1. ## 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.

2. There are $\displaystyle 15$ equal repayments in arrears, so the repayments, $\displaystyle C$ are;
$\displaystyle 75000=C\cdot a_{\overline{15}|j}$

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

So $\displaystyle C=\frac{75000}{12.904565}=$5811.8967$Given that the first payment of the loan is in 5 years time, the price is;$\displaystyle P=(1.05^{-10})\cdot \left[5811.8967 \cdot \ddot{a}_{{\overline{15}|k}\right]$Where$\displaystyle k=(1.05^2)^.25-1=.024695$and$\displaystyle \ddot{a}_{{\overline{15}|k}}=\frac{1-(1+k)^{-15}}{d}$is the payable in advance PV formula with$\displaystyle d=\frac{k}{1+k}$(Since the first payment is in 5 years of loan being created). Therefore,$\displaystyle P=1.05^{-10} \cdot \left[5811.8967 \cdot 12.71565\right]=$45369.44$

3. Originally Posted by blackmaniacs
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?

4. Originally Posted by Wilmer
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 $\displaystyle 75000\cdot 1.08^5=110199.6058$ in 5 years time when the first payment is made.
So using $\displaystyle C=\frac{110199.6058}{\ddot{a}_{{\overline{15}|k}}} =\frac{110199.6058}{13.155256}=$8376.85$And hence$\displaystyle P=1.05^{-10} \cdot \left[8376.85 \cdot \ddot{a}_{{\overline{15}|k}\right]=1.05^{-10}\cdot \left[8376.85 \cdot 12.71565\right]=\$65392.26$

A lil' closer..

5. Originally Posted by Robb
.....the $75000 is worth$\displaystyle 75000\cdot 1.08^5=110199.6058$in 5 years time when the first payment is made. So using$\displaystyle 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 !!