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Microeconomic Principles (ECON111)
Macquarie University
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MACQUARIE UNIVERSITY
Macquarie Business School
ACST2001: Financial Modelling
WORKSHOP PROBLEMS (Week 06)
HORIZON ANALYSIS: BONDS & BILLS
A 10% 5-year Treasury bond is bought at $92 (market yield-to-maturity is 12% p.). Assuming a reinvestment rate of j 2 = 10%, what is the holding period yield if the bond is sold after three years, when the market yield is 9%?
How sensitive is the holding period yield (HPY) in Problem 1 to a change in the sale yield? (Try an increase of ten basis points in j 2 .)
What annualized yield would be earned over a 15-day holding period if a $500, 180 -day bank bill was bought at a yield of 13%, and assumed to be sold at 12%?
How sensitive is the holding period yield (HPY) in Problem 3 to an increase of five basis points in the sale yield?
Split the dollar yield (i., the difference between the sale price and the purchase price) in Problem 3 into:
(a) interest component; and
(b) capital gain/loss component.
MACQUARIE UNIVERSITY Macquarie Business School
ACST2001: Financial Modelling
SAMPLE SOLUTIONS TO WORKSHOP PROBLEMS (Week 06)
HORIZON ANALYSIS: BONDS & BILLS
1. $P
$92 $5 $5 $5 $5 $5 $
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When this 5-year 10% T-bond is sold after 3 years, it will then be a 2-year 10% T-bond, so its sale price will be the present vale (at the sale yield, j 2 = 9%) of the remaining future cash flows (four half-yearly coupons of $5 each, plus the maturity amount of $100).
Sale price = 5 a 4 + 100 v 4 at 4%
= 17 + 83.
= $101 (per $100 face value)
Check on price: Yield (9%) is less than coupon rate (10%), so price on an interest coupon payment date (exactly 2 years before maturity) should be greater than maturity amount ($100) ... and it is ($101) ... so it seems reasonable
If each of the six coupons is reinvested, from the date it is received until the sale date, at the rate of j 2 = 10%, then the accumulated value of the coupons at sale is:
5(1+i) 5 + 5(1+i) 4 + 5(1+i) 3 + 5(1+i) 2 + 5(1+i) 1 + 5 where i = 0.
5 s 6 at 5% = $34.
Total accumulated value of the bond investment on sale date
= accumulated value of reinvested coupons + sale price
= 34 + 101 = $135.
This is made up of:
- capital gain (101 – 92) 9 (21%)
- total amount of coupons received (6 u $5) 30 (70%)
- interest earned on reinvested coupons (34 – 30) 4 (9%) $ 43.
- return of original outlay (purchase price) 92. $135.
What yield has the investor actually earned? Let the yield be i per half-year. Then:
Comment: In other cases the HPY will be more sensitive to a change in the sale yield. In other words, a small change in sale yield will cause a larger change in HPY. Try some different combinations to see this for yourself.
3.
Buy ($P 1 ) Sell ($P 2 ) |_________________________________________| m⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ 15 days⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→
Purchase price (P 1 ) = 500,000 / {1 + 0 u (180/365)}
= $469,876.
Sale price (P 2 ) = 500,000 / {1 + 0 u (165/365)} = $472,449.
The dollar yield (or dollar return) from the investment is:
472,449 – 469,876 = $2,572.
Holding period yield (HPY) as a rate of simple interest is:
(472,449 – 469,876) 365 ----------------------------------- u ----- u 100 = 13% pa 469,876 15
Comment: This is not a new type of problem. We are already familiar with how to calculate the holding period yield for a bank bill investment. This problem is included here simply to provide a platform for the next problem, where we explore the sensitivity of the HPY to a change in the sale yield (just as we did in Problem 2 for a bond investment).
- If the sale yield is 5 basis points higher (ie 12% + 0% = 12%), then we have:
Sale price (P 2 ) = 500,000 / {1 + 0 u (165/365)}
= $472,348.
Check on price: Yield is higher (12% to 12%), so price should be lower than $472,449, and it is, so it looks reasonable.
The dollar yield (or dollar return) from the investment is:
472,348 – 469,876 = $2,471.
Holding period yield (HPY) as a rate of simple interest is:
(472,348 – 469,876) 365
----------------------------------- u ----- u 100 = 12% pa 469,876 15
So, an increase of 5 basis points in the sale yield causes a decrease of 52 basis points (13% - 12%) in the holding period yield (HPY). In this case, the HPY is very sensitive to a change in the sale yield.
- The dollar yield (difference between sale price and purchase price), also known as the dollar return, from the investment in Problem 3 can be thought of as made up of two parts:
(a) some interest, based on the dollar yield that would be earned if the market yield remained unchanged from purchase to sale; and
(b) a capital gain/loss, resulting from a change in the market yield between the time of purchase and the time of sale.
(a) Finding the interest component
If the market yield was unchanged from purchase to sale, then the sale price would be:
500,000 / {1 + 0 u (165/365)} = $472,247.
Interest component = Sale price at purchase yield - Purchase price = 472,247 - 469,876. = $2,370.
(b) Finding the capital gain or loss
This is the difference between the actual sale price and the sale price calculated at the purchase yield. It expresses the change in the sale price caused by a change in the market yield between purchase and sale. In this case, the yield fell from 13% to 12% between the purchase date and the sale date, and we have:
Capital gain/loss = Sale price - Sale price at purchase yield = 472,449 - 472,247. = $201.
Summarising, we have: • Interest component 2,370.
- Capital gain/loss 201. $2,572.
Note: The sum of the two components is the dollar yield (that is, the amount – in dollars – earned from the investment). While the interest component will always be positive, the capital gain/loss can be positive (gain) or negative (loss). In this case it was positive because the sale yield (12%) was lower than the purchase yield (13%).
Httpscontent.ilearn.mq Httpscontent.ilearn.mq.edu.au0d460d46dea7a2e8394c4bb30124247577769Gggggggggggggggggggggggggggggggggg e03fbaeresponse-content-disposition=inline 3Bfilename 3 11
Course: Microeconomic Principles (ECON111)
University: Macquarie University
