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Math Help - Fixed Income

  1. #1
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    Fixed Income

    Hey guys help me out with question...
    A loan of 75,000 dollars is to be issued bearing interest of 8% per annum,payable quarterly in arrear.The loan will be repaid at par in 15 equal instalments.The first instalment is paid 5 years after the issue date.Find the price to be paid on the issue date by purchaser of whole loan,who wishes to realise a yield of 10% per annum convertible semi_annually.
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  2. #2
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    There are 15 equal repayments in arrears, so the repayments, C are;
    75000=C\cdot a_{\overline{15}|j}

    Where  j=1.08^{.25}-1=0.0194265 is the effective quarterly rate and a_{\overline{15}|j}=\frac{1-(1+j)^{-15}}{j} is the payable in arrears PV annuity formula

    So C=\frac{75000}{12.904565}=$5811.8967

    Given that the first payment of the loan is in 5 years time, the price is;
    P=(1.05^{-10})\cdot \left[5811.8967 \cdot \ddot{a}_{{\overline{15}|k}\right]

    Where k=(1.05^2)^.25-1=.024695 and \ddot{a}_{{\overline{15}|k}}=\frac{1-(1+k)^{-15}}{d} is the payable in advance PV formula with d=\frac{k}{1+k} (Since the first payment is in 5 years of loan being created).

    Therefore,
    P=1.05^{-10} \cdot \left[5811.8967 \cdot 12.71565\right]=$45369.44
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  3. #3
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    Quote Originally Posted by blackmaniacs View Post
    Hey guys help me out with question...
    A loan of 75,000 dollars is to be issued bearing interest of 8% per annum,payable quarterly in arrear.The loan will be repaid at par in 15 equal instalments.The first instalment is paid 5 years after the issue date.Find the price to be paid on the issue date by purchaser of whole loan,who wishes to realise a yield of 10% per annum convertible semi_annually
    Very "unclear". Like, is an interest payment made quarterly during the 1st 5 years?
    Is it 5 years after issue date, or more logically 5 years and 1 quarter?

    To illustrate how this "could" be handled, I'll change your problem slightly:

    A loan of 75,000 dollars is to be issued bearing interest of 2% quarterly.
    The loan will be repaid in 15 equal QUARTERLY payments, the first payment
    being due 5 years and 1 quarter after the issue date. There will be no payments
    during the 1st 5 years, the interest accumulating at above rate of 2% quarterly.
    Find the price to be paid on issue date by a purchaser of the whole loan, who
    wishes to realise a yield 2.5% quarterly.

    (I'll round out to closest dollar)
    The $75,000 will accumulate to $111,446 as at end of 20th quarter:
    75000(1.02)^20 = 111446
    The required quarterly payment (for 15 quarters; quarter 21 to 35) = $8,673:
    11446(.02) / (1 - 1/1.02^15) = 8673

    For the purchase price @ 10% annual cpd. quarterly (2.5% quarterly):
    Present value (as at end of 20th q.) of the 15 payments of $8,673 = $107,384:
    8673(1 - 1/1.025^15) / .025 = 107384
    Present value of $107,384 (as at issue date) = $65,533:
    107384 / 1.025^20 = 65533

    So purchase price to realise 2.5% cpd. quarterly = $65,533

    I see that's quite different from Robb's $45,369.
    Even if I changed my rates to match Robb's (I agree with Robb's calculations
    as far as getting precise effective rates), my $65,533 would not change by much.

    So one of us is "way off!" .... probably me...
    What sayest thou, Robb?
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  4. #4
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    Quote Originally Posted by Wilmer View Post
    What sayest thou, Robb?
    Ah yea, I'm way off

    I took it to be the loan is for $75,000 payable quarterly in arear (ie. issued at time 0, first payment at time 1) however its issued 5 years before the first payment and forgot to work out the future value for when the first payment is made.
    I took the 8% p.a to be effective, so worked out the quarterly effective rate, and 5% half yearly effective rate of retrun.
    So my mistake was: for the original question, the $75000 is worth 75000\cdot 1.08^5=110199.6058 in 5 years time when the first payment is made.
    So using C=\frac{110199.6058}{\ddot{a}_{{\overline{15}|k}}}  =\frac{110199.6058}{13.155256}=$8376.85

    And hence
    P=1.05^{-10} \cdot \left[8376.85 \cdot \ddot{a}_{{\overline{15}|k}\right]=1.05^{-10}\cdot \left[8376.85 \cdot 12.71565\right]=\$65392.26

    A lil' closer..
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  5. #5
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    Quote Originally Posted by Robb View Post
    .....the $75000 is worth 75000\cdot 1.08^5=110199.6058 in 5 years time when the first payment is made.
    So using C=\frac{110199.6058}{\ddot{a}_{{\overline{15}|k}}}  =\frac{110199.6058}{13.155256}=$8376.85
    Agree. Personally, I prefer:
    75000 * 1.08^4.75
    Then call for 1st payment 3 months later.
    Then, being lazy, I don't have to explain to student the "immediate annuity" formula

    I always wondered why "immediate" is sometimes used with loans;
    like, borrow $3000: assume $100 monthly payment called for immediately;
    well, why not borrow $2900 instead!!
    Seems silly to pick up your $3000 cheque, cash it, come back BEFORE MIDNIGHT to make a $100 payment !!
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