Question

Today is 1 July 2019. John is 30 years old today. He is planning to purchase an apartment with the price of $800,000 on 1 January 2024. John believes that, at the time of purchasing the house, he should have savings to cover 20% of the house price (i.e., $160,000) on 1 January 2024. John has a portfolio which consists of two Treasury bonds and a bank bill (henceforth referred to as bond A, bond B and bank bill C). There are 200 units of bond A, 300 units of bond B and 400 units of bank bill C.

a)

• Bond A is a Treasury bond which matures on 1 July 2030. One unit of bond A has a coupon rate of j2 = 3.95% p.a. and a face value of $100. John purchased this Treasury bond on 15 February 2017. The purchase yield rate was j2 = 3.85% p.a.

• Bond B is a Treasury bond which matures on 1 January 2026. One unit of bond B has a coupon rate of j2 = 3.7% p.a. and a face value of $100. John purchased this Treasury bond on 1 July 2016. The purchase yield rate was j2 = 4.1% p.a.

• Bank bill C is a 180-day bank bill which matures on 1 September 2019. One unit of bank bill C has a face value of $100. John purchased this bank bill on 15 April 2019. The purchase yield rate was 3.05% p.a. (simple interest rate).

Calculate:

• The purchase price of one unit of bond A

• The purchase price of one unit of bond B

• The purchase price of one unit of bank bill C

b. John decides to sell each of the three security types today. Both the bonds are sold at a yield rate of j2 = 3.2% p.a. and the bank bill is sold at 3% p.a. (simple interest rate).

Calculate • The sale price of one unit of bond A

• The sale price of one unit of bond B

• The sale price of one unit of bank bill C

Round your answer to three decimal places. Then calculate the total sale price of the portfolio (round your answer to the nearest dollar). Note that the sale of each bond occurs after a coupon payment.

c. John plans to use $80,000 of the sale proceeds calculated in part b to invest into a fund today. John predicts that the return rate of this fund will be 1 July 2019 to 30 June 2021 j2 = 5.1%, 1 July 2021 to 31 December 2023 j2 = 5.3%. Calculate the accumulated value of John’s fund investment on 1 January 2024.

d. To save for remaining required amount on 1 January 2024 (the difference between the 20% of the house price and the accumulated value from part c), John plans to deposit z% of his annual after-tax salary into a saving account on 1 July of each year from 2019 to 2023 (5 deposits in total). The saving account rates are assumed to be 0.2% per month. Assume that John’s after-tax salary is $90,000 p.a. Find the value of z (expressed as a percentage and rounded to two decimal places).

e. From John’s perspective, draw a carefully labelled cash flow diagram to represent the above financial transactions of parts c and d.

Answer 1

Facts of the question :						(All amounts are in $ )
Portfolio of John
Particulars	No.of units	Face value	Coupon rate	Purchase yield rate	Date of purchase	Date of maturity
Bond A	200	100	3.95%	3.85%	15-Feb-17	1-Jul-30
Bond B	300	100	3.70%	4.10%	1-Jul-16	1-Jan-26
Bank bill C	400	100	3.05%	3.05%	15-Apr-19	1-Sep-19	180 day bill
Solution :
A) Purchase price :
Purchase yield rate is arrived by dividing the annual interest payments by purchase price. Accordingly, in the below table we shall compute the purchase price.
Purchase yield rate = Annual interest payment/ Purchase price
Therefore, purchase price = Annual interest payment/ Purchase yield rate
Particulars	No.of units	Face value	Coupon rate	Purchase yield rate	Annual interest payment	Purchase price of one unit
Bond A	200	100	3.95%	3.85%	3.95	102.597
Bond B	300	100	3.70%	4.10%	3.70	90.244
Bank bill C	400	100	3.05%	3.05%	3.05	100.000
Note : Annual interest payment is always computed using coupon rate and face value
Yield is computed by dividing the annual interest payment by the current price
B) Going by the same analogy mentioned in A, the sale price computation is as follows.
Particulars	No.of units	Face value	Coupon rate	Yield rate on the date of sale	Annual interest payment	Sale price of one unit	Total sale price
Bond A	200	100	3.95%	3.20%	3.95	123.438	24,687.500
Bond B	300	100	3.70%	3.20%	3.70	115.625	34,687.500
Bank bill C	400	100	3.05%	3.00%	3.05	101.667	40,666.667
Total sale price of the portfolio							100,041.667
C) Accumulated value of John’s fund investment on 1 January 2024 :
Period		No.of years	RoI	Principal amount	Amount at the period end date
Start date	End date
1-Jul-19	30-Jun-21	2	5.10%	80,000.000	88,368.080
1-Jul-21	31-Dec-23	2.5	5.30%	88,368.080	100,579.880
Accumulated value of John’s fund investment on 1 January 2024 :					100,579.880
The fund value is computed using Compound interest formula, (i.e., interest earned is reinvested)
Using compound interest, the value of an investment at the end of a specific period is
found using the formula
A = P* (1+i)^n , where
P = Principal amount
i = R/100
n = No. of years
Note : On 1-Jul-21, $88,368.080 is considered as the principal amount as the it is a continuous investment.
Hence, the fund value at the end of the previous period is reinvested.
D) Value of z% :
Particulars	Amount
20 % of house price	160,000.000
Less : Accumulated fund value as found in Part C
	(100,579.880)
Remaining amount required	59,420.120
Date	Savings account rate	Present value factor 0.20%
1-Jul-19	0.20%	1.0000
1-Jul-20	0.20%	0.9980
1-Jul-21	0.20%	0.9960
1-Jul-22	0.20%	0.9940
1-Jul-23	0.20%	0.9920
Total (Annuity factor)		4.9801
Since, John is depositing z% of his annual after tax salary , it will be annuity as uniform cash flows are made over a specific period of time.
In the case of an annuity , the amount of annuity can be found by dividing the required amount by annuity factor
where, annuity factor is the sum of present value factors. (In this case, annuity factor is computed in the above table)
Hence, going by the above explanation, the amount of investment to be made every year is found as follows.
Particulars		Amount
Remaining amount required	(a)	59,420.120
Annuity factor	(b)	4.980
Annuity amount	(c) = (a) / (b)	11,931.560
Annual after tax salary	(d)	90,000.000
Value of Z %	(e) = (c) /(d) *100	13.26%