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In my notes, I have an example:

Question: Suppose that John is self-employed and wants to save for his retirement in 20 years. From the time of his retirement onwards he wants to withdraw £1,000 every month at the beginning of each month for 30 years. What amount of money does he have to save every month at the beginning of each month for the next 20 years to fund his retirement? Suppose that the nominal interest rate is 6% compounded monthly.

Solution: The idea is that the forwarded value of the deposits at the end of year 20 should be equal to the discounted value of all the withdrawals at the beginning of year 21......

Can someone explain the differences between 'forwarded value' and 'discounted value' in this context? Please do so in laymans terms, since I'm quite new to all this finance jargon (I know what present value is btw). I did a Google search for 'discounted value' and it comes up with results on 'present value', so are these the same? For 'forwarded value', results come up as 'forward price' but I don't understand what that is

YinWai Tse
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1 Answers1

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As a general principle, the term forwarded value is used to express the future value of some cash-flow. What is being forwarded is money, and it is being forwarded in time, while accumulating interest. On the othe hand, the term discounted value goes in the opposite time-direction: you know that you will accept some cash in the future, and you want to know how much it is worth now. The operation of discounting is the arithmetic operation to convert future value to present value.

Suppose you deposit one pound every month for twenty years, at a nominal rate of $6\%$ compounded monthly. Then you will have forwarded this amount: $$\sum_{k=1}^{240}(1+0.06/12)^{240-k}$$ because the first pound you deposit a month from now will earn interest for the remaining $239$ months, the second pound will earn interest for the remaining $238$ months, and so on. So the forwarded value of monthly deposits of $x$ pounds will have been forwarded to the expression above, multiplied by $x$.

We want this sum (multiplied by some $x$, denoting the fixed monthly deposit) to fund exactly $30$ years of monthly withdrawals of $1,000$ pounds each, starting $20$ years from now. What is the value of these monthly withdrawals exactly $20$ years from now? It is the discounted value of the withdrawals, namely: $$1000\times\sum_{k=1}^{360}(1+0.06/12)^{-k}$$ and so the equation you need to solve is:

Forwarded value of deposits = discounted value of withdrawals which leads to the solution:

$$x=\frac{1000\times\sum_{k=1}^{360}(1+0.06/12)^{-k}}{\sum_{k=1}^{240}(1+0.06/12)^{240-k}}$$

So John should deposit about about $361$ pounds every month for the next $20$ years in order to insure $30$ years of monthly withdrawals of $1,000$ pounds each. Unfortunately, the interest rates right now are very far from $6\%$, so in reality, John will have to deposit much more.

uniquesolution
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