imported>Doug Williamson |
imported>Doug Williamson |
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| The zero coupon rate is also known as the [[zero coupon yield]], spot rate, or spot yield.
| | ''Education - diversity and inclusion.'' |
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| | Lifelong learning is the concept of individuals continuing to develop over their entire lifespan. |
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| | It may include informal learning, as well as academic and professional development. |
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| ==No arbitrage conversion principles==
| | It is not confined to immediately practical knowledge and skills. |
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| If we know the zero coupon rates (yield curve) for a given risk class and set of maturities, we can calculate both the [[forward yield]]s and the [[par yield]]s for the same maturities and risk class.
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| | :<span style="color:#4B0082">'''''There are some important things we'd all benefit from knowing - OECD'''''</span> |
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| The conversion process and calculation stems from the '[[no-arbitrage]]' relationship between the related yield curves.
| | :"Lifelong learning occurs in formal, non-formal and informal settings. |
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| This means - for example - that the total cumulative cash flows from a two-year investment must be identical, whether the investment is built:
| | :It involves individuals both young and old, with different experiences, motivations and attitudes. |
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| * '[[Outright]]' from a two-year zero coupon investment
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| * Or as a [[synthetic]] deposit built using a forward contract, reinvesting intermediate principal and interest proceeds at a pre-agreed rate
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| * Or using a par investment, reinvesting intermediate interest to generate a total terminal cash flow
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| | :These differences influence not only whether and how people learn, but also their learning objectives. |
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| ==Zero coupon rate to forward rate conversion==
| | :It is therefore essential to diversify the content and provision of learning, to account for the inherently heterogeneous nature of learning and learners. |
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| <span style="color:#4B0082">'''Example 1: Converting from one and two-period zero coupon yields to forward yields'''</span>
| | :However, heterogeneity in learners’ profiles, learning objectives, socio-demographic characteristics, attitudes towards learning and learning goals should be managed so that it does not lead to a strong separation in learning pathways, with important risks to social cohesion and inclusiveness." |
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| Periodic zero coupon yields ('''z''') are:
| | :''OECD Skills Outlook 2021 : Learning for Life.'' |
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| z<sub>0-1</sub> = 0.02 per period (2%)
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| z<sub>0-2</sub> = 0.029951 per period (2.9951%)
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| The cash returned at Time 2 periods in the future, from investing £1m at Time 0 in a zero coupon instrument at a rate of 2.9951% per period, is:
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| £1m x 1.029951<sup>2</sup>
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| = £'''1.0608'''m
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| Under no-arbitrage pricing conditions, the identical terminal cash flow of £1.0608m results from investing in an outright zero coupon investment of one period's maturity, together with a forward contract for the second period - for reinvestment at the forward market yield of '''f<sub>1-2</sub>''' per period, as follows:
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| £1m x (1 + z<sub>0-1</sub>) x (1 + f<sub>1-2</sub>) = £'''1.0608'''m
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| Using this information, we can now calculate the forward yield for 1-2 periods' maturity.
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| 1.02 x (1 + f<sub>1-2</sub>) = 1.0608
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| 1 + f<sub>1-2</sub> = 1.0608 / 1.02
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| f<sub>1-2</sub> = (1.0608 / 1.02) - 1
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| = 1.04 - 1
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| = '''0.04''' per period (= 4%)
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| This is the market forward rate which we would enjoy if we were to pre-agree today, to make a one-period deposit, committing ourselves to put our money into the deposit one period in the future.
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| ==Is terminal cash the same in each case?==
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| The no-arbitrage relationship says that making such a synthetic deposit should produce the identical terminal cash flow of £1.0608m.
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| Let's see if that's borne out by our calculations.
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| Investing the same £1m in this synthetic two-periods maturity zero coupon instrument would return:
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| After one period: £1m x 1.02 = £1.02m
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| Reinvested for the second period at the pre-agreed rate of 0.04 per period for one more period:
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| = £1.02m x 1.04
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| = £'''1.0608'''m
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| ''This is indeed the same result as enjoyed from the outright zero coupon investment, as expected.
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| (The one period forward rate f<sub>0-1</sub> represents the identical deal to the one period zero coupon rate z<sub>0-1</sub>. For this reason the rate is also identical = 2% per period.)
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| ==Zero coupon rates to par rates conversion==
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| <span style="color:#4B0082">'''Example 2: Converting from zero coupon rates to par rates'''</span>
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| Again using the given zero coupon rates ('''z'''), the par rates ('''p''') can also be calculated.
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| The periodic zero coupon yields ('''z''') are:
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| z<sub>0-1</sub> = 0.02 per period (2%)
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| z<sub>0-2</sub> = 0.029951 per period (2.9951%)
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| The no-arbitrage relationship between par rates and zero coupon rates is summarised in the formula:
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| p<sub>0-n</sub> = (1 - DF<sub>n</sub>) / CumDF<sub>n</sub>
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| ''Where:''
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| p<sub>0-n</sub> = the par rate for maturity n periods, starting now
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| DF<sub>n</sub> = the discount factor for 'n' periods maturity, calculated from the zero coupon rate (z<sub>n</sub>)
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| CumDF<sub>n</sub> = the total of the discount factors for maturities 1 to 'n' periods maturity, again calculated from the zero coupon rates (z<sub>1</sub> to z<sub>n</sub>)
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| ''Applying the formula:''
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| p<sub>0-2</sub> = (1 - DF<sub>2</sub>) / CumDF<sub>2</sub>
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| p<sub>0-2</sub> = (1 - 1.029951<sup>-2</sup>) / (1.02<sup>-1</sup> + 1.029951<sup>-2</sup>)
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| = 0.029803 (= 2.9803% per period)
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| This is the theoretical fair (no-arbitrage) market price for the par instrument.
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| It is the calculated rate of interest payable on a two-period investment on par rate terms. This means that 2.9803% interest will be paid on the amount of the original investment, at Times 1 and 2 periods. In addition, the original investment will be repaid at Time 2.
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| (The one period par rate p<sub>0-1</sub> represents the identical deal to the one period zero coupon rate z<sub>0-1</sub>. For this reason the rate is also identical = 2% per period.)
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| ==Terminal cash is still the same==
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| In theory, an investment of £1m in a par instrument should produce the same terminal cash flow as a zero coupon instrument or forward instruments. On these figures (forward rates of 2% and 4% for 0-1 and 1-2 periods maturity, respectively) we saw that was £1.0608m for an original investment of £1m.
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| Let's see if it's the same for the par instrument, assuming we arrange for the re-investment of any intermediate cash flows at today's forward rates.
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| Cash flows from the two period par instrument, paying periodic interest of 2.9803% per period, assuming an initial investment of £1m:
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| Interest coupon at Time 1 period = £1m x 0.029803 = £<u>0.029803</u>m
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| Principal + interest at Time 2 periods = £1m + 0.029803m = £'''1.029803'''m
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| The coupon receivable at Time 1 period is reinvested at the pre-agreed forward rate of 4% (0.04) for the maturity 1-2 periods.
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| So the Time 2 proceeds from the reinvested coupon received at Time 1 are:
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| £0.029803 x 1.04
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| = £'''0.030995'''m at Time 2
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| The total terminal value at Time 2 periods is:
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| 0.030995 + 1.029803
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| = £'''1.0608'''m (as before)
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| The par rate we have calculated is indeed consistent with the no-arbitrage pricing relationship.
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| ==Present value is the same too==
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| No-arbitrage pricing also says that the present value of the par instrument should be the same as the present value of the zero coupon and forward instruments.
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| Present values are calculated from discount factors based on the periodic zero coupon rates.
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| Continuing with our examples above, the zero coupon rates are:
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| 0-1 period: 0.02 (2%) per period
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| 0-2 periods: 0.029951 (2.9951%) per period
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| The related discount factors are:
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| 1 period: 1.02<sup>-1</sup>
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| 2 periods: 1.029951<sup>-2</sup>
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| The cash flows from the zero coupon instrument and the forward instrument are a cash inflow of £1.0608m at Time 2 periods.
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| The present value is:
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| £1.0608m x 1.029951<sup>-2</sup>
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| = £'''1.0000'''m
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| <span style="color:#4B0082">'''Example 3: A par bond trades at par'''</span>
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| A two-period par instrument pays periodic coupons of 2.9803%.
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| Prove that a £1m face value bond has a total present value of par (£1m), using the figures above.
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| ''Solution''
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| Cash flows from the two period par instrument, paying periodic interest of 2.9803% per period, with a face value of £1m:
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| Coupon at Time 1 period = £1m x 0.029803 = £<u>0.029803</u>m
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| Principal + coupon at Time 2 periods = £1m + 0.029803m = £'''1.029803'''m
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| The present values of these cash flows are:
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| Time 1: £0.029803 x 1.02<sup>-1</sup> = 0.02922
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| Time 2: £1.029803 x 1.029951<sup>-2</sup> = 0.97078
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| Total = £'''1.0000'''m
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| The 'par bond' trades at par, as expected.
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| The pricing is consistent with no aribtrage pricing conditions.
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| == See also == | | == See also == |
| * [[Zero coupon yield]] | | * [[ACT Diversity and Inclusion Calendar]] |
| * [[Forward yield]] | | * [[Continuing education]] |
| * [[Converting from forward rates]] | | * [[Continuing professional development]] |
| * [[Par yield]] | | * [[Continuing professional education]] |
| * [[Converting from par rates]] | | * [[Corporate governance]] |
| * [[Bootstrap]] | | * [[Corporate social responsibility ]] |
| * [[Par bond]] | | * [[Diversity]] |
| * [[Coupon]] | | * [[Equitable]] |
| * [[Spot rate]] | | * [[Further education]] |
| * [[Yield curve]] | | * [[Higher education]] |
| * [[Zero]] | | * [[Inclusion]] |
| * [[Zero coupon bond]] | | * [[Learning and development]] |
| * [[Flat yield curve]] | | * [[Organisation for Economic Co-operation and Development]] (OECD) |
| * [[Rising yield curve]] | | * [[SDG 4]] - inclusive & equitable quality education & lifelong learning opportunities for all |
| * [[Falling yield curve]]
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| * [[Positive yield curve]]
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| * [[Negative yield curve]]
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| ==Other resources== | | == External link == |
| [[Media:2013_09_Sept_-_Simple_solutions.pdf| The Treasurer students, Simple solutions]] | | [https://www.oecd-ilibrary.org/sites/f37893af-en/index.html?itemId=/content/component/f37893af-en OECD Skills Outlook 2021 : Learning for Life] |
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| [[Category:Financial_management]] | | [[Category:Self_management_and_accountability]] |
| [[Category:Corporate_financial_management]] | | [[Category:Knowledge_and_information_management]] |
| | [[Category:The_business_context]] |
| | [[Category:Identify_and_assess_risks]] |
| | [[Category:Manage_risks]] |
