold it on this day, what annual rate of return would she have earned? If an investor had purchased the security at market on March 28, 2018, and held it until it matured, what annual rate of return would she have earned?
Explanation / Answer
Price Paid for one security on March 28, 2008 = $23,399
Price paid at the maturity of secutiy i.e. on March 28, 2039 = $100,000
It is equivalent to a zero coupon bond which does not pay any coupon and pay at maturity
To calculate the rate of interest we need to discount the value at muturity to its present value that investor paid in March 2008
Present value = future value /(1+r)n
where r is interest rate, and n is time period
Present value = $23,399, Future value = $100,000, n = 31 years
23,399 = 100,000/(1+r)31
(1+r)31 = 100,000/23,399
1+r = (100,000/23,399)1/31
r = (100,000/23,399)1/31 - 1 = 4.797%
So the company is paying interest rate of 4.797%
If the security is priced at $41,680 on March 28, 2018
Then to calculate the rate of interest earned we need to discount this value to its value on March 28, 2008
In this case, present value = $23,399, future value = $41,680 and n = 10 years (from March 28, 2008 to March 28, 2018)
Therefore,
23,399 = 41,680/(1+r)10
(1+r)10 = 41,680/23,399
1+r = (41,680/23,399)1/10
r = (41,680/23,399)1/10 - 1 = 5.943%
So interest earned is 5.943%
If a person buy the security on March 28, 2018 and sell it on March 28, 2039 and getting the maturity price we will calculate the interest earned same as in previous case but now the present value will be the value on March 28, 2018 and future value will be value on March 28, 2039
Time period will be 31-10 = 21 years
In this case, present value = $41,680, future value = $100,000 and n = 21 years (from March 28, 2018 to March 28, 2039)
Therefore,
41,680 = 100,000/(1+r)21
(1+r)21 = 100,000/41,680
1+r = (100,000/41,680)1/21
r = (100,000/41,680)1/21 - 1 = 4.255%
So interest earned is 4.255%
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