The Time Value Of Money Notes

What is money?

• Problematic, because there can be different definitions in law, in economics, and in finance.

• An asset that is generally accepted as payment for goods and services or repayment of debt.

  • Medium of exchange. Facilitates the exchange of goods. Greater convenience, because no need for barter trading as in the past.

  • Unit of account. Money as an instrument for measuring the (economic) value of things?

  • Store of value. This is slightly controversial.

• Commodity monies.

  • Things of intrinsic value, such as gold, silk, whale teeth, etc.

• Fiat monies.

  • Value due to government decree.

  • The value of a 10 note comes from governmental decree. But how does it work? The government issues legislation saying that the currency issued by the central bank is legal tender; legal tender means that, even if the currency becomes worthless because of inflation, it can nevertheless still be used to pay taxes.

• The underlying idea of money appears to be that of debt: I owe something to you for some service or goods that you have provided me with in the past.

  • It used to be that, if I have nothing that you want right now, I could owe you a favour that you can call on sometime in the future.

  • But, as society became more complex, I may no longer know my blacksmith or butcher personally, and a centralised means of exchange therefore became necessary.

• Forms of money:

  • Cash (in the form of bank notes and metal coins);

  • Central bank reserves; and

  • Bank deposits.

    • What happens to a bank’s balance sheet when a customer deposits 50?

    • 50 in the customer’s bank account is a 50 debt owed by the bank to the customer. So, the bank has 50 more in cash, but also has 50 more in debt, the result being that its balance sheet has grown and its firm value has grown by 50.

    • Banks can increase the money supply in society by granting more loans.

  • Note that bank deposits constitute 97.4% of the money used in the economy, while cash accounts only for 2.6%.

  • What of cryptocurrencies?

    • If a coffee shop accepts Bitcoin as payment, Bitcoin then acts as a medium of exchange.

    • It has been said that there is too much fluctuation in the value of cryptocurrencies for them to be used as a store of value.

    • The are not legal tender, and neither are they fiat money, because they backed by the government.

The Time Value of Money

“A Dollar today is worth more than a Dollar tomorrow”. But why?

• The opportunity cost of lending: Interest as a compensation for the opportunities lost during the time the loan was outstanding.

  • The forgone opportunity of using the money to say, go to the cinema, because I chose to lend that money to you instead.

• We are therefore faced with two questions:

  • How much money will I receive at a future date if I invest a certain amount today?

  • How much is a future cash flow worth in today’s money? This is the more important question from a finance perspective.

• Notations:

  • t: a certain moment in time (t(0) = starting period; t(1) = end of the first period, etc.)

  • c(t): generic cash flow at time t.

  • f(t): alternative cash flow at future time t.

  • p(t): alternative cash flow at present time t.

  • r: rate of return.

• Rate of return:
  The gain or loss in the value of a financial asset over some specified period of time as a percentage of the initial value.

  • r_0,t = (f_t-c₀)/c₀ = f_t/c₀ = 1.

  • If you invest 100 today and are promised to be paid back 150 in one year’s time, what is the (promised) return, net return, and the 1-year rate of return?

    • Promised return = 150.

    • Promised net return = 50.

    • Promised rate of return = 50%.

• Note on basis points:

  • What does it mean to say that the interest rate has increased by 5%?

    • 10% + 5% = 15%?

    • 10% x (100% + 5%) = 105% of 10% = 10.5%?

  • Basis points avoid this ambiguity:

    • 100 basis points = 1%.

    • 10% to 15% = An increase of 500BP.

    • 10% to 10.% = An increase of 50BP.

• Compounding:
  If you invest 500 today in a loan that promises a fixed interest rate of 7%, what will your return be in one year from now, in 5 years, in 10 years, and in 30 years? We will disregard the risk of default for now.

  • One-year holding period:

    • Rate of return formula: r(0,t) = (f(t)-c(0))/c(0).

    • Solve for the future value at time t:

      • r.c₀ = f_t - c₀

      • f_t = r.c₀ + c₀ = c₀.(1 + r)

  • For variable rates of return: f_t = c₀.(1 + r_0,1).(1 + r_0,2) …

  • For fixed rate of return: f_t = c₀.(1 + r).(1 + r) … = c(0).(1 + r)ⁿ

  • Multiperiod holding rate:

    • If you have a savings account that pays you 1% interest per month, what is the annual interest rate? It is 12.68%, and not 12%. This is because you receive interest on your interest; this is precisely what “compounding” means.

    • An investment promises to return 12% per year. What is the monthly return? r = 1.12^1/12 - 1 = 0.95%

  • Compounding is important in law as well. See Littlewoods.

• Discounting:
  An investment will return 100 in 5 years. How much should you be willing to pay for this investment today if the prevailing interest rate in the economy is 10% per year?

  • p₀ = 100/(1 + r)⁵ = 62.1.

    • If the project is offered to you for more than 62.1, you should reject it. This is because you can get more returns by investing your money elsewhere.

    • If the project is offered for less than 62.1, you should invest it.

    • ...

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